On constructing zeta elements for Shimura varieties
Syed Waqar Ali Shah


§ ABSTRACT We present a novel axiomatic framework for establishing horizontal norm relations in Euler systems that are built from pushforwards of classes in the  motivic cohomology of Shimura varieties. This framework  is uniformly applicable to  the Euler systems of  both algebraic cycles and Eisenstein classes. It also applies to non-spherical pairs of groups that   fail  to  satisfy  a  local multiplicity one hypothesis, and thus lie beyond the reach of existing methods. 
A key application of this work is the construction of an  Euler system for the spinor Galois representations arising in the cohomology of Siegel modular varieties of genus three,  which is  undertaken  in two companion articles.
    

Speaker and Style Disentanglement of Speech Based on Contrastive Predictive Coding Supported Factorized Variational Autoencoder

    Yuying Xie1, Michael Kuhlmann2, Frederik Rautenberg2 , Zheng-Hua Tan1, Reinhold Haeb-Umbach2

    1. Department of Electronic Systems, Aalborg University, Denmark 

    2. Department of Communications Engineering, Paderborn University, Germany 



    
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§ INTRODUCTION
    


Euler  systems are objects of  an arithmetic-algebraic-geometric nature that are designed   to provide a  handle on  Selmer groups   of p-adic     Galois representations    
and  play a crucial role in linking these  arithmetic groups to special values of  L-functions.    Though Euler systems are Galois theoretic objects,  the tools involved in their   construction     are often  of  an   automorphic nature.  A typical setup of its kind    
starts  by  identifying the Galois  representation in the cohomology of a  Shimura  variety  of a    reductive    group  G. The Galois representation is 
required to be automorphic, i.e.,    its L-function  
matches that of a corresponding  automorphic   representation.   The class at the bottom of such a hypothetical system is   taken to be the pushforward  of a  special element  that  lives in the   motives of a sub-Shimura variety arising from a  reductive  subgroup H  of     G. Two common types of special elements are  
fundamental cycles  and Eisenstein classes, and their  respective Euler systems are   often   distinguished based on this dichotomy.   The desire to construct an Euler system via such pushforwards is  also  motivated by a corresponding period integral (p-adic or complex)  which     provides a link between   L-values and the bottom class of this hypothetical system.  For this reason, the classes in an Euler system are  sometimes  also  referred to as `zeta elements' (<cit.>).  Once such a setup is identified,  the problem of constructing the deeper (horizontal) layers of  zeta elements  is   often      tackled by  judiciously picking   special elements of the same type   in  the   motives at  deeper levels of H and pushing them  into motives of G along   conjugated   embeddings.     
   This turns out to be a  rather  challenging problem in  general.    At the moment, there is no  known   general    method   that illuminates  what levels and conjugated embeddings   would yield the desired  norm relations in any  particular  setting.       











The primary   goal of this article is to describe an  axiomatic machinery  that specifies   a   precise criteria  for   constructing the   deeper  layers  of zeta elements  in   the       aforementioned    settings. It is also designed to handle the potential failure of the so-called multiplicity one hypothesis,   which is a crucial requirement for the technique of local zeta integrals introduced in <cit.>.  This failure does arise in practice, 
most notably in the situation studied in <cit.>,  where the relevant period integral  unfolds to non-unique models. An immediate new application of our work is the construction of a  full Euler system for GSp_6 envisioned in loc.cit., which is carried out in <cit.>, <cit.> using the framework presented here.  Other forthcoming applications include <cit.> and <cit.>.  



 §.§ Main  results
   To describe our  main  results,    it is convenient to work in the   abstract setup of locally profinite groups as used in  <cit.> (cf. <cit.>). 
Let H = ∏_v ∈ I' H_v, G = ∏_v ∈ I'G_v, 
be respectively the restricted products of locally profinite groups     H_v, G_v taken with respect to compact open subgroups U_v⊂ H_v, K_v ⊂ G_v. We assume that H_v is a   closed 
subgroup of G_v and that U_v = H ∩ K_v. 
Let Υ_H, Υ_G be suitable      non-empty     collections of compact  open subgroups of  H, G.   
Let 
    N : Υ_H→_p-Mod,      M : Υ_G→_p-Mod
 be mappings  that associate to each compact open subgroup a _p-module  in a           functorial         manner  mimicking   the   abstract        properties of cohomology of Shimura varieties over varying levels.  More precisely, it is assumed that for each K_1⊂ K_2 in Υ_G and g ∈ G, there exist  three maps;   ^* : M(K_2) → M(K_1) referred to as restriction, _* : M(K_1 ) → M(K_2) referred to as induction  and [g]^* : M(K_1) → M(gK_1g^-1) referred to as conjugation and  that these maps   satisfy     certain  compatibility conditions.     Similarly for N.   We also require that there are   maps  ι_*  :  N(K_1∩  H) → M(K_1)  for all K_1∈Υ_G.    
These model the   behaviour of pushforwards induced by embeddings of Shimura varieties.   


Fix for each v ∈  I  a     compact open normal subgroup L_v of K_v.  By K, L, U, we denote   respectively  the products of K_v, L_v, U_v over all v and assume that L, K ∈Υ_G and U ∈Υ_H.      
Let 𝒩 denote the set of all finite subsets of I. For ν∈𝒩,  we denote G_ν = ∏_v ∈ν G_v, G^ν      =   G / G_ν and use similar notations for H, U,    K, L.  Set   K[ν] = K^ν L_ν for ν∈𝒩.   Thus K[μ]  ⊂    K[ν] whenever  ν⊂μ  and  we  denote by   _μ,ν,* the induction map M(K[μ]) → M(K[ν]).    
For each v ∈ I, let ℌ_v be a finite _p-linear combination  
of characteristic functions of double cosets in K_v\ G_v / K_v.  Then  for any   pair of disjoint  μ, ν∈𝒩, there are   linear   maps 
    ℌ_μ,* : M(K[ν]) → M(   K [ ν ] )

given essentially by sums of tensor products of Hecke correspondences in  ℌ_v for    v ∈μ.     For  v ∈ I, let g_v,1, …, g_v,r_v∈ G be an arbitrary but fixed  set of representatives  for  
    H_v\ H_v·( ℌ_v) /K_v .
    For i = 1, …, r_v, let H_v, i : = H_v∩ g_v,i K_v g_v,i^-1, V_v,i = H_v∩  g_v,i L_v g_v,i ^-1⊂ H_v,i and   let     𝔥_v,i  =  𝔥_g_v , i be the function h  ↦ℌ_v((-)g_v,i) for h ∈  H_v.     Then we have induced _p-linear maps 𝔥_v,i,* : N(U) → N(H_v,i U^v   ) for each i   given by  Hecke  correspondences. Given x_U∈ N(U), our goal is to be able to construct      classes y_ν∈  
M (K[ν]) such that y_∅  =  ι_*(x_U) ∈ M(K) and 

    ℌ_μ∖ν  ,  *   ( y_ν )   =  _μ ,  ν,  * ( y_μ )

for all  ν , μ∈𝒩  satisfying  ν⊂μ.   A classical example of norm relation in this   format is  <cit.>. 
See   
<cit.> for an exposition of Heegner point scenario  in  a  similar spirit.      


[Theorem  <ref>]   Let x_U∈ N(U). Assume that N equals a restricted tensor product ⊗'_v N_v with respect to x_U_v∈ N_v   ( U _ v ) (see below) and x_U =  ⊗'_v  x_U_v. Suppose that  for each v ∈ I and 1 ≤ i ≤ r_v, there exists x_v, i∈ N_v(V_v,i) such that 
    𝔥_v,i ,  *   (x_U_v) = _V_v,i, H_v,i,*(x_v,i)
    Then there exist classes y_ν∈  M
(K[ν])  for  all  ν∈𝒩 such that y_∅ =  ι_*(x_U) and      (<ref>)   holds for all ν , μ∈𝒩   satisfying   
     ν⊂μ.   
         


That N = ⊗'_v N_v 
means the following. 
For each v ∈ I, there are  functorial _p-Mod valued mappings N_v on compact open subgroups of H _v   
and  there  are    elements x_U_v∈ N_v(U_v) such that for any  compact open subgroup   W = ∏_v  W_v∈Υ_H that satisfies W_v = U_v for all but finitely  many v,  N(W) equals  the restricted tensor product ⊗'_v N_v(W_v) 
taken with respect to x_U_v. 


In the case where special elements are taken to be fundamental cycles, the situation can be modelled by taking N(W) = _p· 1_W where 1_W denotes the   fundamental  class of the Shimura variety of level W  ∈Υ_H. Then N is trivially a restricted tensor product. 
In this case,  Theorem <ref>  reduces to  verifying   certain      congruence     conditions between degrees of Hecke operators. Here we define 
the degree  of a double coset operator  T =  (W h W')  to be |  Wh W'/ W' |  and that of T_* to be | W\ W h W' |, and extend this notion to linear combinations of such operators in the obvious way. Set  d_v,i = [ H_v,i :  V_v,i ]. Note that d_v,i divides the index [K_v : L_v]. 

[Corollary <ref>]   Let N be as above and  x_U = 1_U∈ N(U). If for each v ∈ I and 1  ≤ i  ≤  r_v, the degree  of   𝔥_v,i,* lies in  d_v,i·_p,  
 there exist classes  y_ν∈ M( K [ν] ) for each ν∈𝒩   such that y_∅ =  ι_*(x_U) and  (<ref>)  is  satisfied for all ν⊂μ in 𝒩.        


There is a generalization of such congruence criteria that applies  
to   Eisenstein classes. 
For each v ∈ I, let X_v be a locally compact Hausdorff totally disconnected topological  space endowed with a continuous right  action   X_v×  H_v→ X_v. Let Y_v⊂ X_v be a compact open subset invariant under U_v.  Let X = ∏_v  ' X_v be  the  restricted topological product of X_v taken  with  respect to Y_v.  Then we get a  smooth  left action of H on the  so-called Schwartz  space 𝒮_X of all locally constant  compactly  supported  _p-valued  functions on X. For our next result, we assume that for each    W of the form ∏_v ∈ I W_v∈Υ_G,   we have   N(W) equals 𝒮_X(W), the _p-module   of all W-invariant functions in 𝒮_X.  Then N is a restricted tensor product of N_v with respect to ϕ_U_v = (Y_v) ∈ N_v(U_v)  where N_v(W_v) for  a compact open subgroup W_v⊂ H_v   is  the set of all  W_v-invariant  Schwartz  functions on X_v.  
Given compact open subgroups  V_v, W_v⊂ H_v  such that V_v⊂ W_v 
and x ∈ X, we denote by V_v(x,W_v)   
the subgroup of W_v    generated by V_v and the stabilizer Stab_W_v(x) of x in W_v.   
[Theorem 
 <ref>]         Let ϕ_U = ⊗ ' ϕ_U_v∈ N(U) = 𝒮_X(U). 
Suppose that for each v ∈ I and 1 ≤ i ≤ r_v,    

    (𝔥_v,i,*( ϕ_U_v  )    ) (x)  ∈  [ V_v,i(x,H_g,i): V_v,i] ·_p
    for all x ∈    ( 𝔥_v,i, * (ϕ_U_v  )       ).  Then there exist y_ν∈ M(K[ν]) for all ν∈𝒩  such that y_∅ = ι_* ( ϕ_U )     and   (<ref>)  is satisfied for all ν⊂μ in 𝒩. 


If  X is  reduced to a point {pt}, 
one recovers   Theorem   
<ref>  since for all v and i,    V_v,i(pt,H_v,i) = H_v,i and the action of 𝔥_v,i,* is via multiplication by  its degree. 
 
While it is conceivable to prove our main result in a more direct fashion (see Remark <ref>), we have chosen to develop our approach from the point of view of specifying a  “best possible test vector" that yields a solution to (<ref>). Let us explain this. It is possible to recast  the  relations
 (<ref>) in terms of  intertwining maps of  smooth representations of H × G by passing to the inductive limit over all levels. More precisely,  let *NN, 

*NM     denote the inductive limits of N(V) ⊗__p_p, M(K') ⊗__p_p over all levels V ∈Υ_H, K' ∈Υ_G  with   respect  to  restrictions. 
These are naturally   smooth representations of H, G respectively. Let ℋ( G ) denote the _p-valued  Hecke algebra of G with respect to a suitable Haar measure on G. We can  construct  a map 
    ι̂_* : *NN⊗__pℋ(
    G 
    ) →*NM
 
of H × G representations with suitably 
 defined actions on the source and the target.     
One can then take an arbitrary finite sum of twisted pushforwards from N to M of classes at arbitrary  `local' levels of H_v and ask whether the element given by this sum satisfies the norm relation  (<ref>), say for ν = ∅, μ = { v }.  
In terms of the map ι̂_*, this becomes a problem of specifying  a “test vector” in N⊗__pℋ(G)  that satisfies certain integrality properties and whose image under ι̂_* equals ι̂      _*(x_U⊗ℌ_v ). This leads to a notion of integral test vector given for instance in <cit.>, analogues of which also appear in several other recent    works.  If such an integral test vector lies in the H_v-coinvariant class of ι̂_*(x_U⊗ℌ_v), we refer to it as an  abstract zeta element.      
[Theorem  <ref>]  An abstract  zeta element at v exists if and only if  the  norm relations (<ref>) hold for  
1 ≤ i ≤   r_v up to _p-torsion.  
 
This result connects our approach to the one pursued in <cit.> (cf. <cit.>), which seeks such integral test vectors by means of local zeta integrals. However, it   provides no mechanism on how one may find them in the first place. Our approach, on the other hand, pinpoints an essentially unique test vector in terms of the Hecke polynomial to check the norm relations with. Another key advantage of our approach over theirs is that ours is inherently integral, as no volume factor normalizations show up in the criteria above.  Crucially, it also has broader applicability, since  it remains  effective even in cases where the so-called ‘multiplicity one’ hypothesis fails to hold.







 §.§ Auxiliary results
  The  execution of our approach  hinges on  explicit  description of   the Hecke polynomials ℌ_v and   their    twisted restrictions  𝔥_v,i. This requires among other things a description of left or right cosets contained in  double cosets of parahoric subgroups.   It is possible to exploit the   affine  cell decompositions   of flag varieties  to  specify  a    “geometric"    set of representatives which makes  double coset manipulations a much more  pleasant task.       In <cit.>,  Lansky derives such a  decomposition recipe for double cosets of parahoric subgroups of split Chevalley groups by studying the structure of the underlying Iwahori  Hecke  algebras.    Though the class of groups we are interested in is not covered by Lansky's results, the ideas therein are completely adaptable. 
We generalize Lansky's recipe by  axiomatizing it in the language of  generalized   Tits systems as follows. 

Let 𝒯 =  (G,B,N) be a Tits system and let φ : G  →G̃ be  a  (B,N)-adapted inclusion. Let W = N / B be the Weyl group of 𝒯  and   S the generating  set of reflections in W determined by 𝒯. We assume that B s B / B is finite for each s ∈ S. Then B w B / B is finite for each w ∈ W. For each s ∈ S, let _s⊂ G denote a set of representatives for BsB/B.  For X  ⊂  S, let W_X denote the subgroup of W generated by X and K_X = B W_X B ⊂ G the corresponding parabolic subgroup.  Let B̂ be the normalizer of B in G̃, Ω = B̂ / B and W̃  =  W ⋊Ω be the extended Weyl group. 
For any X, Y ⊂ S,  let [W_X\W̃   /  W_Y ] denote the set of all w ∈W̃ whose length  among elements of W_X w W_Y is  minimal. For   a  reduced  decomposition  w = s_1… s_mρ where s_i∈ W, ρ∈Ω,   let  _w = _s_1×…_s_m, ρ̃∈B̂ a lift of ρ  and 𝒳_w : _w→ G the map which sends κ⃗ = (κ_1, …, κ_m)  ∈_w to the product κ_1…κ_mρ̃.  
Then the image of 𝒳_w modulo B only depends on the element w.     
   
[Theorem     <ref>] 
For any X , Y ⊂ S and w ∈ [ W_X\W̃  /  W_Y   ], we have 
    K_X w K_Y =    _τ_κ⃗∈_τ  w 𝒳_τ w  (  κ⃗  ) K_Y

where τ∈ W_X overs minimal length representatives of W_X  /   (  W_X∩ w W_Y w^-1   ).
    
   


The images of the maps 𝒳_w defined   above  can  be  viewed  as  
 affine  generalizations  of   the more  familiar Schubert cells one encounters in the geometry of  flag varieties. See  <ref> for a discussion.        
In practice, the recipe is applied by taking  B to be the Iwahori subgroup of the reductive group at hand, W the affine Weyl group  and W̃  the Iwahori Weyl group. The recursive nature of Schubert cells 𝒳_w proves to be particularly advantageous    for  computing the twsited restrictions of Hecke polynomials.  






 §.§ Other approaches
 The framework presented here focuses on ‘pushforward-style’ constructions in cohomology, motivated by period integrals where cusp forms on a larger group are integrated against (some gadget on)  a smaller group. Recently, a new ‘pullback-style’ approach has been proposed by Skinner and Vincentelli (<cit.>), opening up the possibility of using “potentially motivic" classes such as the Siegel Eisenstein class constructed in   <cit.>.   
Another approach, developed by Eric Urban, uses congruences between Eisenstein series to intrinsically construct Euler system classes in Galois cohomology <cit.>, <cit.>.   Both of these approaches differ fundamentally from ours and do not seem applicable to the  various  settings that can be explored using our method, e.g., <cit.>.    



For Euler systems of fundamental cycles, an earlier approach developed by Cornut and his collaborators also aims to prove norm relations in the style of (<ref>). This approach involves studying the Hecke action on the corresponding  Bruhat-Tits buildings, e.g., see <cit.>, <cit.>, <cit.>. However, it was observed  in <cit.> that the Hecke action used in these works is not compatible with the geometric one. Cornut has informed us however  that this issue can be resolved. It is our expectation  that insights from studying actions on Bruhat-Tits buildings  may  provide a more conceptual explanation for  computations in our  own work.



 §.§ Organization
   This  article  is  divided into two  parts, where the first develops our approach  abstractly   and the second executes it in concrete situations.  We briefly outline the contents of each  section  within both.  

In  <ref>, we revisit and expand upon the abstract formalism   of   functors      developed  in <cit.>. Our motivation  here is  partly  to develop a  framework for Hecke operators that works well in the absence of Galois descent.  We prove several basic results that normally require a passage to inductive limits.    We also introduce the notion of mixed Hecke correspondences that allow us to relate   double  coset  operators   of a  locally  profinite group  to those of a closed subgroup.  These play a crucial role in establishing the aforementioned   norm  relation     criteria. 
We  end  the  section by outlining how the formalism applies to Shimura varieties even in the absence of Milne's (SV5) axiom, which was assumed in <cit.>.   

In <ref>, we develop our machinery from the point of view of  abstract zeta elements.  We  strive for maximum possible generality in defining these and establish a structural result  in Theorem <ref>. This  
allows    us to focus attention on a specific type of such elements; the one given by  twisted  restrictions of Hecke polynomials. This comes with the added benefit of eliminating all normalizations by volume factors, giving a highly canonical crtieria that we are  able to upgrade in <ref> to finite levels.  To handle Euler systems of Eisenstein classes, we have included an axiomatic study of traces in arbitrary Schwartz spaces of totally disconnected spaces   in <ref>. 
Finally,  a toy example of CM points on modular curves is included  to illustrate our machinery in a simple case.    

In <ref>, we collect several  facts about  Satake transforms and Hecke polynomials.  Everything  here   can  be   considered     well-known   to   experts        and  we make   no  claim      of     originality. 
We have  however  chosen to   include  
proofs of a few results, partly because  we could not find a  satisfactory reference that covers the generality we wish to work in  and partly because   conventions    seem to differ from one reference to  another. Some of these results 
play a crucial role in our  computations. A few results are included (without proofs) to provide a check on our computations. For instance, certain congruence properties of Kazhdan-Lusztig polynomials are  not necessary for the computations done in this article  but are invoked  in <cit.>.  



In <ref>,  we develop from scratch  another important   ingredient    of our approach. After    justifying all the necessary facts we need on (generalized) Tits systems, we prove a recipe for decomposing  certain    double cosets following  the method   of    Lansky.   We briefly review some facts from Bruhat-Tits   theory        that allow us to apply this recipe in   practice.   The results of this section also complement the content of <ref> in the sense that we can often  use the decomposition recipe 
to efficiently invert  various Satake transforms for Hecke polynomials, though we note that this step can often be skipped.      







Part II of this article  is devoted to examples. Its primary purpose is to provide concrete evidence that the abstract criteria proposed in Theorem <ref> does hold in practice. We study the split case of unitary Shimura varieties GU_1,2m-1 for arbitrary m in <ref> and the inert case for m = 2 in <ref>. The inert case for general m is the subject of a later work. In both these scenarios, we  show that along anticyclotomic towers, the criteria of Theorem <ref> holds for pushforwards of fundamental cycles of products of two sub-Shimura varieties. The split case of our results for these  Shimura varieties   strengthens  <cit.> and also applies  to   certain     CM versions of these varieties.    An interesting  observation in the split case is that our criteria fails to hold if one considers the full abelian tower   (Remark  <ref>). This is consistent with the  well-documented   observation that Heegner points do not “go up"   cyclotomic extensions.   Another  interesting  observation  is that  the degrees of the various restrictions of the Hecke polynomials turn out to be q-analogues of the binomial expansion (1-1)^k for k a positive integer. This alludes to an intimate relationship between  twisted   restrictions of Hecke polynomials and    factors of  Satake  polynomials.       

In <ref>, we study the case  of genus  two Siegel modular varieties. Here we establish that the criteria of Theorem  <ref> holds for pushforwards of cup products of Eisenstein classes for modular     curves.  This yields the  “ideal"   version of the horizontal norm relations alluded to  in  <cit.>. An interesting observation here is that of the two twisted restrictions of the  spinor   Hecke polynomial, one is essentially the standard _2-Hecke polynomial  for a diagonally embedded copy of _2 and the corresponding trace computation is reminiscent of <cit.> for Kato's Euler system.



 §.§ Acknowledgements
This article is based on the author’s thesis work done at Harvard University. The author is sincerely grateful to Barry Mazur for all his advice and encouragement; Christophe Cornut for valuable feedback; Antonio Cauchi for bringing the author’s attention to several new applications of this work and for carefully explaining an unfolding calculation; and Wei Zhang for several useful conversations. Many ideas of this article have their roots in an earlier joint work, and the author is thankful to his collaborator Andrew Graham for their continued discussions. The softwares MATLAB® and SageMath proved particularly helpful in carrying out and verifying numerous computations that arose in the course of this project.  A part of this work was also completed when the author was affiliated with the University of California, Santa Barbara. The author extends his sincere gratitude to Francesc Castella, Zheng Liu, and Adebisi Agboola for their mentorship and support.   
  
Part 1. General theory  


§ PRELIMINARIES
     
In this section, we recall   and   expand   upon     the  abstract  formalism of  functors  on compact  open   subgroups of locally profinite groups  as  introduced in  <cit.>[which in turn was inspired by <cit.>)] which we will use in <ref> to study norm relation problems encountered in the settings of Shimura varieties.      We   note     that   a   few   conditions  of <cit.>   have  been  relaxed for generality while others related to vertical norm relations have been dropped  completely since they do not pertain to the questions addressed in this article[though they are needed again in <cit.>)].    Note also that   the terminology in a few  places   has been modified to   match  what seems to be the standard in pre-existing   literature on such  functors  e.g.,    <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.  The material in this section can however be read independently of all of these  sources.         





One of the reasons    for developing this formalism further   (besides convenience and generality)  is to address the failure of Galois descent in the  cohomology of Shimura varieties with integral coefficients.  This failure  in particular means that the usual approach to Hecke operators   as seen in  the theory of smooth representations is no longer available as  cohomology at finite level can no longer be recovered after passage to limit by taking invariants. See  <cit.> for a discussion of a similar issue  that arises when defining cohomology with support conditions.    
In our development, the role of Galois descent is  primarily  played by what is known in literature as  Mackey's decomposition formula which was used in <cit.> to study vertical norm relations. This formula  can serves as a replacement for Galois descent and allow us to derived many results that hold when coefficients are taken  in a field.        One may thus  view this  formalism as  an integral counterpart of the ordinary theory of     abstract     smooth representations.   





 §.§ RIC functors
  For G a locally profinite group,    let Υ = Υ_G be a non-empty    collection of compact open subgroups of G satisfying the following conditions
      

  (T1) For all g ∈ G, K ∈Υ, gKg^-1∈Υ.

  (T2) For all K, L ∈Υ,   there exists a K ' ∈Υ such that K ' ◃ K, K'  ⊂ L.     

  (T3) For all K, L ∈Υ,  K ∩ L ∈Υ.
 
Clearly the set of all compact open subgroups of G satisfies these properties.  Let  ℱ be any collection of compact open subgroups of G.  We let Υ(ℱ ) denote the collection of all compact open subgroups of G that are obtained as finite intersections of conjugates of elements in ℱ. We refer to Υ(ℱ) as the     collection generated by ℱ. 
        For any ℱ  as  above,  the collection   Υ( ℱ  ) satisfies satisfies (T1)-(T3). In particular, any  collection that satisfies (T1)-(T3) and contains ℱ must contain Υ(ℱ ).       


   Axioms (T1) and (T3) are automatic for Υ(ℱ)  and we need to  verify (T2). Let K , L ∈Υ (ℱ ).  Pick a (necessarily finite)  decomposition K L  =  _γγ  L and define  K ' : = K ∩⋂    _γγ  L  γ^-1∈Υ(F)    Then K ' ◃ K and K' ⊂ L and so (T2) is satisfied.    
    
To any Υ as above, we associate a category of compact opens   𝒫(G)    =    𝒫(G, Υ) whose objects are elements of Υ and whose morphisms are given by  Hom_𝒫(G) ( L, K ) =  { g ∈ G   |    g^-1 L g  ⊂  K   } for L , K ∈Υ with 
 composition  given by  
    (L  K  ) ∘  (  L '  L ) =  (L'  L  K )  =  ( L '   K ) .
  A morphism  (L  K ) will be denoted by [g]_L,K, and if e denotes the identity of G, the inclusion (LK) will also denoted by _L,K. Throughout this section, 
 let  R denote  a commutative ring with identity.      
     An   RIC functor M on (G, Υ ) valued in    R-Mod  is a pair of covariant functors 
    M ^  *  :  𝒫(G) ^ op→   R-Mod              M  _  *  :  𝒫(G)  →  R-Mod

satisfying   the   following   three   conditions:         
  

  (C1) M^*(K) = M_*(K) for all K  ∈Υ. We will denote the common R-module  by M(K).

  (C2)For all L, K   ∈Υ such that g ^-1 L g = K, 
    (L  K)^* =  ( K  L )_*∈Hom(M(K),M(L)) .

Here, for ϕ∈𝒫 (   G   ) a morphism, we denote ϕ_*  : = M_*(ϕ), ϕ^*  : = M^*(ϕ). 

  (C3) [ γ ]_K,K,* : M(K) → M(K) is the identity map  for all K ∈Υ, γ∈ K.
    
We refer to the maps ϕ^* (resp.,   ϕ_*)    in (C2) above as the pullbacks  (resp., pushforwards) induced by ϕ. If moreover ϕ =  [e], we also refer to ϕ^* = ^* (resp.,     ϕ_* = _*) as  restrictions (resp.,  inductions).    We  say that a functor M  is   -torsion  free  if M(K) is   -torsion free  for all K ∈Υ. Moreover, we  say that  M is   
  

  (G) Galois if for all L , K ∈Υ, L ◃ K, 
    _L,K^* : M (K)  M(L)^K/L .

Here the (left) action K/L × M(L) → M(L) is given by (γ, x) ↦  [ γ ]^*_L,L (x).

  (Co)  cohomological if for all L , K ∈Υ  with   L   ⊂ K,  
 
    ( L    K )   _  * ∘   (  L    K    ) ^  *    =  [K:L] ·   (  K         K   )^* .

That is,   the composition is multiplication by index [K:L] on  M(K).   

  (M) Mackey   
if for all K, L, L' ∈Υ with L ,L' ⊂ K, we have a commutative diagram 

    [column   sep = large]
     ⊕_γ M(L_γ) [r,"∑_*"]     M(L) 
    
     M(L') [r,"_*",swap] [u,"⊕  [γ]^*"]     M(K) [u,"^*",swap]
   
where the direct sum in the top left corner is over a fixed choice of coset representatives γ∈ K of  the  double   quotient     L \ K / L' and L_γ =  L  ∩γ L' γ^-1∈Υ. The  condition  is then satisfied by  any such  choice   of     representatives of L \  K /  L '.     
   
If M satisfies both (M) and (Co), we will say that M is CoMack.  If S is  an   R-algebra, the mapping K ↦ M(K) ⊗ _R  S is a S-valued  RIC functor, which is cohomological or Mackey if M  is   so.  
    


In what follows, we will often say that M : G →  R-Mod is a functor when we mean to say that M is  a RIC functor on (G, Υ) and suppress Υ if it  is clear from  context.
 The acronym  RIC stands  for restriction, induction, conjugation     and   the terminology is  borrowed from  <cit.>. Cf.  <cit.> and <cit.>.    
    

 A morphism φ : N  →   M of RIC   functors  is  a  pair  of  natural  transformations φ    ^  *  :  N ^  *  →  M^ *    ,   φ_* : N _*→  M    _*
such that φ_* ( K )  = φ^* ( K ) for all K ∈Υ.  We   denote  this   common  morphism   by  φ(K).  The category of R-Mod valued RIC functors on (G,Υ) is denoted RIC_R(G, Υ) and the category of CoMack functors by CoMack_R(G, Υ). 
 


We record some  straightforward   implications.    Let M : 𝒫(G,Υ) → R-Mod be  a RIC functor. 

        The functor  M is  Mackey if and if only  if   for all K, L, L' ∈Υ with L ,L' ⊂ K, we have a commutative diagram  
    [column sep = large]  ⊕_ δ M ( L _ δ '  )   [r,  "∑ [δ ]_*"]        
        M ( L  )  
     
        M ( L' )  [r, "_*"]   [u, "⊕ ^*"]      M (K)   [u,"^*", swap]
 where the direct sum in the top left corner is over a fixed choice of coset representatives δ∈ K of L ' \ K / L and L_δ ' = L' ∩δ  L   δ    ^-1∈Υ. 
    

  If γ∈ K runs     over representatives of L \ K / L ',  then  δ  =  γ ^-1 runs      over a set of representatives    for L ' \ K / L. For each γ∈ K, we have a commutative diagram

 
      M(L_δ ' )   [r, "[δ]_*"]          M(L) 
 
M(L')   [ru,   "_*" ]      [r, "[γ]^* ",   swap]          M(L_γ )   [ru, swap,  "_*"]    
 

  
where δ =   γ   ^-1.  Indeed, the two triangles obtained by sticking the arrow M(L_γ)  M(L'_δ) in the diagram above are  commutative.  From this,  it is straightforward to see that  diagram (<ref>) commutes if and only if  diagram  (<ref>) does.      
    


  We will refer to the commutativity of the diagram     (<ref>)    as axiom (M').   
 


    We say that M has    injective restrictions       if   _L,K ^ *   :  M ( K )  →   M ( L )       are    injective  for  all  L , K  ∈Υ, L  ⊂    K.    
   

   Suppose M is either Galois or cohomological and -torsion free. Then M has injective restrictions. 
  






 Let L , K ∈Υ with L ⊂ K.  If M is Galois, pick K ' such that K' ◃ K, K' ⊂ L (using axiom (T2)). Then _K', K^*   =     _K', L  ^   * ∘_L, K ^  *   : M ( K )  →  M (  K '  ) is injective by definition which implies the same for _L,K  ^  *. If M is cohomological, then  _L,K,*∘_L,K^* = [K:L] which is injective if M(K) is -torsion free which again implies the same for _L,K^*.    
  





             Suppose M is Mackey.   Let L, K ∈Υ  with L ⊂ K and let K' ∈Υ be such that  K'  ◃ K and K' ⊂ L[such a K' exists by (T2)].    Then 
_K' , K  ^  *   ∘  _  L , K , *     =    
∑_γ  [γ]_K'  ,     L ^* where γ runs over K / L.  

  



 Since K ' ◃ K and K' ⊂ L,     the  right  multiplication action of K'  on L  \  K is trivial i.e.    L  \ K /  K '  = L   \ K.  By  axiom  (M') obtained in Lemma   <ref>,   we   see    that    

 
  [column sep = large]  ⊕_   δ  M   (   K' 
)   [r,  " ∑[  δ]_*"]        
    M (  K '   )  
 
    M ( L )  [r, "_*"]   [u, "⊕ ^*"]      M (K)   [u,"^*", swap]  
    
    
 
    where δ runs over   L  \  K.  Since [δ]_K', K ' , *  = [δ^-1]_K', K'  ^  *, we may replace δ with δ = γ ^-1. Then γ∈ K runs over K / L  as δ runs over L   \  K and  the  claim follows.   
    

          Suppose R is a -algebra. Then M is Galois if it is CoMack.  
 



    For any L , K, Υ with L ◃ K, _L,K^* : M(K) → M(L) is injective by Lemma  <ref>. If x ∈ M(L) is K-invariant, then _L,K^*∘_L,K,*(x)  =   ∑_γ∈ K / L [γ]^*_L,K (x) = [K:L] x by Lemma <ref> and so _L,K^* (y) = x if y = [K : L] ^-1_L,K,*(x) ∈ M(K) . Thus    _L,K^* surjects onto M(L)^K/L.    
 



It is clear how to define the direct sum and tensor product of functors on  finitely  many  groups G_1 , …, G_n to obtain a functor on G_1×⋯× G_n. A more involved construction is that of restricted tensor products which we elaborate on  now. Say for the rest of this subsection only   that G = ∏_ v ∈ I' G_v is a restricted   direct 
product of locally profinite groups G_ v with respect to compact open subgroups K_ v given for each v  ∈    I.     For ν a finite subset of I, we denote G_ν  : = ∏_v ∈ν, G^ν :   = G/G_ν and similarly for K_ν, K ^ν.  For each v  ∈ I, let Υ_ v be a collection of compact open subgroups of G _   v that satisfies (T1)-(T3) and which contains K_v.  Let Υ_I⊂∏_  v  ∈ I    Υ  _  v be the collection of all subgroups of the form L_ν K^ν where  ν  is   a   finite   subset of I and    L_ν∈∏_ v ∈νΥ_v.   Then Υ_I satisfies (T1)-(T3) and contains K.  If  L ∈∏_v ∈ IΥ, we denote L_v its component group at v.    

 Let N_v : 𝒫( G_v   ,  Υ_ℓ )    → R-Mod be a RIC functor  and let ϕ_K_v∈ N_v(K_v) for each v ∈ I.  The  restricted     tensor product  N = ⊗_v' N_v with respect to ϕ_K_v  is the RIC functor M : 𝒫(G , Υ_I) →  R-Mod given by L ↦⊗'_v N(L_v) where ⊗ '_v denotes the restricted tensor product of R-modules N(L_v) with respect to ϕ_K_v. 
  
We elaborate on the definition above. Fix L ∈Υ_I and  write L = L_ν K^ν. For each finite subset μ of I with μ⊃ν, denote N_μ : =  ⊗_v ∈μ N_v (L_v ) the usual tensor product of R-modules.   If μ_1⊂μ_2 are two such sets, there is an induced map N_μ_1→ N_μ_2 of R-modules that sends x ∈ N_μ_1 to x ⊗⊗_v ∈μ_2∖μ_1ϕ_K_v.  Then N(L) =   _μ   N    _μ   where  the  inductive  limit is  over the directed  set of all finite subsets μ of I that contain ν.




 §.§ Inductive Completions
    
Let (G, Υ) be as in   <ref>. The category CoMack_R(G ,Υ) is closely related to the category of smooth G-representations. We show that
when R is a field and Υ is the collection of all compact open subgroups of G, there is an equivalence between the two. When R is not a field however, axiom (G) can fail and the former category requires a more careful treatment. 

   Let π be a left module over R[G]. We say that π is a  smooth   representation of G if for any   x ∈π, there is a compact open 
 subgroup   K ⊂ G  such that x is fixed under the (left) action of K. A  morphism of smooth representations is a R-linear map respecting the G-actions. The category of smooth representations of G is denoted SmthRep_R(G). 
  

Suppose π∈SmthRep_R(G). For K ∈Υ, let M_π : G  → R-Mod be the functor given by K ↦π^K. For g ∈ G and (L  K ) ∈𝒫(G,Υ),    let 
  [g]  ^  *  : M(K)       →M(L)       [g]_ * :  M(L)        →M(K) 
  
 x      ↦g ·x        x       ↦∑_γ∈K / g^-1 L g   γg ^-1 ·x 

Here, g · x ∈π in the mapping on the left  above is indeed a well-defined   element  of   M(L) as it is invariant under L ⊂ gKg^-1 and similar remarks apply to the expression on the right above. In particular, the map [e]^*_L,K : M(K) → M(L) is the   inclusion π^K↪π^L.  The following is then straightforward.    
 The mapping M_π is a RIC functor that is CoMack and Galois. 

    We refer to  M_π  as  the RIC functor associated to π. If π = R is the trivial representation, we denote the associated functor by M_triv and  refer to  it as the  trivial functor.      
   

   Let M : G  →  R-Mod  be a functor.  
The   inductive  completion M is defined to be  the  limit  _K ∈Υ  M ( K ) where the limit is taken over all    restriction  maps.     We let j_K  : M ( K )  →M denote the natural map.   
    There is an induced smooth   action  G ×M→M ,  (g,x) ↦ g· x where g · x is defined as follows. Let  K   ∈Υ, x_K∈ M(K) be such j_K(x_K) = x. Then g · x  is defined to be the image of x_K under the composition  
    M(K)     M (  g K  g ^  - 1 )  →M .
     It is a routine check that is well-defined. The action so-defined is smooth  as the image of  j_K  :  M(K)  →M(K) is contained in the K-invariants M^K.   If   M is  also Galois, j_K  identifies M(K) with  M^K.
Moreover    if φ :  M → N is a  morphism of functors, the induced map φ : M→N is G-equivariant.    

 
        

Suppose M is   cohomological.  Then  ( j_K  ) is contained in M(K)_-tors. In particular, if R is a field of  characteristic  zero,  j_K is injective.     
 



   Let  x ∈ (   j_K ).  By  definition,  there exists L ∈Υ, L ⊂  K such that _L,K^* (x) = 0.  Since _L,K,*∘ _ L, K ^*  = [ K : L ], we must have [K:L] · x = 0.  
    

The following result  seems originally due to <cit.> for finite groups.         

   Let R be a -algebra and Υ the collection of all compact open subgroups of G.     Then the  functor SmthRep_R(G)   →CoMack_R(G  ,  Υ    ) given by π↦  M_π 
induces  an  equivalence of categories with (quasi)    inverse given by M ↦M. 
    

  By Lemma  <ref>, any CoMack functor valued in a -algebra  is Galois and therefore one can recover a functor M from the representation M.  Similarly, _ K ⊂ Gπ^K = π by smoothness of π.     
    


 §.§ Hecke operators
 
    

A smooth representation comes equipped with an action of algebra of measures known as Hecke algebra. In this subsection, we briefly review the properties of this action and fix conventions.   For  background material on Haar measures and further reading,  the reader may consult <cit.>.     
Let (G, Υ) be as in <ref>. 


        Let μ be a left invariant Haar measure on G  valued in R[e.g.,  if μ is -valued and R is a -algebra]  and let K ∈Υ.    
The   Hecke algebra ℋ_R( K  \  G /  K  ) of level K is defined to be the convolution algebra locally constant K-bi-invariant functions valued in R.  The convolution product is denoted by *.  
The  Hecke algebra of G over Υ  is   defined  to  be     ℋ_R(G)  =  ℋ_R(G, Υ) =   ⋃ _  K ∈Υℋ_R ( K \ G /  K  ). The transposition  on ℋ_R ( G , Υ ) is the mapping ξ↦ξ^t = ( g ↦ξ(g^-1)  ), ξ∈ℋ_R(G).    
  
The convolution    ξ_1 * ξ_2 where ξ_1 , ξ_2∈ℋ_R(G, Υ) is given by    
    (      ξ_1  * ξ_2  ) ( g ) = ∫_x ∈ G ξ_1(x)  ξ_2(x^-1 g)      d μ(g) .
  In particular, if ξ_1 =  (  α K  ) for α∈ G, K ∈Υ     and ξ_2 is right K-invariant, then ξ_1 * ξ_2 = μ(K) ξ_2 ( α ^-1 ( - ) ).   
If G is unimodular, then one  also has (  ξ _1 *  ξ_2 ) ( g)  =  ∫_Gξ_1 ( g y^-1 ) ξ_2(y)       d μ(y) obtained by  substituting x with g y ^-1.      The transposition map is an anti-involution of ℋ_R(G) i.e. ( ξ_1 * ξ_2 ) ^  t    = ξ_2   ^    t        * ξ_1  ^t for all ξ _1 , ξ_2∈ℋ_R(G). It stabilizes    ℋ_R(K \ G / K ) for any K ∈Υ.



The  Hecke  algebra  ℋ_R ( K  \  G / K   ) has   an   R-basis  given by the characteristic functions of double cosets K σ K for σ∈  K \ G /  K   denoted  ( K σ  K ) and  referred    to   as     Hecke operators. The degree     of ( K σ K ) is defined to be | K σ K / K | or equivalently, the index [ K : K ∩σ K σ^-1].       The  product    (Kσ K) * (K  τ   K )   is  supported on K  σ K τ K and can  be described  explicitly as a function on G / K    as  follows: if  K σ K = _iα _ i  K, K  τ  K =  _jβ_j   K, then 

    ch(K σ K) *  ch(K τ   K)            = μ ( K ) ·∑    _  i  ( α_i K τ K )   =   μ(K)  ·∑     _ i , j ch ( α_iβ _j K )
   
On the other hand, the value of the convolution at υ∈ G equals μ ( K σ K ∩υ K τ^-1 K  ). Thus the convolution above can be written as   μ(K)  ·∑_υ c_σ, τ ^ υ ( K υ K ) where υ∈    K \ K σ K τ K / K  and    

    c_σ , τ ^υ  = |  (  K  σ K  ∩υ K τ ^-1 K  ) / K  |
    
If  μ( K ) = 1, then ℋ_R(K \ G / K ) is unital and   the  mapping ℋ_R( K \ G  / K )  → R given by  ( K σ K ) ↦ | K σ K / K  | is a homormorphism of  unital    rings. 

Any smooth left representation π∈SmthRep_R(G)  inherits a  left  action of the Hecke algebra ℋ_R(G, Υ). 
The   action of ch(K' σ K  ) ∈ℋ_R ( G,   Υ   ) on an element x ∈π invariant under K  is given by ch(K ' σ  K) · x =  μ( K )  ∑ _α∈ K  '  σ    K / K α· x.   
Similarly, if K ∈Υ, the R-module π ^K is stable under the action of ℋ_R(K \ G / K ) and is therefore a module over it. 
In particular, if   M is a  RIC functor, then  M is  a module over ℋ_R(G, Υ) and if M is Galois, M(K) =  M^K is naturally a module over  ℋ_R(   K   \  G  /  K    ).     


We note that ℋ_R(G , Υ ) is   itself   a smooth  left representation   of     G under both right and left translation actions.  It   is therefore a  (left) module   over   itself in two   distinct ways.  Let   
  λ: G ×ℋ_R ( G , Υ)       →ℋ_R ( G , Υ)           ρ:  G ×ℋ_R(G, Υ)        →ℋ_R(G ,  Υ)    

(g, ξ)        ↦ξ( g^-1 (-) )            (g , ξ)           ↦ξ( (-) g )

When ℋ_R(G  ,   Υ    ) is considered as a G-representation under λ, the induced action of ℋ_R(G,  Υ ) on itself is that of the convolution product *.  When ℋ_R(G  ,  Υ   ) is considered as a G-representation under ρ, the induced action of ℋ_R(G,  Υ ) will be denoted by  *_ρ. 
There is a relation between * and *_ρ that is   useful    to   record.      


        For  ξ_1 ,  ξ_2∈ℋ_R(G, Υ ),    ξ_1  *  _ ρξ_2  = ξ_2  * ξ _1 ^  t.      
       By definition,  we have  for all g ∈ G   
    ( ξ_1  * _ ρξ_2  )  ( g)  
        = ∫_Gξ_1( x )   ξ_2(gx)     d μ(x)   
        = ∫_Gξ_2(y)  ξ_1( g^-1 y )       d μ(y )  =  ∫_Gξ_2(y)  ξ_1^t( y^-1 g )    d μ(y)  
         =    (  ξ_2 * ξ_1^t       )  ( g)
 
where in the second equality, we used the change of variables x = g^-1 y.    
    
        


 §.§ Hecke correspondences
    
On RIC functors, one may abstractly  define correspondences  in the same manner as one does for  the  cohomology  of  Shimura varieties. We explore the relationship between such correspondences and the action of Hecke algebra defined in <ref>. More crucially, we need to  establish the usual properties of Hecke correspondences in the absence of axiom (G).      
   
Let M   G   → R-Mod be a  functor.    For every K, K' ∈Υ and σ∈   G, the Hecke  correspondence  [ K '  σ K ]    is defined to be the composition

    [K ' σ K]  M(K)  M(K ∩σ^-1 K ' σ)  M(σ K σ^-1∩ K ')  M(K') .

If 𝒞_R (K  '  \ G / K ) denotes   the free R-module on functions ( K ' σ K ), σ∈ K' \ G / K, there  is a R-linear  mapping 𝒞_R ( K '  \ G / K  ) →Hom_R( M(K) , M(K') ) given by ( K ' σ K ) ↦ [K' σ K ]. The transpose of [ K ' σ  K ] is defined to be the correspondence 
    [K' σ K]_* =  [K σ ^-1 K ' ]   : M ( K' ) →  M( K )
 which we also refer to as the covariant action of [K σ K' ].    The  degree of [K' σ K ] is defined to be the cardinality of K ' σ K / K or equivalently, the index [ K'  : K' ∩σ K σ^-1 ]. The degree of [K' σ K]_* is the degree of [K σ^-1 K'].  



     Let M  : G → R-Mod be a  Mackey functor and let K , K ', L ∈Υ with L ⊂ K.  Suppose  that   K 
'  σ K   =   _  i   L σ_i K.  Then  _L, K'  ^  * ∘  [ K '  σ K  ]   = ∑ _ i    [L σ _  i   K   ].   
 
 Denote L  '    : =K ' ∩σ K σ^-1∈Υ.      As     K' / L' → K ' σ K / K, γ   L '    ↦γσ K is a bijection, so is the induced map L \ K' / L ' 
   → 
 L  \ K'  σ K / K   and  we   may  therefore    assume that σ = γ_iσ   where  γ _i  form  a    set of representatives for  L \ K ' /  L '.  Set L_i   :  =  L  ∩γ_i L '  γ_i  ^  - 1.    Since M is Mackey, we see that the square in the diagram


[sep = large,  /tikz/column 3/.style=column sep=0.1em]      
 ⊕_  i       
 M ( L_i )   [r, " ∑_*"]                   M(L)                  

 M( K ) [r, "[σ]^*"']   [ur ,   "[σ_i]^*"]  
 [rr, "[K'  σK]"', bend right, swap]             M ( L ' )  [r, "_*"'] [u, "⊕[γ_  i      ]^*"']     M(K') [u, "^*"']


    
commutes and  therefore  so does the whole diagram.  Noting that L_i =  L ∩σ_i K σ_i^-1, the 
 claim  follows  from the  commutativity of the diagram  above.       
 


 
        Let M : G →  R-Mod be a Mackey functor  and  μ be a Haar measure on G. Let K , K ' ∈Υ  be  such that μ (  K  ) ∈ R.  Then for any σ∈ G,  the actions    of [K  '  σ  K ] and (K  ' τ K) on   M agree up to μ(K). That is,      for all x ∈ M(K),  
    μ(K) ·  j_K∘ [K  '  σ K](x) =  ch(K '  σ     K ) ·  j_K (  x    )    .

In particular  if M(K) →M is injective  and μ(K)  = 1,  the R-linear mapping ℋ_R(K \ G / K )  →End _R M(K)   given   by     ( Kτ K ) ↦ [K τ K ]  is  an  R-algebra  homomorphism.   
   

 
By (T2), there exist L  ∈Υ such that L ⊂σ   K  σ^-1, L ◃ K '. Then K ' σ K / K  =   L 
 \ K ' σ K / K and   [  L γ K   ]  = [ γ ] _ L , K  ^  * for any γ K ⊂ K '  σ K/ K.  So we get the first claim by    Lemma  <ref>.  
 The second claim    then      follows  by  the first and  eq. (<ref>).      



 See Corollary  <ref> where the map ℋ_R(K \ G  / K ) →End_R  M(K) is shown to be an algebra  homomorphism under the assumption that  M is  CoMack.           
    


          Suppose that G = G_1× G_2, σ_i∈ G_i and  K_i, L_i⊂ G_i are compact open subgroups such that  K_1 K_2, L_1   L_2, K_1L_2 and L_1 K_2  are all  in Υ. Let   M : G → R-Mod  be a Mackey functor.  Denoting  τ_1 = (σ_1 , 1), τ_2 =   ( 1,  σ_2 ),  we  have   
    [ (L_1 L_2)  τ_1 (K_1 L_2) ] ∘ [(K_1 L_2)  τ_2   (  K_1  K_2  )    ] =  [(L_1L_2)τ_1τ_2 ( K_1K_2) ] =   [ (  L_1 L_2 )  τ_2   (  L_1 K_2  ) ] ∘[  ( L_1  K_2  )  τ_1    (K_1K_2  )  ]
     
as morphisms   M(K_1 K_2 ) →  M    (L_1 L_2    ).  We also denote this morphism  as [L_1σ_1 K _1 ]  ⊗  [L_2σ_2 K _2  ] and refer to it as the tensor product.     A similar fact holds for tensor products of  a finite number of    Hecke correspondences  in  restricted topological product of groups. 
  

 For i = 1, 2, denote    P_i = σ_i K_iσ_i^-1∩  L_i.   Then 

    τ_1 (  K_1K_2 )  τ_1^-1∩ (L_1K_2)       = P_1K_2 , 
    τ_2  (L_1 K_2) τ_2^-1∩ (L_1L_2)     =L_1 P_2  , 
    τ_1τ_2  (  K_1K_2 
     )  (τ_1τ_2 ) ^-1∩  (L_1L_2)         = P_1 P_2
 and all of these  groups  are in Υ.  Since  τ_2^-1  (L_1P_2   )   τ_2\   L_1 K_2  / P_1 K_2   =   {  1_K} and  M is Mackey,  we get a   commutative  diagram


[row sep = large, column sep = tiny]
           M(P_1P_2) [rd, "pr_*"]                                                                               

    M(P_1K_2) [rd, "pr_*"] [ru, "[τ_2]^*"]                                     M(L_1P_2) [rd, "pr_*"]          

M(K_1K_2) [rr, "[(L_1K_2)τ_1(K_1K_2)]"'] [ru, "[τ_1]^*"]                                                M(L_1K_2) [rr, "[(L_1L_2)τ_2(L_1K_2)]"'] [ru, "[τ_2]^*"]                                              M(L_1L_2)

    
which implies  that     [   L_1 L _2τ_2     L_1  K_2   ] ∘ [  L_1 K_2τ_1 K_1K_2] = [L_1L_2τ_1τ_2  K_1K_2]. By interchanging  the roles of τ_1,  τ_2, we get the second  equality.  




 §.§ Mixed  Hecke     correspondences
   
    
In  the   situations that we are going to  consider, the classes used for constructing Euler systems  are  pushforwarded from a functor  associated with a  smaller  (closed)  subgroup. Here we study this scenario  abstractly and introduce some terminology that will be used extensively in the next section.     Let ι  : H ↪  G  be a closed  subgroup, and Υ_H , Υ_G be  a collection of compact open subgroups of H, G respectively  satisfying (T1)-(T3) and such that the collection ι^-1 ( Υ_G) :    = { K ∩ H   |   K ∈Υ_G} is contained in  Υ_H.   Note that ι^-1( Υ_G ) itself satisfies (T1)-(T3) for H and we refer to it as the pullback of Υ_G to H.        
         
We say that
(  U, K ) ∈Υ_H×Υ_G forms a compatible pair if U ⊂ K.  A morphism of compatible pairs h : (V,L) →  (U,K) is a pair of morphisms 
(V     U ), (L    K ) for some h ∈ H.  Let  M   _   H, M_G be R-Mod     valued functors on H, G respectively. A  pushforward M _ H →  M_G  is a family of morphisms ι_U, K, * : M_H(U) → M_G(K) for all compatible   pairs (U,K) ∈Υ_H×Υ_G  such that ι_U,K,*, ι_V,L,* commute with the pushforwards [ h ] _  * on M_H, M_G   induced by  any   morphism h : (V,L)  → (U,K) of compatible pairs.   We  say that ι_*  is   Mackey if for all U ∈Υ_H, L , K   ∈Υ _  G   satisfying   U , L ⊂ K, we have a commutative diagram 


      [column sep = large]  ⊕_  γ M_H ( U _ γ   )   [r,  " ∑[γ]_*"]        
    M_G ( L  )  
 
    M_H ( U )  [r, "ι_*"]   [u, "⊕ ^*"]      M_G(K)   [u,"^*", swap]  
    


where γ∈ U \ K  /  L is a fixed  set of representatives, U_γ = U ∩γ L γ^-1 and  [γ]_* : M _  H  (  U  _ γ  )  →  M_G ( L )  denotes  the    composition M_H ( U_γ)   M _G (  γ L γ ^-1 )     M_G    (   L   ).
 
If φ_G : N_G→ M_G is a morphism of functors, then it may be viewed as a pushforward in the sense of Definition  <ref>. We will say that φ is Mackey if it is so as a pushforward. 
 If M_G is Mackey, then so is any morphism φ_G : N_G→ M_G. 

 If M is Mackey, then M satisfies the axiom (M') given in Lemma <ref>. Using its notation, the commutativity of <ref> implies that     

    [column sep = large]  ⊕ _ δ  N _  G  ( L' _  δ  )  [r,  "φ_G"]     ⊕_ δ M _  G   ( L _ δ '  )  
        
     
       N  _  G     (L') [r, "φ_G" ,swap]  [ u , "⊕^*" ]      M  _  G       ( L' )  
        [u, "⊕^*"']

is   commutative  as   well.      
   

      Let ι _  *   : M _  H → M_G be a pushforward.    For   U ∈Υ _H, K ∈Υ_G and σ∈ G,    the mixed Hecke correspondence  [ U σ K ] _   * is defined as 
    [U σ K ]_* :  M_H (U)   M_H  ( U ∩σ K σ ^   - 1  )    M_G( σ K σ ^-1 )   M_G(K) .
 One can verify that  [Uσ K]_ * depends only on the double coset U σ K. The  degree of [Uσ K]_* is defined to be the index [H ∩σ K σ^-1 : U ∩σ K  σ ^-1  ].       
    

   
   Suppose that H = G, ι = id and  ι_* : M_G→ M_G is the  identity  map.   Then one can verify that ι_* is Mackey iff M_G is. Moreover if U , K ∈Υ_G   and σ∈ G,   we  have      [U σ K]_* =  [ U  σ  K ] ^t =  [K σ^-1 U   ] agrees with the covariant action introduced before  and the degrees   of [ U σ K ]_*, [K σ^-1 U] also agree.     The `*' in the notation  of mixed Hecke  correspondence  is meant to emphasize its `pushforward nature' and its dependence  on  ι_*.    We    note  that [U σ K   ]  _  *  is    however      independent  of  ι_*.         
          

   Let ι_* : M_H→ M_G be a pushforward and let  σ∈ G,  U ∈Υ_H, K  ∈Υ_G.  For h ∈ H, g ∈  G,   denote   U^h : = h U h ^-1, K^g     :  =  g K  g ^  - 1.   Then   
    [ U σ K ] _  *    =  [ U^h  h σ      K    ]   ∘ [ h ] ^ *  _  U   ^h  , U      =  [g]_K^g,  K   ,    * ∘      [ U σ      g ^ - 1     K^g   ]   _*  .
   
Moreover  [ U  σ  K ] _*  =  [ U ^h h σ   K] _ *  =   [ U  σ g^-1   K^g ]_*.  
   



  Let V  :  = U ∩σ K σ  ^-1, V '   :  = U^h∩ h σ K ( h σ )^-1,  L : 
 =  σ K  σ ^-1  and  L' :   = h σ K σ ^-1  h^-1.  By  definition, h V h ^-1 = V ', h  L h ^-1 = L', V ⊂ L and V '   ⊂  L  '.    One  easily verifies that the  diagram   

  
[sep = large] 
M_H(U) [d, "[h]^*"'] [r, "pr^*"] [rrr, "[UσK]_*", bend left = 22]     M_H(V) [d, "[h]^*"] [r, "ι_*"]     M_G(L) [d, "[h]^*"'] [r, "[g]_*"]     M_G(K) 

M_H(U^h) [r, "pr^*"]                                                                      M_H(V') [r, "ι_*"]                             M_G(L') [ru, "[hg]_*"']                               


    
is commutative which implies   [ U σ K ] _  *    =  [ U^h h σ  K    ]   ∘ [ h ] ^ *  _  U   ^h  , U.   By  definition,  [ U σ K ]_* = [ H ∩ L  : V ] and [ U ^h h σ   K ] _*   =[  H ∩ L ' : V' ].   Since  L, L' and V, V' are conjugates under h, [ H ∩ L : V ]  = [ H ∩ L' : V ' ] and so [ U σ K ]_*  =   [ U ^h h σ    K ].  The proof for the second set of equalities  is  similar.    
   


           Let    ι_* : M_H→ M_G be a pushforward and let  σ∈ G, U,  V ∈Υ_H, K, L ∈Υ_G be such that V ⊂   U  ⊂σ K σ^-1, V ⊂σ L σ^-1 and L  ⊂ K. Then  
    [V σ K ]_*   =  [U σ K ] _  * ∘_V , U , *   =   _L,K , * ∘  [ V  σ L  ]   .
 
   


    Since U ⊂σ K σ^-1, [U σ K ]_* is the composition M_H(U)   M_G ( σ K  σ ^-1 )     M_G(K). Similarly [ V σ L ] _* is the composition M_H(V)    M_G ( σ L σ^-1)     M_G(L).  Since pushforwards commute with each other, the claim  follows.    
    

The following result  
is an  analogue of Lemma <ref> for pushforwards.   



     Let  ι_*   :  M_ H →   M_G be a Mackey pushforward and let σ∈ G, U ∈Υ_H, K,  K'  ∈Υ_G with U  ⊂   K.    Suppose that K  σ K '  = _i  U σ _i K '.   Then [ K  σ K  ' ]   _   *   ∘ι_U , K ,  *     =  ∑ _ i   [ U σ _ i  K '] _  *.
 

  Let L  := K ∩σ K'  σ^-1. As     K / L → K σ K ' / K ', γ L ↦γσ K' is a bijection, so is the induced map U \ K /  L → U \  K σ K' / L' and    we may thus  assume that σ_i = γ_iσ  where   γ _i∈ K forms    a     set of representatives of U  \ K  / L.  Let    
    K_i   '   :  =  σ _i  K '  σ_i ^-1  ,         L_i   :   = γ_i L γ_i ^-1   ,            U_i  : = U  ∩ K_i  '  .
     Then   L_i = K ∩σ_i K '  σ_i^-1 =  K ∩ K_i ' and   therefore   U _i =   U  ∩ L_i.       As ι_* is Mackey,     we see that     

    _L , K  ^ * ∘ι _  U , K , *   =  ∑  _ i        [  γ _ i   ] _  U _ i   ,  L   ,  *  ∘_U_i , U   ^   *
 where [γ_i ] _ U_i, L , *  : =  [  γ_i ] _  L_i, L , * ∘ι_ U_i, L_i , *   = [U_iγ_i  L ]_* (see the  diagram on the left below). 


[sep = large, /tikz/column 3/.style=column sep=0.1em]   

⊕_  i  M_H ( U_i ) 
[r, "∑[γ_i]_*"]        M_G  (    L)   [dr, "[σ]_*"]  
             M_H ( U_i   )  [dr,  "ι_*", swap]  [r, "[γ_i]_*"]     M_G(L)   
[dr,  "[σ]_*"]      

M_H ( U )   [r, "ι_*", swap]   [u,"⊕  ^*"]       M_G(K)   [u, "^*" ]   [r,"[KσK']_*" , swap ]       M_G(K ')              M_G ( K _i' )      [r, "[σ_i]_*" , swap    ]               M_G(K')   


  
As     
ι_U_i, K_i',* 
= [γ_i  ^  - 1 ]_L,K_i',*∘  [   γ_i ]_U_i, L , * for each i, we see that 








    [σ]_L,K',*∘   [γ_i]_U_i,L,*    =       ( [     σ_i    ] _ K_i',  K   ' , * ∘  [ γ _  i  ^-1 ] _ L, K_i   '    , *  )   ∘   [γ_i ]_U_i, L, *
         =  [  σ_i ] _K_i ' , K   '    ∘ι_U_i, K  _   i     '    ,*
  
(see the diagram on the right above).   
Using [K σ K ' ]_*  =    [ σ ] _   L , K  '    , * ∘ _  L , K    ^   *  in conjunction with eq. (<ref>) and   eq.  (<ref>), we see that  

    [  K σ K  '   ]  _   *   ∘ι_U, K ,  *      =   [ σ   ]  _ L  , K  '   ,   * ∘∑ _ i       (  [ γ _ i  ] _  U_i , L     ,   * ∘ _  U _ i   , U  ^  *       )        
        = ∑ _ i  (  [ σ   ]    _L,K   '    ,  * ∘   [ γ_i ] _  U_i , L , *  )  ∘_  U _ i   , U    ^     *    
        =  ∑  _  i   (  [  σ _  i  ] _   K_i  '  , K  '  ,  * ∘ι _ U _  i   ,  K_  i     '     )      ∘ _    U _ i  , U   ^  * 
         =   ∑ _  i   [ U σ   _ i   K  '] _  *

 









We end this subsection by showing that any two  (contravariant)   Hecke correspondences compose in the usual way.  For  K _1, K_2, K_3∈Υ_G,  the convolution of double cosets is the -linear homomorphism  

    ∘    :  𝒞_( K_3\ G / K_2 )  ×𝒞_(K_2\ G / K_1 )     →𝒞_( K_3\ G / K_1 )
  
given by (K_3σ K_2)   ∘(K_2τ K_1 )    = ∑ _υ c^υ_σ, τ (K_3υ  K_1)  
where c^υ_σ, τ = | (K_3σ K_2∩υ K_1τ^-1 K_2 ) /  K_2 |. 
       Let M  =  M_G be a  CoMack     functor  on G, K_1, K_2, K_3∈Υ_G and σ, τ∈  G. Then [K_3σ  K_2 ]  ∘  [ K _2τ   K_1 ]  ∈Hom_R ( M(K_1   )    , M(K_3) ) is a sum  of   Hecke correspondences obtained by the convolution of double cosets as above.    
 
  Let L  =  τ  K_1τ^-1∩  K_2∈Υ_G and suppose that K_2σ  ^-1 K_3  =  _i L  σ _i  ^-1 K_3 for some σ  _i∈ G.  Since  M is  Mackey,  we  see by   Lemma  <ref>   that    

    [K_3σ  K_2]  ∘ [ K_2τ  K_1 ]       =  [K_3σ   K_2 ] ∘_L,K_2,*∘  [  τ    ]  _   L  ,  K _1 ^*
         =    (  [K_2σ^-1  K_3 ]_*∘_L,K_2,* )  ∘  [ τ  ] _L,K_1  ^  *  
         =    ∑    _i      [L  σ _  i  ^   -1   K_3 ]_*∘  [ τ  ]^*_L,K_1       
    = ∑ _ i   [K_3σ _  i  L]  ∘    [  τ   ] _  L , K_1   ^  *   .

 For each i, let  d_i : =  [ σ_iτ K_1 (σ_iτ)^-1∩ K_3 :  σ_i  L  σ_i  ^  - 1 ∩   K_3    ]. Since   M is cohomological, we see that the diagram   


            M ( σ_i L σ_i^-1 ∩K_3   )    [dd,   "_*  " ]  [dr, "_*"]        
 
    M(K_1)  [ur,   "[σ_i τ]^*"]    [dr, "d_i ·[σ_i τ]^* "']              M(K_3   ) 

             M (   σ_i τK_1 ( 
   σ_i τ)   ^-1 ∩K_3 ) [ur, "^*"']         
     


is commutative. So  [ K_3σ_i  L ] ∘  [  τ   ]_L, K_1  ^*  =  d_i·     [K_3σ_iτ K_1 ] as maps M(K_1) → M(K_3) (take the two routes in the diagram above) and therefore     
    [  K_3σ  K_2 ]   ∘   [K_2τ K_1  ]  =    ∑    _i   d_i   [  K_3σ_iτ  K_1  ] .
     
To show that ∑_i d_i (K_3σ _iτ K_1 ) equals the convolution product,   
take M = M_π to be the functor associated with the smooth left G-representation π where π =  ℋ_(G, Υ)  with  G acting via left translation λ (<ref>).    
Let  μ be a -valued left Haar measure on G. By  Corollary  <ref>, 

    μ(K_2) μ(K_1)  ·  j_K_3∘ [K_3σ K_2 ] ∘ [K_2τ K_1]   ( (K_1)    )   =  (K_3σ K_2 ) * (K_2σ K_1  )
    
as elements of    M_π   = π =   ℋ_(G, Υ_G). 
   Similarly, 
    μ (K_2 )  μ(K_1 )    ·      j_K_3∘∑    _i d_i   [ K_3σ_iτ K_1 ] · (  (K_1)   ) =   μ(K_2 )     ∑    _i d_i (K_1σ_iτ K_3)  .

As the LHS of these two equalities are equal by the above   argument,    we must have   
    (K_3σ K_2 ) * (K_2σ K_1 ) =    μ(K_2)  ∑    _i d_i (K_1σ_iτ K_3 ) .
   But the coefficient of ( K _ 3 υ K_1) in   (K_3σ K_2 ) * (K_2τ K_1 ) equals μ (  K_3σ K_2∩υ  
K_1τ ^-1 K_2 )   =  μ(K_2 )  c_σ  ,  τ  ^ υ. Therefore ∑_i d_i( K_1σ_iτ K_3  ) must equal the convolution of double cosets.                 
    




 §.§ Completed pushforwards
    
Let ι : H → G be    as  in  <ref> and assume  moreover  that  H, G are  unimodular.     Let μ_H, μ_G Haar measures on  H, G  respectively  with μ_ H  (  Υ_H ) , μ_G ( Υ_G)   ∈ R ^ ×.  
Let ℋ _  R ( G , Υ  _  G   )  denote the Hecke algebra of G  over   Υ   _  G. 

 

  
Given  smooth  representations τ of H, σ of G,  we consider  τ⊗ℋ_R(G ,  Υ _  G  ) and σ smooth representations  of H × G under the following  extended action. 
       

  * (h, g) ∈ H × G acts on x ⊗ξ∈τ⊗ℋ _  R  (G   ,   Υ   _  G      ) via x ⊗ξ↦  h x  ⊗ξ ( ι ( h )  ^  - 1   ( - )  g ).   

  * (h,  g ) ∈  H ×  G   acts  on y ∈σ via y ↦ g ·  y. 
          
An    intertwining map Ψ : τ⊗ℋ_R(G,  Υ    _  G    )   →σ is  defined  to  be  a  morphism of H × G  representations.  
    


        Let Ψ : τ⊗ℋ_R(G,  Υ    _  G    )   →σ be an intertwining map. 
For any ξ _1 , ξ_2∈ℋ_R( G , Υ _  G   ) and  x  ∈τ,  
    ξ_1·Ψ (  x  ⊗ξ_2 )   =  Ψ  (   x ⊗ξ_2 * ξ_1^t ) ,
 
where ξ_1 ^t is the transpose of ξ_1.   
 
    Since Ψ is an intertwining map,  it is also a morphism of ℋ_R(G, Υ_G )  modules  under the induced  actions. Thus, ξ_1·Ψ  ( x ⊗ξ_2) =  Ψ ( x ⊗ξ_1 *_ρξ_2 ).  But   Lemma  <ref> implies that ξ   _  1        *_ρξ_2  =  ξ_2 * ξ_1^t. 
  

The proofs for the next two results are omitted and can be found in <cit.> (cf.  <cit.>).          

  Let σ ^ ∨ denote the smooth dual of σ and ⟨·  ,  ·⟩  : σ  ^ ∨×σ→  R denote the induced pairing. Consider τ⊗σ ^ ∨ as a smooth H-representation   via h (x ⊗ f   )  = hx ⊗ι(h) f.   Then for any intertwining  map  Ψ as above, there is a unique morphism ψ :  τ⊗σ^∨→ R of  smooth  H-representations such that 
    ⟨  f ,   Ψ ( x ⊗ξ )  ⟩  =  ψ (  x ⊗ ( ξ· f ) )
    
for all x ∈τ, f ∈σ  ^ ∨ and ξ∈ℋ_R(G,  Υ_G ). The mapping Ψ↦ψ  thus   defined    induces a bijection   between     Hom_H × G  ( τ⊗ℋ_R(G, Υ_G ) , σ ) and Hom_H ( τ⊗σ^∨ ,  R ).
    
  


  
Suppose  M_  H, M_G are RIC functors  with M_H CoMack and  M_G  Mackey.  Consider M_H⊗ℋ_R (G  ,  Υ  ) and M_G   as smooth H × G representations via the extended action.  Then for any pushforward ι_*  : M_H→  M_G,   
there is a unique    intertwining map of H  ×  G representations 

    ι̂ _  *  : M_H⊗ℋ _  R  (  G ,  Υ  _     G        )      → M  _G
 satisfying the following  compatibility    condition:  for all compatible pairs (U,K)  ∈Υ _H×Υ_G, x ∈ M_H(U), we  have ι̂ _* ( j_U(x ) ⊗(K)   ) =  μ_H  ( U   )  j_K (   ι_U,K,* ( x )    ).  Equivalently any ι_* determines a unique   morphism ι̃_*  :    M_H⊗   (M_G)^∨→  R of H-representations    such that  ι̃_* (  j_U(x ) ⊗ f    )    = μ_H(U)    f  ( j_K∘ι_U,K,*(x)  ) for U, K, x  as above  and f ∈   ( M_G   )     ^∨           



          Let ι̂_* be as above. For any U ∈Υ_H, K  ∈Υ_G, g ∈ G and x 
∈ M_H(U),     
    ι̂_* ( j_U(x_U ) ⊗ ( g K  )  ) = μ_H ( U ∩ g K g^-1 ) ·  j_K∘  [ U g K ] _  *   ( x  
    ) .

  


    In the definition of ι̂_*, we may replace ℋ_R(G) with 𝒞_R( G ) which is ℋ_R(G) considered as a R-module with G-action given by right translation, since the definition of ι̂_* does not require the convolution operation. In particular, ι̂_* is independent of μ_G.    
 



 §.§ Shimura   varieties
 
In this subsection, we   briefly  outline how  the abstract formalism  here applies to the  cohomology of   general Shimura varieties.  We refer the  reader to <cit.> for terminology which we will be used freely in what  follows.      

Let (, X) be a Shimura-Deligne (SD) datum and let 𝐙 denote the center of . For any neat compact open subgroup K ⊂(_f), the double quotient 
    _(K)() := () \ [ X ×(_f)/K]
  is  the   set of -points  of a  smooth  quasi-projective variety over .   If    (, X ) satisfies (SD3)  or  if  (, X) admits an embedding into a  SD datum which satisfies (SD3),   then _(K) admits a canonical model over its reflex field.  For two neat compact open subgroups K' , K ⊂(_f) such that K ' ⊂  K,    it is not true in general  that _(K')() →_(K) is a covering map of degree [K' : K ] unless (SD5) is also  satisfied (<cit.>). However  one can establish the following (cf. <cit.>).            

          Let K,  K'  ⊂ ( _f   ) be  neat  compact open subgroups  such that K ' ⊂ K and  K ∩𝐙() =   K'  ∩𝐙().    Then the natural  map  _K', K :  Sh_(K')( )     ↠Sh_(K)   ( )  of smooth -manifolds     is  an  unramified  covering map   of degree [K :  K' ]. 
    
    
Suppose   that    there exists x ∈ X, g ∈𝐆(_f ) and k ∈  K such that [x, g]_K'  = [x , g k   ]   _    K'  in  
Sh_(K')(). Let K_∞  denote  the  stabilizer of x in 𝐆( ).     By  definition,    there exists a  γ∈() ∩ K_∞ such that 

    g k  =  γ g k '
    for  some   k '  ∈   K '  ⊂  K. Then     γ   = g k (k')^-1 g^-1 is an element of Γ : =  () ∩  g K g^-1. Since () is discrete in (     ),  we  see  that    Γ is discrete in () and so   Γ∩  K_∞ is discrete in  K_∞.  In  particular,    the group C : = ⟨γ⟩⊂Γ∩ K_∞  generated by γ is  discrete in K_∞.        By  <cit.>, the  quotient   K_∞  /  (   𝐙 (  )  ∩  K _ ∞ ) is  a  compact    group. Since C / ( 𝐙() ∩ C )  is a (necessarily closed)   discrete subgroup of this quotient,  it  must  be  finite.  There is  therefore  a  positive  integer n   such that 
    γ ^n∈𝐙(  )    ∩   C     ⊂𝐙 ()  .
   Since Γ⊂() is neat, its image Γ̅⊂𝐆^ad() under the natural map 𝐆 (    )   →𝐆^ad (  ) is also neat   <cit.>.  Thus   Γ /  ( 𝐙 ( ) ∩Γ  ) = Γ / ( 𝐙() ∩Γ) ⊂Γ̅ is neat as well   and in  particular   torsion  free.    So  it must be the case that  γ∈𝐙(). From  (<ref>), we   infer that   k = γ k ' and this makes γ an element  K ∩𝐙().  As K  ∩𝐙() = K' ∩𝐙(), we see that   k = γ k ' ∈ K'  .     
The upshot   is  that   the fiber of _   K' ,K above  [x,g]_K is of cardinality [K : K ' ].   

Now  let L ⊂ K' be normal in K. By replacing L with L    ·     (𝐙() ∩ K), we may assume that L  ∩𝐙()    = K ∩𝐙().   Applying the same argument to L, we  see that Sh_(K)() 
is a quotient of Sh_(L)()   by  the free action of   K / L,    
hence the natural quotient map is an unramified covering of  degree [K : L].   Similarly for K'. This implies the claim since [K : L ] = [K  : K   ' ] ·  [K' : L].   
   
      Let L    , L   ' ,  K be neat  compact open subgroups of (_f) such that L  , L'  ⊂ K and all three have the same intersections     with 𝐙().         For   γ∈ K, let L_γ = L ∩γ L' γ ^-1.  
Then  the diagram  below   

   
    _γ Sh(L_γ )  [r]  [d ,  "⊔[γ]",  swap ]      Sh(L)   [d,   ]     

Sh(L')  [r]        Sh(K) 


 
where γ∈ K runs over representatives of L  \ K / L '  is  Cartesian  in the category  of smooth -manifolds.    
    
If canonical models exist for Sh_(K), then the following lemma allows us to descend the Cartesian property above to the level of varieties.   
            Let W , X,  Y , Z be geometrically reduced locally of finite type schemes over a field k of characteristic zero  forming a commutative diagram 

   W   [r, "a"] [d, "g",  swap]     
 X  [d, "f"]   

Z  [r, "b"]       Y 


 
such that f ,    g are  étale. Suppose for  each  closed point z ∈ Z  the map a  : W → X is injective on the pre-image g^-1(z) and surjects onto the pre-image of f^-1(b(z)). Then the diagram above is Cartesian in the category of k-schemes. 

  Suppose that 𝒲  = X × _ Y  Z is a pullback. Let p_X : 𝒲→ X, p_Y  :   𝒲→ Z be the natural  projection  maps and γ : W  →𝒲 the map induced  by the universal property  of 𝒲. 
As  f  is  étale, so is  p_Z  and since p_Z∘γ  =   g   is  étale, so is γ.   Let   k̅ denote  the separable closure of k.  Since 𝒲(k̅       ) = { (x,z) ∈ X(k̅) ×  Z ( k̅     )   |     f ( x ) =   b     (  z    )  },  the  condition on closed points (i.e.  k̅-points)  implies that  γ :  W(k̅)  →𝒲(k̅) is a bijection. The result follows since an étale  morphism between such schemes that is bijective on k̅ points is  necessarily    an  
isomorphism.
  

We now assume for the rest of this subsection that _(K)() admits a canonical model for each neat level K. We let
Υ be any collection of  neat compact open subgroups of (_f) such that the intersection of any  K ∈Υ with 𝐙()  gives a subgroup of 𝐙() that is independent of  K. For instance,  we may take Υ = Υ(K_0) for any given neat level K_0 where Υ(K_0) is the set of all finite intersections of conjugates of K_0. By Lemma <ref>, such a collection satisfies (T1)-(T3) and clearly, the intersection of any group in Υ(K_0) with 𝐙() equals K_0∩𝐙(). Now   let {ℱ_K}_K ∈Υ  be a collection of _p-sheaves ℱ_K on _(K)  that are equivariant under the pullback action of (_f ). 
More precisely, for any σ∈(_f) and L, K ∈Υ such that σ^-1 L  σ⊂ K, we assume that there are natural isomorphisms φ_σ :  [σ]_L,K^*ℱ_K≃ℱ such that φ_τσ = [τ]^*_L',L∘φ_σ for any L ' ∈Υ satisfying τ^-1 L ' τ⊂ L. 
For any integer i ≥ 0 and K ∈Υ, let 
    M(K) :=  H^i_( _(K), ℱ_K)
    denote Jannsen's continuous étale cohomology.  Then   for  any  morphism ( L  K ) ∈𝒫(G, Υ), there are induced _p-linear maps [σ]_L,K^* : M(L)  → M(K) and [σ]_L,K^*  : M(K)  → M(L)  that make M a  RIC   functor  for  𝒫(G, Υ) (see <cit.>).       
 M is  a  cohomological Mackey functor.   
    

 Lemma <ref> and <cit.>) imply that M is cohomological.   Corollary     <ref> and <cit.>  imply that M is Mackey.    
    

 
Using similar arguments, one may establish that an injective morphism (, Y) ↪ (, X) of Shimura-Deligne data and a collection on sheaves of the two sets of varieties that are compatible under all possible pullbacks  induce  a Mackey pushforward on the corresponding cohomology of varieties over the reflex field of (, Y). See e.g.,  <cit.>.  Some care is  required in the case where  the centers of  and  differ and (SD5) is not satisfied for . This is because one to needs to specify a collection of compact open subgroups for (_f ) which contains the pullback of Υ and which also satisfies the  conditions of intersection with the center of (_f ).        
 




§ ABSTRACT ZETA   ELEMENTS
    
 

In this section, we begin  by  giving  ourselves a  certain  setup that one    encounters in, but  which it is not necessarily  limited  to,  questions  involving  pushforwards  of  elements  in     the cohomology of Shimura  varieties   and  we formulate  a  general  problem in  the style of Euler  system  norm  relations within that setup. We then propose an abstract resolution for it by defining a notion we refer to as  zeta elements and study its  various properties.   An example involving CM points on modular curves is provided in  <ref> and the reader is encouraged  to refer to it 
 while   reading   this section.  We note  for the convenience of the reader that in the said example, it is the group  denoted  `G̃' (resp., `K̃') that plays the role of the group denoted `G' (resp., `K') below. 


 §.§ The setup
   
 
Suppose for all of this  subsection that we are given 
         

  ∙        ι : H ↪ G a closed immersion of unimodular  locally profinite  groups,   

  ∙ Υ_H, Υ_G   non-empty    collections of compact open  subgroups satisfying (T1)-(T3) and    ι^-1(Υ_G) ⊂Υ_H,       

  ∙   an  integral  domain  with field  of fractions a -algebra,

  ∙ M_H, : 𝒫(H,   Υ_H   ) →𝒪-Mod, M_G,      : 𝒫(G,   Υ_G    ) →𝒪-Mod CoMack  functors,    

  ∙   ι _  *   : M _ H , →  M _  G , a pushforward,   


  ∙ U  ∈Υ _H, K ∈Υ_G compact opens such that U = K  ∩  H referred to as  bottom levels,   


  ∙   x _    U ∈  M _  H ,   (  U  ) which we call the source bottom class,         


  ∙  ℌ∈ 𝒞    _      (   K  \  G    /  K   ) a  non-zero   element which  we call the Hecke  polynomial,    


  ∙ L ∈Υ _   G, L   ◃     K a  normal compact open subgroup referred to as a layer extension of degree d =  [ K : L ].

As in   Definition    <ref>, ℌ induces a -linear map ℌ_* =  ℌ^t  :  M_G ,  ( K ) → M_G,  ( K ). Let y_K : =  ι_U, K ,  *  (  x _  U     )  ∈  M_G,  (   K ) which we call the target bottom class.      

           
Does there exist  a  class y   _   L   ∈  M_  G ,    ( L  ) such  that  
    ℌ _* (y_K)  =      _  L , K  ,  *  ( y _  L     )

 as  elements  of    M _   G  ,     ( K )?   


          Let us first  make a  few  general  remarks.  First note note is that if  d ∈^×, the class d^-1·_L,K^*(y_K) ∈ M_G,(L) solves  the   problem   above.   Thus  the non-trivial case occurs only when d is not invertible in , and in particular when  is not a field. In  Kolyvagin's bounding argument, the usefulness of such a norm relation  is indeed where d is taken to be non-invertible e.g.,  
= _p and d = ℓ - 1 where  ℓ≠ p is a prime such that a large power of p divides ℓ  -   1.  


Second, Problem <ref> is meant to be  posed as a family of  such  problems where one varies L  over a prescribed lattice of compact open subgroups of K  (which correspond to layers of certain abelian field extensions) together with the other parameters  above  and the goal is to construct  y_L  that satisfy such    relations  compatibly  in  a   tower.       This is typically achieved by breaking the norm relation problem into  `local'    components and varying the parameters componentwise. More precisely, H and G are in practice  the  groups of adelic points of certain  reductive  algebraic  groups over a number field and the class x_U  has the features of a  restricted    tensor  product.    The problem above   is then   posed     for each   place in   a      subset of  all  finite  places of the number field.  Thus Problem <ref> is  to be seen as one of a local  nature  that  is            extracted      from  a   global setting. See  <ref> for an  abstract formulation of  this   global     scenario.                


Third, the  underling premise of <ref> is that y_K is the image of a class x_U that one can vary  over the levels of  the   functor M_H, and for which   one has a better description   as     compared to their counterparts  in   M _      G  ,.  If  ι_* is  also  Mackey, then Lemma   <ref> tells us  that ℌ_*  (y_K) is the image of certain mixed Hecke correspondences. The    class y_L  we are  seeking is  therefore    required   to be of a similar form.   As  experience  
suggests,  we assume that  y_L =    ∑ _  i = 1    ^  r  [V_ig_i L]_*   ( x_V_i )   where
     

  * g_i∈ G,

  * V_i⊂ g_i L g_i ^  -1, V_i∈Υ_H

  * x_V_i∈  M_H,  ( V_i )  
 
are unknown quantities that we need to pick to obtain the said equality.    
If we only require equality up to -torsion (which suffices for applications, see  <ref>), then one can use Proposition <ref>     to guide  these  choices. More precisely, let μ_H be a  -valued   Haar  measure on H, Φ a field containing , and M_H, Φ, M_G, Φ denote the functors obtained by tensoring with Φ. Let  M _ H , Φ,   M  _ G , Φ be the completions of M_H, Φ,   M _  G , Φ    respectively. For V ∈Υ_H, let  j_ V      : M_H,     (  V )  → M  _    H  ,   Φ denote   (abusing  notation)     the natural map and similarly for M_G,.  Let  ι̂ _*     :    M_H, Φ⊗ℋ_Φ(G , Υ_G )         →  M  
  _  G , Φ the completed  pushforward of  Proposition  <ref>.     
As M_G,Φ is cohomological, the kernel of j_K  : M_G,(K) →M_H,Φ is contained in -torsion of M_G, (K). An application of   Corollary  <ref>    then  implies that  ℌ_*  ( y _  K  )   -    _L,K, *  ( y_L ) is -torsion   if and only if 
    ι̂ _ *  (  j_U ( x _U ) ⊗ℌ  )    =   ι̂ _* ( ∑ _ i = 1   ^   r   μ_H(U)/μ_H ( V_i ) (  j_V_i( x_V_i )    ⊗ch(g_i K  )  )     )
 
as elements of the Φ-vector space M_G, Φ  (see the  proof of     Proposition  <ref> below). Thus we  are seeking a  specific “test vector"   
  in M_H, Φ⊗ℋ_Φ(G , Υ_G )  ^K  containing the data of certain  elements in  M_H,  
whose image under ι̂_*  coincides with that of j_U(x_U)⊗ℌ.  Any such test   vector can equivalently be seen as a right K-invariant compactly supported function ζ : G →M_H, Φ.  The shape of the  element inside ι̂_* on the RHS of  (<ref>)    forces upon us  a notion of integrality of such vectors. As ι̂_* is H-equivariant,  a natural way of    enforcing    (<ref>) is to require that the two functions  in the inputs of ι̂_* have equal H-coinvariants with respect to the natural H-action on the set of such functions. If a test vector  satisfying these two conditions exists,  Problem <ref> is solved modulo -torsion. In fact, such a vector solves the corresponding problem (modulo torsion) for any pushforward emanating from M_H, to a functor on 𝒫(G,Υ_G), since the two  aforementioned properties  are    completely   independent of ι_*.
Under certain additional conditions, the  resulting norm    relation can be upgraded    to an equality.    
See <ref>            

We now formalize the discussion   above. For τ an arbitrary group, we let 𝒞 (G/K, τ  )  denote the  set  of all  compactly supported functions ξ : G →τ that are invariant under right translation by K on the source. Here the support of ξ is the set of elements that do not map to identity element in τ.   If τ is abelian  and  has the structure of  a Φ-vector space,   
𝒞 (G/K, τ ) is a Φ-vector space.  If τ is in addition a Φ-linear  left H-representation,  so is 𝒞(G/K, τ) where  we let  h ∈ H act on ξ∈𝒞   (G/K,   τ  ) via   ξ↦ h ξ  : =  h   ξ ( h^-1 ( - ) ).  In this case, we denote by 𝒞(G/K, τ ) _H the   space of H-coinvariants and  write ξ_1≃ξ_2 if ξ_1, ξ_2∈𝒞(  G  / K ,   τ ) fall in the same H-coinvariant class.   Given a ϕ∈𝒞   (G/K, Φ) and x ∈τ, we let x ⊗ϕ∈𝒞    (G/ K ,   τ   ) denote the  
function given by  g ↦ϕ(g) x. Fix a   -valued     Haar  measure μ_H on H. For V_1, V_2∈Υ_H, we denote 
[V_1 : V_2 ] : = μ_H(V_1 ) /  μ_H(V_2).  This is then independent of the choice of μ_H.         

 An  element    ξ∈𝒞  (G/K , M_H,Φ)  is said to be  -integral at level L   if for  each g ∈ G, there exists a finite collection { V_i∈Υ_H | V_i⊂ gL g^-1}_i ∈ I and  classes x_V_i∈  M_H,  ( V_i) for each i ∈  I  such that 
    ξ(g) =   ∑    _i ∈  I     [U:V_i]      j_V_i( x_V_i    ) .
     A  zeta element for (x_U, ℌ, L) with coefficients in Φ   is  an  element ζ∈𝒞  (   G    / K ,   M  _H, Φ ) that  is     -integral at level L and lies in the H-coinvariant class 
of          j_U(x_U) ⊗ℌ 
 
 

 This notion of integrality  appears in <cit.>. Cf. <cit.>.       
 

Let ζ be   a      zeta  element for (x_U ,  ℌ , L  ). Then we may   write ζ as a (possibly empty if ζ = 0) finite   sum ∑_α [U : V_α]     j_V_α (x_V_α )  ⊗(g_αK)  where  for  each α,   V_α⊂ g_α L g_α^-1 and x_V_α∈  M_H,  ( V_α ).    
Given such a presentation of ζ, we refer to 

    y_L   : =  ∑       _  α   [V_  α g_α L   ] _   *   (x_V_α)  ∈  M_G,  ( L )
    as an associated class for ζ under ι_*. 
It depends on the choice of the  presentation for ζ.        


        Suppose   there  exists a zeta 
element for (x_U , ℌ , L ).   
Then  for    any  associated class y_L∈ M_G, (K),   the   difference     ℌ_*(y_K )  -  _ L , K , *   ( y _  L ) lies in the -torsion of M_G, ( K ).    


 As above, let  ζ =   ∑_α [U : V_α ]         j_V_α( x_V_α )   ⊗(g_α K ) be  a choice (of a presentation)  of  a zeta element to which y_L  is   associated.    
Let j_K   :  M_G,  (K)  →M_G , Φ denote  the natural map and  let     μ_G be a Haar measure on G such that μ_G(K) = 1.    
Corollary   <ref> and the properties of ι̂_*  as 
 an   intertwining map (Lemma <ref>)  imply      that      
    μ_H(U)  ·  j_K(  ℌ_*(y_K) )   =  ℌ^t·ι̂ _ *  ( j_U ( x_U ) ⊗(K)    )    =   ι̂    _* ( j_U( x_U ) ⊗ℌ   )  .
 Since ι̂_* is H-equivariant, its restriction to M_H,Φ⊗ℋ(G, Υ_G)^K≃𝒞  
(G/K,  
M_H, Φ ) factors through the space  of  corresponding 
    H-coinvariants. Since  ( j_U(x_U   )   ⊗ℌ  
   )    ≃ζ   
by  assumption,       
    ι̂_* ( j_U(x_U ) ⊗ℌ )  =   ∑    _α [U:V_α]  ι̂_*     ( j_V_α(x_V_α) ⊗(g_α K )  ).
    Corollary <ref> 
allows us to rewrite each summand on the right hand side above  as   μ_H(U)            j_K∘ [ V  _  α  g_α K ]_*(x_V_α ).  By  Lemma  <ref>, [ V_α g_α K ]  _ *  = _L, K , * ∘  [ V_α g_α L ]  _   *.  Putting everything together, we get that         
    μ_H(U)   ·  j_K ( ℌ_*( y_K )  )       =   μ_H ( U )   ·      j_K∘_L, K , * ( ∑    _α   [V_α  g_α L ]  _ *  ( x_V_α )    )   
        =   μ_H ( U )  · j_K  (   _L,K,* ( y_L )        )

Thus j_K ( ℌ_*(y_K) - _L,K,*( y_L ) )  = 0.  This implies the claim since the kernel of j_K  :    M_G,   ( K ) → M_G, Φ ( K ) →M_G, Φ is  M_G, (K)_-tors  by   Lemma <ref>.    
  

We next  study how  a given  presentation of a   zeta element  may be  modified. 
    Given f ∈𝒞   (G/K, H) and ξ∈𝒞 (G/K, τ ) for τ any   left H-representation over Φ, we define f  ξ∈𝒞      (G/ K,  τ ) by g ↦ f(g) ξ( f(g)^-1 g ).    
  


        If ζ is a zeta element, so is f  ζ for any f ∈𝒞    (     G/K, H).  Moreover the set of associated classes for the two elements under any    pushforward are equal.           
 


   Clearly  
fζ lies in 𝒞  (    G/K,   M_H,  Φ ) and   f ζ≃ζ.      Say  ∑_α   [ U  :   V_α ]    j_V_α ( x_V_α ) ⊗( g_α   K ) is a  presentation for ζ.    
Set h_α : = f (
g_α ), V_α' : = h_α V_α h_α^-1 and x_V_α' ' : = [h]_V_α', V_α^* ( x_V_α). Then  
    f     ζ =  ∑    _α  [ U  :  V_α   ]          j_V_α' ( x_V_α' ' )  ⊗( h g_α K ) .

Since [U : V_α ]  =  [ U :  V_α' ]  by   unimodularity of H,  f ζ is integral at   L.   That the sets of   associated  classes for ζ and ηζ under a pushforward   are equal   follows  by  Lemma   <ref>.       
 

 Let ζ be a zeta element.   A  presentation ζ  =     ∑_α  [U : V_α]          j_V_α(x_V_α) ⊗(g_α K ) 
is said to be  optimal if   V_α = H ∩  g_α L g_α^-1 for all α  and the    cosets     H g_α K are pairwise   disjoint. We say that ζ is  optimal if it has an  optimal  presentation.   
    
    If there exists a zeta element, there exists an optimal one and such that the set of associated classes of the latter element under any pushforward contains those of   the  former.    
   

    Let ζ =  ∑ _α∈ A [U  : V_α ]        j_V_α(x_V_α   )     ⊗( g_α  K )  be  a   presentation of a zeta element. Say there is an index β∈ A such that V_β≠   H ∩   g_β   L g  _β    ^-1.  Temporarily denote    V_β ' :  =  H ∩ g_β L g _ β and x_V_β' = _V_β,  V_β' ,  *   ( x_V_β ).  Then 
    [U : V_α ]       j_V_β  ( x_V_β )  ⊗( g _β K ) ,          [U : V_β ' ]    j_V_β' (  x_V_β ' )  ⊗( g_β   K )
 are equal in  𝒞( G/ K ,  M_H, Φ)  _   H    by  Lemma  <ref>.   So the element ζ ' obtained by replacing the summand indexed by β in ζ  with [U : V_β ' ]    j_V_β' (  x_V_β ' )  ⊗( g_β  K )  constitutes a  zeta   element.        Since [V_β  ' g_β L   ]  ∘_V_β , V_β ' , * = [V_β  g_β L ]_* by  Lemma  <ref>, the  associated classes    for  the  chosen    presentations of ζ and ζ' are equal.    So we  can   assume that there is no such index β in our chosen ζ. But then  f ζ for any f ∈𝒞(G/K,  H ) is a zeta element (Lemma <ref>) which also  possesses the same properties.    By choosing f suitably, we can  ensure that H g _α K  are  pairwise            disjoint. 
     


    
The   terminology
`zeta element'  is inspired by <cit.> and motivated by the fact that  Hecke polynomials specialize to 
zeta functions of Shimura varieties <cit.>,  <cit.>. 
 


 §.§ Existence Criteria
     
In this section, we  derive a necessary and sufficient criteria   for the     existence of zeta elements  that can be applied   in   practice.        


Retain the  setup of <ref> and the notations introduced  therein. For X ⊂ H a group and g ∈ G, we will often denote by X_g = X_g,K the intersection   X ∩ g K g^-1. For the result below,  we denote   by  τ be an arbitrary  left   H-representation over Φ.     
For ξ∈𝒞( G/ K , τ ), we  let    f_ξ∈𝒞(G/K, H ) be an element  satisfying the following condition: supp  (   f_ξξ  )  =  ⊔_i  g_i K and H g_i     K are pairwise  disjoint (see Notation <ref>).  It is clear that a  f_ξ exist for each ξ.        


    The class [ξ]_H∈𝒞(G/K, τ)_H   vanishes   if and only for each g ∈ G, the class of (f_ξξ)(g)  ∈τ in the space τ_H_g of H_g-coinvariants   vanishes.      
    
 
For each α∈ H \ G / K,  fix a choice  g_α K ∈ G / K such that H g_α K =  α and set  
    ℳ : = ⊕_α∈ H \ G / K τ_H_g_α  .
   We are going to define  a Φ-linear map φ :  𝒞(G/K, τ ) →ℳ. Since 𝒞(G/K, τ) ≃⊕_gK ∈ G/ Kτ, it suffices to specify  φ on simple tensors.  Given x ⊗(gK) ∈𝒞(G/K, τ), let α : =  H g K and  pick h ∈ H such that h g K = g_α K. Then we  set φ (  x ⊗(gK )     )  ∈ℳ to be the element whose component at any index β≠α vanishes and at α equals   the      H_g_α-coinvariant class of hx.   
It is straightforward to verify φ is well-defined  and factors through the quotient  𝒞(G/K, τ)_H.

We now prove the claim. If ξ = 0, the claim is obvious, so assume otherwise.   Since [f_ξξ]_H = [ξ]_H, we may replace ξ with f_ξξ and assume    wlog   that elements of ( ξ)/K ⊂ G/K represent distinct cosets in  H \ G  / K. Say ξ = ∑_i       x_i⊗(g_iK ). Denote α_i : = H g_i K and let h_i∈ H be such that h_i g_i = g_α_i K. If [ξ]_H  vanishes, 
so does φ(ξ)  which in turn implies that the class of h_i x_i in H_g_α_i-coinvariants 
of τ vanishes for each i. By conjugation,   this is equivalent to the vanishing of H_g_i-coinvariant  class   of x_i for each i. This proves the only if direction. The  if   direction is straightforward since the vanishing of H_g_i-coinvariant  class  of x_i∈τ readily   implies    the same for the  H_g_i-coinvariant 
 class  of x_i⊗(g_i K )  ∈𝒞(G/K , τ ).  


  For g ∈ G, the g-twisted H-restriction or the (H,g)-restriction of ℌ is the function 
    𝔥_g :  H →𝒪

given by h ↦ℌ(hg) for all h ∈ H. 
 

      For each α∈ H \  H ·(ℌ ) / K, choose a representative g_α∈ G for α. We denote (abusing notation)  H_α = H ∩ g_α K g_α^-1, V_α = H ∩ g_α L  g_α^-1, d_α = [ H_α  :  V_α ]  and 𝔥_α = 𝔥_g_α denote  the (H, g_α)-restriction of ℌ.  
  

         There exists a  zeta  element for ( x_U ,   ℌ    ,  L  )  if  and  only if there exist classes x_V_α∈ M_H, 𝒪(V_α) for all α∈  H  \ ( H    ·ℌ  ) / K  
such that 

    𝔥_α^t·  j_U(x_U)   =  j_H_α∘_V_α, H_α,*(x_V_α)

in M  _H, Φ.      
Moreover if y_L is an   associated  class for a given zeta element under ι_* and M_H, is -torsion  free,    the classes x_V_α  satisfying the criteria   above can be picked  so  that    _L, K, *  ( y _   L   )        =    ∑_α [V_α  g_α K ]_* (x_V_α ) where the sum runs over 
   α∈ H  \ ( H ·ℌ  ) / K.         
     
      For notational convenience, we will  denote by A   : =  H \  H  · (   ℌ )     / K and x : = j_U(x_U)  
in the    proof.   Since U  \ G / K 
 → K \ G  / K is surjective and  has finite fibers,  we have a natural injection 𝒞_( K \ G / K )  ↪𝒞_  (  U \ G / K ) given by  ( K  σ K )  ↦∑ ( U τ   K ) where τ runs over U \ K σ K / K. Via this map, we  consider ℌ as an element in 𝒞_( U \ G / K ). We  assume that  

    ℌ   =  ∑  
       _ j  ∈ J  c_j   ( U σ _j K )
    where J is a finite indexing set, U σ_j K are pairwise disjoint double cosets and   c_j≠ 0 for all j ∈  J. For α∈  A, let J_α⊂ J be the set of all j ∈  J     such that H σ _j K =  α. For each j ∈ J_α, let h_j∈ H 
be such that σ_j K =  h_j 
 g_α K.  Then   U h_j  H_α∈  U  \ H / H_α is independent of the choice of h_j and  
    𝔥_α : = ∑    _ j ∈  J_ α c_j  (   U    h_j   H_α    )     ∈𝒞 _      ( U \ G /  H_α 
     ) .
 


()    Assume that there exist  x_V_α∈  M_H,  ( V_α   ) satisfying the equality in the statement.   
We claim that  
 ζ=∑_α∈   A    [ U : V_α ]      j_V_α( x_V_α  )  ⊗( 
 g _ α K ) is  a   zeta element.  As  ζ is clearly -integral, we need to  only  show that x 
 ⊗ℌ≃ζ.  To this end,  note  that     
 
    x    ⊗ℌ   ≃∑_ j ∈  J      c_j      [ U : U_σ_j ]    (  
        x    ⊗ (   σ_j  K )   )   
       ≃∑_j ∈ J   c_j      
     [ U :  H_σ_j ]     
     [U    σ_j K ]_* (  x  ⊗ ( σ_j K  )    )  
       ≃∑_α∈   A 
            [ U  : 
     H_α ] ∑_j ∈ J_α c_j     [U σ _j K]_*  (   h_j^-1   x   ⊗  (  g_α  K  )    )

where    the third relation uses that μ_H ( H_σ_j)  = μ_H( H_α ) for 
j ∈ J_α.  For each α∈   A, let θ _α∈Φ [H_g_α] denote the sum  over  
 a set of representatives of H_α / V_α and for each j ∈  J_α,  let ς_j∈Φ[H_α] denote the sum over  a set of  representatives for H_α /  ( h_j^-1 U _σ_j  h_j  ).  By  Lemma <ref> and Lemma  <ref>, we have 
    𝔥_α^t· x =   ∑_ j ∈ J_α  c_jς_j h_j^-1  x,                  
    j_H_g_α∘_V_α, H_g_α,*(x_V_α)   =     θ_α         j_V_α ( x_V_α )

Now note that ς_j = [H_α :  h_j^-1  U_σ_j h_j   ] =     [U σ_j  K ] _* where degree of an element in a group algebra denotes its image under the augmnetation map.  Thus for each α∈ A,                  

    ∑  
    _  j ∈ J_α c_jdeg  [U σ _j K]_*  (  h_j  ^  -  1     
     x ⊗ (  g_α K   )      )     ≃∑  _ j ∈  J_α      
     c_j (   ς_j h_j ^-1  x   ⊗ (  g_α   K   )      )     
        =   ( 𝔥_α^t·   
      x ) ⊗ ( g_α K )  
        =   ( j_H_α∘_V_α, H_α, * (x_V_α )  )   ⊗( g_α  K ) 
        =   ( θ_α     j_V_α ( x_V_α )   )   ⊗( g_α K  )   
       ≃    [H_α : U_α ]   ·     j_V_α (  x_V_α   )     ⊗( g_α K )

Putting everything together, 
we deduce that   
    x ⊗ℌ   ≃∑ _α∈  A   
       [U : H_α] [ H_α : V_α ]  ·    j_V_α  ( x_V_α  ) ⊗(g_α K )  
        =  ∑_α∈ A  [U : V_α ] j_V_α(x_V_α ) ⊗( g_α K )  = ζ .
  
This completes the proof of the if direction.  
   




()   Suppose that ζ is a zeta element for (x_U, ℌ, L ). Invoking  Lemma 
<ref>, we may assume that ζ is optimal. Say ζ  = ∑  _ β∈ B [U : V_β ] j_V_β(x_V_β) ⊗( g_β K) is an optimal presentation where B is some finite indexing set. We identify    B   with   a subset of H \ G / K by identifying β∈   B with H g_β K  ∈  H \ G / K.   Extending B by adding zero summands to ζ if necessary, we may assume that A  ⊂  B. 
Lemma <ref> allows us to further assume that { g_α   K    |  α∈ A    }⊆{ g_β   K    |  β∈  B        }.   

We claim that x_V_β for β∈  A    are the  desired elements.   Set 𝔥_β : = 0  and d_β : = [H_g_β     : V_β ] for β∈  B  
 ∖   A.  Our calculation in the first part and the fact that  ζ≃  x ⊗ℌ  imply   that      

    ∑_β∈  B  [U : H_g_β ]   (  
     d_β  j_V_β (  x_V_β   )    - 𝔥_β ^t ·  x    ) ⊗(g_β K )   ≃  0 .

Lemma <ref> now implies that  H_g_β-coinvariant  class  of d_β j_V_β (x_V_β ) -    𝔥_β x vanishes   for each β∈ 
 B. Fix a β∈   B for the remainder of this   paragraph     and write  

    d_β   j_V_β ( x_V_β  )   -   𝔥_β^t· x =  ∑_i=1^k (γ _i -1) x_i
   
where γ _i∈ H_g_β and x_i∈  M _H,Φ. Let W ⊂  H_g_β be a normal compact open subgroup contained in V_β  
such that W fixes x_i for each i = 1, …, k. If β∈  A,  
we require in addition  that   W ⊂ h_j  ^-1  U h_j∩ g_β K g_β^-1 for each  j ∈ J_β.  
Let Q  = H_g_β/W and 
e_Q∈Φ[H_g_β] denote    |Q|^-1 times the sum over a set of representatives in H_g_β for Q.  Then left multiplication of elements in M_H,Φ by e_Q annihilates (γ_i - 1) x_i for all i = 1, …, k,  stabilizes    𝔥_β^t·  x ∈ (M_H,Φ)^H_g_β         and sends d_β j_V_β (x_V_β ) to  j_H_g_β∘_V_β, H_g_β, * ( x_V_β ). Thus multiplying  (<ref>) by e  _  Q on both sides yields an equality involving  𝔥_g_β and x_V_β. The equalities for β∈  A are the ones  sought  after. This completes the proof of the only if   direction.   
    
  
























It remains to prove the seoncd claim. So assume that  M_H, is -torsion free and let y_L  be   an associated class for an arbitrary  zeta element.   By the second part of  Lemma <ref>, we may pick the optimal presentation for ζ (as we did at the start of the if direction above) to  further     ensure that  
    y_L =   ∑_β∈  B        [V_β g_β L]_* ( x_V_β     )  .
     Since  
_L, K, *∘ [ V_β g_β  L ] =  [V_β g_β K ] _*   for  each  β∈  B (see Lemma <ref>),   it suffices to show that [V_β g_β K]_* ( x_V_β   ) = 0 for each β∈  B  ∖   A to establish the second claim.  Torsion freeness   of  M _ H ,  implies that  

    j_ H_g_β        : M_H,    ( H_g_β  )   →M_H, Φ
  is injective for all β.  
Thus the conclusion of the  previous  paragraph gives  _V_β , H_g_β, *  ( x_V_β  ) = 0 for all β∈  B  ∖   A      (recall that 𝔥_β = 0 for such β).   Invoking  Lemma  <ref>   again, we get that     [V_β g_β K ]_*(   x_V_β ) =  [H_g_β g_β K ] ∘_ V_β , H_g_β  *  ( x_V_β  )  = 0 
which finishes the proof.    
    


 
A  zeta element   for ( x_U , ℌ ,   L )   exists      over Φ if and only if one   exists   over  Frac  (    ).   


   The if direction is trivial and the only if direction follows by  Theorem  <ref> and injectivity of M_H, Frac (    )   ( V )  →  M_H, Φ(V) for any V   ∈Υ _     H.    
    

 Suppose in the notation introduced at the start of the proof of Theorem <ref> that for each α∈ H \  H ·(ℌ) / K,   
       

  * j_U(x_U) ∈M_H, Φ lifts    to  a  class  in   M_H,  ( H _  α )[this condition is automatic if H_α⊆  U], 
 
  * for each j ∈ J  _  α,   we  have     h_j^-1· j_U(x_U) =  a _  j   j_U(x_U )    for  some  a _  j ∈Φ.  
   
 Then there exists a zeta element  for  ( x_U  ,  ℌ   ,  L  )     if ∑   _j ∈ J_  α  c_j  a_j [ U σ_j  K ] _*∈ d_α𝒪 
 for all α.        

 For  each  α, let    x_H_α∈ M_H,  ( H_α    ) denote an element satisfying 
  j_H_α ( x_H_α ) =  j_U( x_U ). Then   
( H_α h_   j ^-1 U ) · j_U(x_U) equals a_j  [U σ_j K ]_*   j_U(x_U) for each j ∈ J_α.   
So by  Theorem <ref>, a zeta element exists in this case if (and only  if) there exist x_V_α∈  M_H,  ( V_α    )  for  each α     such that   

    ∑    _j ∈ J_α (  c_j  a_jdeg [U σ_j K ]_*   )  
      j_U ( x_U    )     =    j_H_α∘pr_V_α, H_α    , *(x_V_α)    .
    
But if ∑_j ∈  J_  α  c_j  a_j  [ U σ _  j  K  ] _*   
= d_α f _α for some f_α∈,   
 (<ref>) holds with   
 x_V_α : =  
 _V_α ,  H_α ^ *  (  f_α     x_H_g_α    ).      
    


         Suppose that M_H, 𝒪 is the trivial functor on H and x_U∈  M_H,  (U)  = is an invertible element.     Then a   zeta  element   exists  for   ( x_U  ,    ℌ    ,  L    )     if and only if  
    (  𝔥_α^t   ) 
     ∈   d_α
  for  all α∈     H \   H ·  (ℌ   )    
 / K.   



    The if part is clear since the conditions of Corollary <ref>  are satisfied  with a_j  =   1 and the sum ∑_ j ∈ J_α c_j [U σ_j K]_* equals 
 𝔥_α.       
For the only if part, note  that (<ref>) in the previous  proof  is also 
necessary.  
So we may  assume that there exist   x_V_α∈ M_H, (V_α)  =   such that (<ref>) holds. Now x_U = x_H_α∈,   
x_V_α = f_α x_ U where f_α = x_V_α· 
 x_U^-1∈ and   _V_α , H_g_α , *   (x_V_α  ) = d_αf_α  x_U.  Thus  
(<ref>)  is equivalent to   
    ∑   
     _ j ∈ J_α  (   c_j [U σ_j K ]_*    )  
     x_ U       = d_α  f_α x_U
   
and    the claim follows by multiplying by x_U^-1 on both sides.        
   
The next result is included for completeness and shows that if the norm relation problem <ref> is trivial, so is the existence of a zeta element. 
 If   𝔥_ α^t· j_U(x_U  ) ∈M_H, Φ lifts to class in d_α·     M_H,  ( H_α  ) for all α∈  H      \     H   ·  (  ℌ  )      / K,     a zeta element  exists  for (x_U ,  ℌ,  L ).  The   lifting  condition holds  automatically for  an  α  if d  _  α  is  invertible  in   .  In particular, a zeta element exists   unconditionally     if d is invertible in .       
 

  Let  x_H_α∈ M_H,  ( H_α )  be  such that j_H_α      ( d_α  x_H_α     )  =    
 𝔥_α ^t· j_U ( x_U  ). The   criteria  of   Theorem     <ref> is 
 then satisfied by taking x_V_α : = _V_α , H_g_α ^*   ( x_H_α )    ∈ M_H,(V_α).  By Lemma  <ref>, 
 we have 
    j_H_α   (  [ U h_j H_α  ]_* ( x_U )  )  =   ( H_α h_j^-1 U )    ·  j_U ( x_U ) .

Since [H_α h_j^-1 U ] ( x_U )   ∈  M_H,  (H_α   ), the class 
  𝔥_α^t· j_U(x_U)   always lifts to an element in M_H,   (  H_g_α  ). In particular when  d_α∈  ^ ×, this class   lifts to an element in d_α·  M_H ,   (H_g_α  )    = M_H,  ( H _  g_α      ).  


As noted in the proof above,  each  𝔥_α gives rise to an -linear map 
    𝔥_α,* :    M_H, (U)  →  M_H, (H_g_ α)

given by Hecke correspondences in the covariant convention. 
Theorem <ref> then says that  constructing a  zeta  element  amounts to finding  x_V_α∈ M_H,  (V_α) such that   
 
    j_H_g_α∘𝔥_α  ,  *     
     (  x_U   ) = j_H_α∘_V_α , H_α , *      (    x_V_α ) .

 Using this, we can record the following version of Theorem <ref>.   
         Suppose M_H, is -torsion free. Then a zeta element exists for (x_U , ℌ, L) if and only if there exist x_V_α∈ M_H, (V_α ) such that 
    𝔥_α, * (x_U) = _V_α , H_α     , *  ( x_V_α  )
 
for all α∈ H \ 
  H  · (   ℌ    )      / K. 
   Moreover if y_L is an associated class for a zeta element under ι _*, the classes x_V_α can be picked to ensure that _L,K,*(y_L) = ∑_α [V_α g_α K]_*  ( x_V_α  ).           
 


 
 
 
     Let  ℌ  '  ∈𝒞_ ( K \ G / K )  such   that ℌ - ℌ ' ∈  d  ·𝒞_ ( K \ G / K ). Then  there   exists a zeta element for (x_U   ,   ℌ    , L ) if and only if there exists one for  ( x_U   ,  ℌ  ' , L )   
 For g ∈ G, let 𝔥_g' ∈𝒞_𝒪 ( U \ H / H_g ) denote the (H,g)-restriction of ℌ'. Then 𝔥_g - 𝔥_g' ∈  d ·𝒞_𝒪(U\ H / H_g ). 
Since  d ·  M_H,  ( H ∩ g K g^-1   ) is in the image of the trace map from M_H, 𝒪(H ∩ g L g^-1), the claim follows.   
    




      The motivation  behind these  criteria 
is that in practice   the source  functor M_H, is    much better understood (e.g.,  ones    arising from   cycles or   Eisenstein  classes)     than the  target  M_G,  and  the results above   provide   a means for parlaying this  additional   knowledge (and that of the Hecke polynomial)   for  Euler  system  style   relations. In fact in all the cases that  we will consider, M_H,  will      be a space of functions on a suitable topological space that would parametrize classes in the cohomology of Shimura varieties. In <ref> we study the trace  map      for     such spaces  in detail.    
  
 
For the case of cycles  coming from a sub-Shimura datum, the collection of fundamental classes of the  sub-Shimura  variety   constitutes the trivial functor on H (see  <cit.> for a concrete instance).  In this case,  Corollary  <ref>  applies   and  proving norm relations amounts to verifying certain congruence conditions. One may of course use the finer structure of the connected components of a Shimura variety as   prescribed by the reciprocity laws  of  <cit.>.   See   <cit.>  where   a general      formula   for  the    action of Hecke operators  is provided. 
We  however  point out   that  for the case considered in <cit.>,  working with the trivial functor turns out to be necessary as the failure of axiom (SD3)   precludes the possibility of describing the geometric connected components of the source Shimura variety.   
  


 While the if direction is the “useful" part of Theorem <ref>, the 
only if direction provides strong evidence that one does not need to look beyond the test vector specified by twisted restrictions, say, in local zeta integral computations.    
 






 §.§ Handling Torsion
  We now address the equality of norm relation asked for in  Problem   <ref>    without forgoing torsion. We retain the setup at the  start of <ref> and <ref>, in particular  Notation <ref>.  



  Suppose  that        ι_* is Mackey and M_H,  is  -torsion  free. 
If a zeta element exists, any associated class y_L∈ M_G,(L)   
satisfies   
     
ℌ_*(y_K)=pr_L, K, *(y_L).  

 
By  Corollary   <ref>,  we can find x_V_α∈   M_H ,  (  V_α )   satisfying  
_L, K , *   ( y_L    )  =   ∑ _ α∈   A        [ V_α  g_α K   ]_* ( x_V_α  ) and   

    𝔥_α, * ( x_U )  = _V_α ,  H_α ,   *  ( x_V_α ).
 In the notation introduced at the start of the proof of Theorem <ref>,  we see using 
Lemma  
 <ref> 
 that 
    ℌ_*(y_K) 
     = ∑_j ∈ J[U σ_j K]_*(x_U)  = 
      ∑_α∈  A    ∑_j ∈ J_α c_j[U σ_j K]_*(x_U).
   
For j ∈ J, let W_j  :    =h_j  ^-1 U_σ_j h_j. Then for j ∈ J_α,  W_j  = h_j^-1  U h_j∩ g_α  K g_α^-1⊂ H ∩ g_α K g_α^-1 = H_α. 
By Lemma  <ref>  and   <ref>,   we  see that       

    ℌ_*( y_K  )     =∑_α∈  A    ∑_j ∈ J_α    c_j[U_σ_jσ_j K]_*∘pr_U_σ_j, U^*(x_U) 
       =∑_α∈  A    ∑_j ∈ J_α   c_j[W_j g_α    K]_*∘[h_j^-1]_ W_j, U ^*(x_U) 
       =∑_α∈   A    ∑_j ∈ J_α c_j[H_α g_α K]_*∘pr_W_j, H_g_α 
     , *∘[h_j^-1]_W_j, U^*(x_U)

Now note that U ∩   h_j H_g_α    h_j^-1 =     U _ σ_j and   h_j ^  - 1   U   
h_j∩ H_α  =   W_j. Thus   
    pr_W_j, H_α     , *∘[h_j^-1]_W_j, U^*(x_U)=[H_α h_j ^-1   U](x_U) = [U h_j H_α]_*(x_U)
 (see   the diagram below) and so  ∑_j ∈ J_α    c_j·_W_j, H_g_α,  * ∘ [ h_j^-1  ]^*  _    W_j  , U  ( x_U )   =   𝔥_α, *   ( x_U ).             

 

      M_H, 𝒪  (U_σ_j)   [r, "[h_j^-1]^*"]     M_H, 𝒪(W_j)   [dr, "_*"]      M_H, 𝒪(V_α  
  )  [d, "_*"]  

M_H,(U) [ur,"^*"]   [rrr, "   [  U  h_j     H_α  ]_*     "   ,  swap     ]                      M_H,   ( H_α    ) 



Therefore by eq.   (<ref>) 
and Lemma  <ref>,  we   have     

    ℌ_*(y_K) 
        =   ∑_α∈  A     [H_α g_α K ]_* ( 𝔥_α,  *   ( x_U )   )  
       =∑_α∈   A    [H_α g_α K]_*∘pr_V_α, H_α, *(x_V_α)      
       =∑_α∈  A [V_α g_α K]_* (x_V_α ) =pr_L, K, *(y_L)
   
which   finishes  the   proof.    
   


        In the proof above, the relation ℌ_*(y_K)=pr_L, K, *(y_L) in Theorem <ref> can also be derived  under the weaker assumption that  the relations eq. (<ref>) is satisfied  modulo the kernel of ι_*  :   M_H,(H_α)  →  M_   G ,  (g_α K  g_α^-1 ) for each α and   even when  M_H, is not -torsion free.   In particular, this result (which is all we really need for Euler systems) can be  stated   
without ever referencing  zeta elements.    However as noted in the introduction, the notion of zeta elements to connects the approach of <cit.>, <cit.> etc., with ours, and also “explains" the nature of integral test vectors chosen in these works.     
    





















                    In   applications to Shimura varieties, one eventually projects the norm relations to a π_f-isotypical component of the cohomology of the target Shimura variety,  where π_f is (the finite part
of) an irreducible cohomological automorphic representation of the target reductive group,  in order to land in the first Galois cohomology H^1 of a Galois representation ρ_π in the multiplicity space of π. The projection step, to our knowledge,   requires the coefficients to be in a field. Thus the information about torsion is lost anyway i.e.,   one apriori obtains norm relations in the image  of H^1 of a Galois stable lattice T_π⊂ρ_π inside H^1 of the Galois representation ρ_π.       One way to retrieve the torsion in the norm relation after projecting to Galois representation   is to to use Iwasawa theoretic arguments e.g.,  see      <cit.>  or 
   <cit.>.        We will however not address this question here.
 








 §.§ Gluing Norm Relations
    We  now  consider  the  `global'          version   of  Problem  <ref>. Let 
I be an indexing set and 
ι_v : H_v→   G_ v be a collection of  embeddings of  unimodular locally profinite  groups indexed by v ∈ I.   We will consider H_ v as  a subgroup of G_ v via ι_v.   For each  v ∈ I,  let     K_v ⊂  G_ v be a  compact open subgroup and set U_v := H ∩ K_v. Let G, H denote respectively the restricted direct product of G_v, H _v with respect to K_v, U_v over all v.  Let K, U denote respectively the products of K_v, U_v over all v.   For any finite subset  ν⊂ I,   we  define  G_ν = ∏_v ∈ν G_v and G^ν = G/G_ν.   If ν = { v }, we denote G^ν  simply  as     G^v.    We  similarly define the H, K and U versions.     


For all but finitely many v  ∈   I, say we are given   L_v    a normal compact open subgroup of K_v. Let I' ⊂ 
I denote the set of  all   such v  and let 𝒩 denote the set of 
 all     finite subsets of I '.   Let Υ_G be   a     collection of compact  open   subgroups     satisfying (T1)-(T3)  that  contains K and the groups L_vK^v  
 
 for all v ∈ I'. 
Let Υ_H be a collection of  compact open subgroups of H satisfying (T1)-(T3) and  which contains ι^-1  ( Υ_G     ).  Fix   an integral domain whose field of fractions Φ is a -algebra. Let 
    M_H ,  : 𝒫(H, Υ_H)  →-Mod,                 M_G ,  : 𝒫(G,  Υ_G ) →-Mod
 be CoMack functors and ι  _  *  : M_H, →  M_G,  be a Mackey pushforward. Let x_U∈ M_H,  ( U )  be  a class and 
  denote      y_K :  = ι_U,K,*(x_U) its image  in  M_G, (K).  Suppose  we are also given for each v ∈ I '  an element   ℌ_v∈ℋ_( K_v\ G_v  / K_v  ).    Given any ν∈𝒩, any K' ∈Υ_G of the form  K_ν K” with K”⊂   G^ν,   we obtain  by  Lemma  <ref> a well-defined -linear endomorphism 
    ℌ_ν,  *    : M_G, ( K '  ) →  M_G,  ( K ' )
  induced by the tensor product  ℌ_ν    :=  ( K” ) ⊗⊗_v ∈νℌ_v.    
For ν∈𝒩,  denote 
K[ν] :  =  K^ν×∏_v ∈ν L_v. If ν = { v }, we denote this group   simply    by K[v]. Note that K[ν] = K if ν = ∅. For ν ,μ∈𝒩  that   satisfy    ν⊂μ, denote the pushforward  M_G,  (K[μ]) → M_G, (K[ν]) by _μ, ν, *.    
           Construct  classes y_ν∈   M_G, (K[ν] ) for ν∈𝒩 such that y_∅ = y_K and for all μ, ν∈𝒩   satisfying     ν⊂μ, we have ℌ_μ∖ν  ,  *     ( y_ν  )  =  _μ,  ν, *  (  y_μ    ).


 

Our “resolution"  to this problem is by assuming the existence of abstract   zeta elements at each v in  suitable RIC functors  whose restricted tensor product parameterizes classes in M_H,.       Let Υ_G_v denote the  collection  of all compact open subgroups that are obtained as finite intersections of conjugates of K_v and L_v.   By Lemma <ref>, Υ_G_v satisfies (T1)-(T3). Then any compact open subgroup in ∏_v ∈ IΥ_G_v whose component at v equals K_v for all but  finitely many v belongs to ∈Υ_G.    Let Υ_H_v = ι^-1_v  ( Υ_G_v     ) and  let   Υ_H, I⊂∏_v ∈ IΥ_H_v denote the collection of all subgroups whose component group at v is U_v for  all but finitely many v. Then Υ_H , I satisfies (T1)-(T3) and Υ_H, I⊂ι^-1 (  Υ_G   ) ⊂Υ_H.     
   












           Suppose  that there exists a 
 morphism φ : N → M_H,  where N : 𝒫(H, Υ_H,I) →-Mod  is a restricted tensor product ⊗'_v ∈ I N_v of  -torsion free functors N_v  :    𝒫(H_v,   Υ_H_v)   →𝒪-Mod      taken  with respect to a collection {ϕ_U_v∈  N_v( U_v ) }_v ∈ I that satisfies φ ( ⊗_v ∈ Iϕ_U_v )  = x_U. If  a  zeta element ζ_v ∈𝒞( G_v/  K_v   ,     N_v,  Φ   ) exists for (ϕ_U_v ,  ℌ_v ,  L_v) for  every v ∈ I',    then    there exist classes y_ν∈  M_G, (K[ν])  for each ν∈𝒩 such that y_∅ = y_L and     
    ℌ_μ∖ν  ,   * ( y_ν )   =  _μ ,  ν,  * ( y_μ )
 for all ν , μ∈𝒩   satisfying   
     ν⊂μ.        
 


  

For v ∈ I ', denote  A_v : =  H _v\  H_v·  (  ℌ_v    )      /  K_v and for each 
α_v∈   A_v,  
let g_α_v∈ G_v be a representative for the class α_v. Denote  H_α_v : =  H_v∩   g_α_v  K_v  g_α  _v ^  - 1, V_α_v  :  =  H_  v    ∩   g_α _v L _ v   g_α  _v  ^ - 1 and  𝔥_α_v∈𝒞_ ( H_α_v\  H_ v   /  U_v   )  be the  (H_v, g_α_v)-restriction 
ℌ_v with  respect to    
 g_α_v.   
By  Corollary  <ref>,   the existence of ζ_v 
is equivalent to the existence of ϕ_α_v∈ N_v(V_α_v) for all α_v∈ A _v    
such that   
    𝔥_α_v, *  ( ϕ_U_v ) =   _V_α_v, H_α_v  , *   (  ϕ_α_v  )    .
  
Denote by _*    : N → M the    pushforward given by the composition ι_*∘φ.  Then _* is Mackey since ι_* is.  Recall that 𝒩 denotes the set of finite subsets of I '.   For  ν∈𝒩, we      denote  A _ν  :=  ∏_   v    ∈ν     A _v. Given a ν∈𝒩 and α    =  α_ν∈  A _ν, we let  
    α_v  denote  the v-th component of α  for  v ∈ν and set  
    H_α :=  ∏_v ∈ν H_α_v ,    V_α = ∏_v ∈ν V_α_v ,          g_α  :  = ∏_v ∈ν g_α_v  ϕ_α = ⊗_v ∈νϕ_α_v∈⊗ _  v ∈ν N(V_α_v ) .
 For  ν∈𝒩, we  let ϕ_U^ν   denote 
   the restricted tensor product ⊗'_v ∉νϕ_U_v 
and  define  
    y_ν : = ∑    _α∈  A _ ν     [U^νV_αg_α L_ν K^ν ]_* ( ϕ _  U ^ ν⊗ϕ_α    ) ∈ M_G, (K[ν])

i.e.,  y_ν is the  sum of classes  obtained by applying  mixed Hecke  correspondences [U^ν V_α g_α L_ν K^ν ]_* : N(U^ν V_α) → M_G,  (L_νK^ν ) = M_G,  (K[ν])  to    ϕ_U^ν⊗ϕ_α∈  N( U^ν V_α ) over all α∈ A_ν. 

We claim that y_ν for ν∈𝒩    are the desired classes.  It is clear that y_∅ = y_K as φ( ϕ_U ) = x_U.  By Lemma   <ref>,  it suffices to prove the norm relation ℌ_μ∖ν, * (y_ν )   = _μ, ν, * ( y_μ ) for ν⊂μ such that μ∖ν = { v }.   To this end, fix  an  α∈  A _ν for the remainder of this proof and 
 consider the inclusion  Υ_H_v↪Υ_H (of sets) given by  W_v↦  W_v   V_α U^μ and the inclusion Υ_G_v↪Υ_G given by K_v ' ↦ K_v' L_ν K^μ. 
 Let 
 N_H_v, α : 𝒫(H_v, Υ_H_v) →-Mod, M_G_v, ν :  𝒫(G_v, Υ_H_v) →-Mod be  respectively      the functors obtained by fixing levels away from ν  as  specified  by  these  inclusions.      We then have a   Mackey     pushforward  
    _ v , α , *  : N_H_v   ,   α→   M_G_v  ,    ν
   where for  a compatible pair (W_v, K_v')  ∈Υ_H_v×Υ_G_v,  
 the map  N_H_v,  α(W_v ) → M_G_v,  ν( K'_v ) is equal to  map     [U^μW^vV_α g_α L_ν K'_v K^μ]_* :   N(U^μW_vV_α)   →      M_G, (L_ν K_v' K^ν).  
 Given ϕ_ W_v∈  N_v(W_v), we denote by ϕ_W_v, α∈   N_H_v   ,  α ( W_v) the element 
ϕ_U^μ⊗ϕ_W_v⊗ϕ_α∈  N_H_v, α ( W_v ) = N(U^μ W_v V_α). Similarly for any β_v∈ A_v,  we  let     ϕ_β_v, α∈  N_H_v,α(V_v) denote  the  element    ϕ_U^μ⊗ϕ_β_v⊗ϕ_α. Then for any β_v∈ A_v, we have 

    𝔥_β_v,* ( ϕ_U_v, α )     = ϕ_U^μ ⊗ 𝔥_β_v ,*  (   ϕ_U_v )  ⊗ ϕ_α
        = ϕ_U^μ ⊗ _V_β_v, H_β_v ,  *    ( ϕ_β_v )  ⊗ϕ_α
          =  _V_β_v, H_β_v, * ( ϕ_β_v, α )      ∈  N_H_v , α (H_β_v )
    
A zeta element for the triple ( ϕ_U_v, α ,  ℌ_v , L_v ) therefore exists in 𝒞( G_v/K_v , N_H_v, α, Φ ) since the ϕ_β_v, α∈ N_H_v, α (V_β) (for β_v∈ A_v) satisfy  the criteria Theorem <ref>.     By Theorem 
 <ref>, we see that  
    ℌ_v, * ∘_v, α, *  ( ϕ_U_v, α  )  =   ∑_β_v∈  A_v   [V_β_v  g_β_v K_v ]_* ( ϕ_β_v , α )

Therefore   

    _ν, μ, *  (y_μ )           = ∑_α∈ A_ν∑ _ β_v∈ A_v   [U^μ V_β_v V_α   g_β_v   g_α     L_μ K^ν]_* ( ϕ_U^μ⊗ϕ_β_v⊗ϕ_α  )     
         =   ∑ _ α∈   A_ν∑_β_v∈ A_v  [V_β_v g_β_v K_v]_* (  ϕ_β_v , α  )    
         =  ∑ _ α∈  A_νℌ_v∘   i    _v, α, * ( ϕ_U_v, α      ) 
        =  ∑_α∈ A_νℌ_v, * ∘   [U^ν V_α g  _  α  L_ν  K^ν ]_* ( ϕ_U^ν⊗ϕ_α )    =   ℌ_v, *  (  y_ν )
  
which  completes  the   proof.   
    


  The  intended  application 
 to    Shimura varieties we have in mind is where we take I to be the set of all places where all groups at hand are unramified and reserve one element v_bad∈ I for all the bad places lumped together i.e.,  if S is the set of all bad places, G_v_bad = ∏_v ∈  S       G_v, ϕ_v_bad = ϕ_U_S etc.,  and we take   I' = I ∖{  v_bad}.    
    












 §.§ Traces in Schwartz    spaces
  Since the machinery   developed  so far 
only allows  us to recast norm relation problem from a  larger group to the smaller one, it is  useful   to have some class of functors  where identifying the image of the trace map is  a more  straightforward check. For instance when M_H, is the trivial functor, the trace map is multiplication by degree and  Corollary <ref> uses this to give us a congruence  criteria involving certain mixed degrees.     This applies to  pushforwards of fundamental cycles. For Eisenstein classes and cycles constructed from connected components of Shimura varieties, the parameter spaces are certain adelic Schwartz spaces. 
In this subsection, we study  the  image  of  the trace  map for  such  spaces and derive  an  analogous     congruence criteria.    
       


Let H be locally profinite group with identity element e  and  X a locally compact Hausdorff  totally disconnected   space  endowed  with a  continuous  right  H-action X ×   H  → X. By definition,  X carries a  basis of compact   open   neighbourhoods.  
For a ring R,  we denote by 𝒮_R(X)  the  R-module of locally constant  compactly  supported  functions on X    valued in R.   Under the right  translation action on functions, 𝒮_R(X)  becomes   a smooth left representation of H.    In  what  follows, we will frequently use the following fact: the set of all compact open subsets of X is closed under finite unions, finite intersection and relative complements.  Moreover if U ⊂ H is a compact open  subgroup,    then the set of compact open subsets  of  X     that are invariant under U is  such a  collection  as well.          


          Let W  ,  V   ⊂ H be  compact open subgroups with V  ⊂   W.  We say that x ∈ X is (W,V)-smooth if there exist a V-invariant  compact  open   neighbourhood  Z of x  such that Z  γ for γ∈ V \ W  are pairwise  disjoint. 
A W-invariant compact  open   neighbourhood     Y ⊂ X is said to be (W,V)-smooth if Y =  _ γ∈  V  \  W  Z_γ such that Z_ e is  a    V-invariant compact  open  neighbourhood     of X and    Z_γ = Z_  e  γ for all γ∈  V   \    W.   

If  Y =  
_γ∈ V \ W   Z_γ is (W,V)-smooth, the points of Z_γ are (W, γ V γ^-1 )-smooth but not  necessarily  (W,V)-smooth unless V ⊴ W. It is clear that any (W,V)-smooth neighbourhood is also  ( W, γ V  γ^-1 )-smooth for all γ∈  W. Smooth neighbourhoods behave well with respect to     finite unions, finite intersections  and   relative   complements.    


           Suppose that Y, S ⊂ X are compact opens such that  Y is  (W , V)-smooth  and S is W-invariant.    Then Y    -  S and Y ∩ S are    (W,V)-smooth. If S is also  (W,V)-smooth, then so is Y ∪ S.     
    
   Let Y =  _ γ∈   V  \ W  Z_γ where Z_e is a V-invariant compact open  and Z_γ = Z_eγ.   Then Y ∩ S is a W-invariant compact open  neighbourhood, Z_e∩ S is  a  V-invariant compact open  neighbourhood  contained in Y ∩ S and Z_γ∩ S =    (Z _e∩ S ) γ. Thus Y ∩ S =   _γ∈  V    \ W  (Z_e∩ S ) γ  which implies (W,V)-smoothness of Y ∩  S.  Similarly    Y  -  S  =   _ γ∈    V  \ W   (Z_e   -    S ).  If S is also ( W, V )-smooth, then  since   
    Y ∪ S =  (  Y - 
     S ) ⊔ ( S ∩ Y )  ⊔  ( S - Y )

is a disjoint union of (W,V)-smooth neighbourhoods,  Y   ∪    S is (W,V)-smooth as well.      
 




 

                  Suppose that S ⊂ X is a W-invariant   compact  open  subset  that  admits a covering by (W,V)-smooth neighbourhoods    of X.       Then S is    (W,V)-smooth.       
    




 
For all x ∈ S, let Y_x⊂ X denote  a  (W,V)-smooth  neighbourhood  around  x. By Lemma  <ref>, Y_x∩ S  is (W,V )-smooth and we may therefore  assume  that    Y_x⊂  S for all x ∈  S.   Since S is compact  and     S = ⋃ _ x ∈ S   Y_x,  we  have     S  =  ⋃ _  i = 1  ^   n           Y_i where Y_1 , …, Y_n   form  a  finite  subcollection of Y_x. Thus S is a finite union of (W, V)-smooth neighbourhoods and is therefore  itself  (W,V)-smooth by     Lemma <ref>.   
  


Next  we   have the following criteria for  checking (W,V)-smoothness of point. For x ∈ X, let Stab_W(x) denote the stabilizer of x in W.         


     
A point  x is (W,V)-smooth if and only if Stab_W (     x   )    ⊂ V. 
 


 The only if direction is clear, so assume that Stab_W(x) ⊂ V. Let U ⊂ V be a compact open subgroup that is normal in W.  For σ∈ W, let C_σ :  =  x  σ   U    ⊂ X denote the U-orbit of x  σ.    Thus  two such subsets   are      disjoint if they  are distinct.   By continuity of the action of H,  C_σ are compact and therefore closed in X. Since U ⊴    W, we have C_σ =  x U  σ and C_στ =  x σ U  ·τ U. Thus  U  \  W  acts transitively on the orbit space  ( xW )/ U =  { C_σ  |  σ∈ W } via  the right  action     (C_σ  ,   U   τ     )  ↦  C_στ. Let U ^∘  denote   the inverse image in W under W  ↠   U   \    W    of the stabilizer of C_ e under this action.   Clearly Stab_W(x) ⊂   U ^ ∘.     If γ∈  U^∘, then x  γ   = x u for some u ∈ U by definition.  This implies that u γ^-1∈Stab_W(x)  ⊂ V and since U ⊂ V, we have γ∈ V. So U^∘ is a   compact open subgroup of W  such  that   Stab_W(x)  ⊂  U^∘⊂   V  .
It therefore suffices to show that x is (W,U^∘ )-smooth.     


Let γ_1 , …, γ_n∈ W be a set of representatives for U ^ ∘\   W,  δ_1 ,   …,  δ_m∈ U  ^ ∘ be a set of representatives for U   \     U^∘ and  denote  C_i  :     = C_γ_i.  Then   C_i for i = 1 ,…, n  are pairwise disjoint  and   each     C_i is stabilized (as a set) by δ_j,i : = γ_i^-1δ_jγ_i for all j   = 1 ,  … ,  m. For any  compact  open neighbourhood T of x, X' :=    T   W is a  compact open      neighbourhood of   X that contains C_i  for   all i.  Since X' is compact Hausdorff, it is normal and we may therefore choose  compact open   neighbourhoods  S_i  contained in  X' 
such that S_i  contains C_i  and S_1 , …, S_n  are pairwise disjoint.    For each fixed k = 1 , … n, ℓ = 1 , …  ,  m,     let   

    Z _ k  ,   ℓ : = S_k      U  δ_ℓ,k  -   ⋃  _      i ≠  k  ⋃_j = 1  ^m S _ i   U   δ_j,i

where i runs over all integers 1 to n except for  k. Since S_i U  δ_j, i =    S_iδ_j, i  U  by normality of U in W  and {   S_iδ_j ,  i        U     |   j = 1 , …, m , i =  1 , … , n  } is a collection of U-invariant compact open  neighbourhoods,  Z_k , ℓ are U-invariant compact open    neighbourhoods  as  well.  By  construction, Z_k,ℓ intersects Z_k' , ℓ' if and only if k = k  '.  We    claim  that       x   δ_ℓγ_k    =  x γ_kδ_ℓ, k ∈    C_γ_k⊆  S_k U δ_ℓ, k is a member of  Z_k, ℓ. Suppose for the sake of deriving a 
 contradiction that x  δ_ℓγ_k∈ S_i  U  δ_j  , i for  some  i , j with  i ≠ k, so that  
    x   δ_ℓγ_kδ_j,  i  ^-1   U  =  C_eδ_ℓγ_kδ_j, i  ^-1  = C_eγ_kδ_ j , i   ^  - 1        =   C_γ_kδ_j, i ^-1
     intersects S_i. As C_γ_i is the only element in { C_σ  |   σ∈ W  }  contained in S_i,  this   can only happen if C_γ_kδ_j,i ^  - 1    = C_γ_i or equivalently if C_γ_k = C_γ_iδ_j,i. But since δ_j,i stabilizes C_γ_i, this means that C_γ_k = C_γ_i   which in turn  implies  
 i = k,  a contradiction.  Thus x δ_ℓγ_k∈  Z_k, ℓ or equivalently, x ∈ Z_k , ℓγ_k ^-1δ_ℓ  ^   - 1.     Now let  
    Z   :  =   ⋂_ k = 1  ^n⋂_ℓ = 1  ^  n          Z    _  k , ℓγ_k^-1δ_ℓ^-1   .
 Then Z is a U-invariant compact open  neighbourhood of x as each Z_k, ℓ is and U ⊴      W.    Since  Z δ_ℓγ_k⊆  Z_k, ℓ,  Z   δ_ℓγ_k and Z   δ_ℓ'γ_k' are disjoint for any 1 ≤ℓ  '   ,  ℓ  ' 
    ≤   m, 1 ≤  k , k ' ≤ n with k ≠ k'.   If we  now    let  Z^∘  : =  ⋃_ℓ = 1  ^  m   Z   δ_ℓ, then  Z^∘ is U^∘-invariant  compact  open  subset of X  such that Z^∘γ_1  , … ,  Z^∘γ_n are pairwise disjoint. Thus   x is (W,U^∘ )-smooth.     



For each x ∈ X, we let V_x  denote the subgroup of W generated by V and  Stab_W(x).  By Lemma <ref> V_x is the unique smallest    subgroup of W containing V such that x is (W,V_x )-smooth.  Let 𝒰 be the lattice of subgroups of W that contain V. For 𝒯⊂𝒰 a sub-collection, we denote by max 𝒯 the set  of maximal elements of  𝒯 i.e.,   U ∈max 𝒯   if no U' ∈𝒯 properly contains U. We have a filtration   
    𝒰 = 𝒰_0⊋𝒰_1⊋…⊋𝒰_N  =  { V  }
    
defined inductively  as    𝒰       _k+1  :  =   𝒰  _k - max 𝒰      _k for k  = 0, … , N - 1.  We let dep : 𝒰→{ 0 , …, N } be the function U  ↦  k  where k is the largest integer such that U ∈𝒰_k i.e.,      k is the unique integer such  that      U ∈max 𝒰_k.   It  is  clear  that dep is  constant  on  conjugacy   classes   of  subgroups.      We let   
    dep = dep_W,V : X    →{  0 ,   1,  2, …, N  }
     
    x      ↦dep(V_x )
 
and refer to   dep(x) as the depth of x.  We   say that  S ⊂ X has depth k if inf{dep(x)   |    x  ∈  S  } = k.   

           If   S  ⊂  X  has  depth  k,
the set of depth k points in S is closed in S.               
    



  Let T ⊂  S be the  set of depth k points. By assumption, the  depth of any point  in    S - T is at least k + 1.     For x  ∈ S   -      T, choose Y_x a (W, V_x)-smooth neighbourhood of x in X. Then each     y ∈ Y_x  is   (W, γ  V   _  x  γ^-1 ) smooth for some γ∈ W. Thus V_y⊆γ V_xγ^-1 by  Lemma   <ref>  and so    
    dep(y) = dep   (  V_y   ) ≥dep   (  γ V_xγ^-1   )  =  dep  (  V_x 
      )     =  dep(x)   >  k
    for all y ∈ Y_x.    Therefore  Y_x∩ T  =  ∅    which makes Y_x∩ S an open (relative to S) neighbourhood of x contained in S  -    T.  As  x was arbitrary, S   -    T is open in S which makes T closed in S.        
    






If V ⊴  W, then V_x = Stab_W(x)   ·         V and [V_x : V ] =  [ Stab_W(x) : Stab_W(x) ∩ V ]. The next result provides a necessary and sufficient criteria for a given function in 𝒮_R(X) to be the trace of a V-invariant  function in terms of these  indices.      
 
                                      Suppose  that V  ⊴     W,  R  is  an integral  domain and ϕ∈𝒮_R(X) ^  W. 
Then there exists    ψ∈𝒮 _   R   (  X ) ^   V such that ϕ = ∑ _γ∈ W  /  V   γ·ψ if and if  only for all x  ∈ ( ϕ ),    ϕ(x) ∈ [V  _  x    :      V   ]  R.
        


 () Let ψ∈𝒮_R(X)^V be an  element satisfying the trace  condition. For x ∈ X,   let   V_x   be as above, γ_1 , …,   γ_n∈  W be a set of representatives  for  W  / V_x    and      δ_1, …, δ _m∈ V_x be a set of representatives of V_x  / V,  so that γ_iδ_j run over a  set    of    representatives for W / V. As V_x = Stab_W(x) V, we may assume that δ_i (and therefore  δ_i^-1) belong to Stab_W(x).  Since  W / V  is a group, δ_j^-1γ_i ^-1 also   run over    a set of representatives for W / V. Therefore 
  
    ϕ(x)  =  ∑       _γ∈ W / V γ·ψ (x)                =     ∑ _  i = 1  ^  n∑_j =1^mψ (  x  δ_j  ^   - 1γ_i ^-1  )  
         =  ∑ _  i     = 1    ^  n        m   ·ψ ( x    γ_i    ^  -  1         )   ∈ [V_x : V ] R .
  
() Set S    :      = ϕ and N : =  dep  V.  By  definition of 𝒮_R(X),    S is a W-invariant compact  open subset X.  We  inductively define a  sequence  S_0 , … , S_N   of W-invariant  compact open  subsets of S such that   
       

  * S = S_0⊔  S_1⊔…⊔ S_N,

  * all depth k points of S are contained in S_0⊔…⊔ S_k for each 0 ≤ k ≤ N,    

  * each  S_k admits a sub-partition _ U ∈max 𝒰_k   S_U where S_U is a   (W,U)-smooth neighbourhood    on which ϕ is  constant and valued in  [U :V ] R. 
    
We  provide the inductive step for going from k - 1 to k which covers base case as well by taking k = 0, S_-1 = ∅.      So assume that for k ∈{0 , …, N-1 }, the  subsets S_0,  …, S_k-1  have been constructed. 
   Let T_  k be the (possibly empty)  set of   all     depth k points in 
    R_k : =  S - _i = 0  ^  k - 1 S_ i
  where R_k = S if k = 0.  By  construction,  R_k is   a W-invariant compact  open subset of S  and depth of R_k is at least k.    By Lemma  <ref>, T_k⊂ R_k is closed and therefore  compact.   For each x ∈ T_k, let  Y_x be a (W,V_x)-smooth neighbourhood of x. By Lemma <ref>, we may assume Y_x⊂   R_k.   
   As ϕ  is   W-invariant and locally constant, x is contained in a W-invariant  compact open    neighbourhood on which  ϕ is  constant.  By intersecting such a neighbourhood with Y_x  if necessary, may also 
 assume that ϕ is constant on Y_x for each x. Since T_k is  compact  and     covered by Y_x, there exist x_1, …, x_n∈ T_k  such that T_k⊆ Y_x_1∪…∪ Y_x_n. Let 
    S_k : = Y_x_1∪…∪ Y_x_n  .
 
   Clearly S_k is   a  W-invariant compact open subset of R_k   since Y_x are and S_k is disjoint  from   S_1 , …, S_k-1.   By construction, all the depth k points of R_k are in S_k and thus all the depth k points of S are in S_1⊔…⊔ S_k. Let Y_i  :   = Y_x_i  -  (Y_x_i+1∪…∪     Y_x_ n    ). Then Y_i are (W, V_x_i )-smooth by Lemma   <ref>  and    S = Y_1⊔…⊔       Y_n.      As x_i∈ T_k, we have  V_x_i∈max 𝒰_k and by construction, ϕ takes the constant value ϕ(x_i) ∈ [V_x_i : V]  R  on Y_i.  For each U ∈max 𝒰_k, we let S_U      :     =  _  V_x_i = U   Y_i.  Then S_k =  _  U ∈max𝒰_k  S_U  is the desired   sub-partition  
   and the inductive step is complete.    

Now for each U ∈𝒰, let Z_U⊂ S_U be a U-invariant  neighbourhood whose U  \ W  translates  partition S_U.   We define ψ : X → R by 
    ψ   ( x  )  =       [U : V ] ^-1ϕ(x)       if  x ∈ Z_U
    
    0      otherwise
 
Then ψ is well-defined since for all x ∈ S_U, ϕ(x)  =    [U:V] · r for a unique r ∈  R -  { 0 }.     As ψ takes  a     non-zero constant value on Z_U and is zero elsewhere, ψ = _ U ∈𝒰  Z_U. As each  Z_U is V-invariant, ψ is V-invariant. Thus ψ∈𝒮_R(X)^V. Let ϕ' = ∑ _ γ∈ W / V ψ. As S is W-invariant and ψ⊆ S,   ϕ ' ⊂ S as well.  Thus ϕ and ϕ' agree on X - S and we show that they  agree on S as well.   For each x ∈ S, there exists  a unique U ∈𝒰  and a unique γ∈  U \  W (both of which depend on x) such that x γ∈ Z_U.  Let γ_1 , …, γ _  n ∈ W be a set of representatives of U \  W 
 and δ_1, …, δ_m∈  U a set of representatives of V   \ U. Then γδ_jγ _i run over set of representatives for V \ W =   W / V. Since x γ∈ Z_U and Z_U is U-invariant,  x γδ_jγ_i∈  Z_Uγ_i for all i, j.   
 Thus  x γδ_jγ_i∈ Z_U  if and only if γ_i represents the identity class in U \ W.  One then easily sees that 
 
    ϕ '  ( x )  = ∑_ i , j ψ ( x γδ_jγ_i  )           =   ∑_ jψ ( x γδ_j )  = [U : V]  ψ (x) =  ϕ( x)   .

Hence ϕ = ϕ'  and  so   ψ  is  the    desired  element.      
   



  Let ϕ∈𝒮    _R(X)^W and  let x_α∈ X for α∈ I  be a set of representatives for (    ϕ   ) /  W. 
 Then ϕ is the trace of an element 𝒮_R(X) ^V for V ⊴   W if and only if ϕ(x_α) ∈  [ V_x_α  : V ] R for all α∈ I.      

  The only if direction is clear by Proposition  <ref>. The if direction also follows from it since any x ∈ x_α W  is (W, γ V_x_αγ^-1 )-smooth for some γ∈ W,  so  that    
    ϕ(x) =  ϕ(x_α)  ∈     [V_x_α : V] R = [V_x : V] R   .
  
 

We   resume  the setup of <ref> and retain Notation <ref>.  Assume   moreover that M_H, is the functor associated with the smooth H-representation 𝒮_(X). In particular, x_U is a U-invariant Schwartz function ϕ_U : X →.   
         Suppose  that p  ∈ϕ_U is an H-fixed point.  Then a zeta element exists for (ϕ_U , ℌ, L) only if ϕ_U(p) ·(𝔥_α,*) ∈ [H_α : V_α]  for all α∈ H \  H  ·(ℌ) / K.  

 By Theorem <ref>, a zeta element exists if and only if 𝔥_α^t·ϕ_U∈  M_H, ( H_α ) is the trace of an element in M_H,  (V_α ) =  𝒮_(X)^V_α.  By  Theorem   <ref>, this can  happen   only if 
    𝔥_α^t·ϕ_U(p) ∈   [H_α : V_α]    .
 Since p is H-fixed, 𝔥_α ^t·ϕ _U(p) = ϕ _U(p) ·(𝔥_α,*).  
    

 For Eisenstein classes, the local Schwartz functions are  characteristic functions on lattices in  certain  vector spaces and the group H at hand acts  via 
linear transformations. The origin is therefore a fixed point for its action  and  Corollary <ref>  provides a quicker initial check[this proved  particularly helpful in <cit.> where the `convolution step'  was quite involved] for applying the  criteria of Theorem  <ref>. 
Incidentally, this is the  same check as in    Corollary  <ref> which applies  to   fundamental cycles.    
    



 §.§ Miscellaneous results
    
We will study zeta elements for groups G that are product of two groups, one of which is abelian and it would be useful to record some auxiliary results that would be helpful in applying the criteria to such groups.  


Suppose for the  this subsection   only    that  G = G_1× T where T is abelian with a unique maximal compact subgroup C. Suppose also     that K = K_1× C, L = K_1× D where K_1⊂ G_1, D ⊂ C (so that d = [C:D])     and that    
    ℌ 
    =  ∑_   k ∈ I    e_ k   ( K γ_  k ϕ_ k    K    )    ∈𝒞_ ( K \ G / K )
 
where e  _    k ∈,  γ   _  k   ∈ G_1 and  ϕ_ k ∈ T.
Let ι_1 : H → G_1, ν : H →  T  denote   the   compositions   H  
 G → G_1, H → G → T respectively. We suppose that ι_1 is injective, so we may consider H, U as a subgroup of G_1 as well as G.  When we consider H, U as subgroups of G_1, we denote them by  H_1, U_1 respectively.   

                        Suppose that K _1γ_ k  K_1 = _ j ∈ J_ k    U_1σ _  j K _1 where J_k is an indexing set and σ  _  j ∈  G_1. Denote  σ_j,k = σ_jϕ_ k and H_1, σ_j = H_1∩σ_j  K _ 1 σ_j^-1  
Then 
      

  * ℌ   =  ∑_ k ∈ I ∑ _ j ∈ J_k   e_k   (  U   σ_j, k   K    ) 
 

  * [U σ _j ,  k    K ]_*   =      [ U_1σ_j K _1  ]_*,  
    

  * [H ∩σ_j  ,  k   K   σ_j , k       : H  ∩σ_j , k   L σ_j, k  ^-1    ] = [ H_1, σ_j  : H_1 , σ_j∩ν^-1(D)  ]. 
    
 
 Since ν is continuous and C is the unique   maximal compact subgroup of T, the image  under ν  of any compact subgroup of H is contained in C.  For (a),  it suffices to note that 
    K_1γ _k K_1 =  _ j ∈ J_k U _1σ_k K_1  K γ_k ϕ_k  K =   _  j ∈  J _k  U σ_jϕ_k  K

since K  γ  _  k  K   =     K_1γ_k K_1×ϕ_k C and ν(U) ⊂ C.  For (b), note that   H ∩σ_j ,  k  K σ_j, k  ^-1  = H ∩σ_j K σ_j^-1 as T is abelian. Since H  _  1 ∩σ_j K _  1 σ_j^-1 is compact, ν ( H _1∩σ_j  K _1σ_j^-1 ) ⊂ C  and  therefore   
    H ∩σ_j, k  K σ_j, k ^-1 =   ι_1^-1 (  σ _j K_1σ_j^-1 )  ∩ν^-1(C) =  H_1∩σ _j K_1σ_j^-1      .
     Similarly  U ∩σ_jϕ_k K  ( σ_jϕ_k ) ^-1  =U _1∩σ_j K_1σ_j^-1 and (b) follows. The  argument for  (c)  is  similar.    
   




 §.§ A prototypical example
    
In this subsection, we show how the machinery above  may be applied to the case of CM points on modular curves to derive the Hecke-Frobenius valued norm relations at a split prime, which is essentially the n = 2 case of the example studied in <ref>.  See also <cit.> for a more thorough treatment.         



Let E be an imaginary quadratic field. Set = Res_E/_m and 
 = _2. Fix a -basis of E and let ι : 𝐇→ be the resulting embedding.  Let 𝐓 be the torus of norm one elements E and set = ×.  Let  ν : 𝐇→𝐓 denote h ↦ h γ(h)^-1 where γ∈  ( E /      ) is the non-trivial  element and let 
    ι_ν : →  ,       h ↦ ( ι (h ),  ν ( h ) )  .

Then both ι and ι_ν is a morphisms of   Shimura datum.    The embedding ι signifies the construction of CM points on the modular curve.         Under Shimura-Deligne reciprocity law for tori,  the extensions corresponding to 𝐓 by class field theory  are  anticyclotomic 
  over .   


Let G_f, G̃_f, H_f,  T_f denote the _f points of , ,  𝐇,  𝐓    respectively.  
Let Υ_G̃_f denote the collection of all neat compact open subgroups of G̃ _f of the form K × C where K ⊂  G_f, C ⊂ T_f and let  Υ_H_f denote the collection of all neat compact  open subgroups of H_f. These collections satisfy (T1)-(T3) and ι_ν^-1   ( Υ_G̃_f          )         ⊂Υ_H_f.  For any rational prime p,        the mappings      N _  _p    : Υ_H_f               →_p -Mod          M   _  _p     :   Υ_G̃_f       →_p -Mod 

U             ↦H^0_ ( Sh_(U) , _p  )         K̃         ↦H^2_  ( Sh_  (  K̃ )    ,  _p   ( 1  )       )  
 
that send each compact open subgroup to the corresponding arithmetic étale  cohomology  of the corresponding   Shimura    varieties over E  constitute CoMack functors. We note that  if K̃  : = K × C, the Shimura variety Sh_  (K̃ )  is the base change of the modular curve over  of level K to the extension of E determined by the compact open subgroup C.    
The embedding ι :  𝐇→𝐆 induces a  Mackey    pushforward ι : N__p→ M__p of RIC  functors. For each U, N__p(U) is the free _p-module on the class of 1_Sh_H(U) and N__p is the trivial functor on Υ_H_f.   
Let ℓ≠   p be a rational prime that is split in E. Then   
    𝐇__ℓ≃_m×_m ,     𝐓__ℓ≃_m
 where  the  isomorphisms are chosen so that the map ν is identified with the map that sends  (h_1, h_2)  ∈__ℓ map to h_2 / h_1∈𝐓__ℓ. The particular choice is so that the action of uniformizer  ℓ∈_ℓ^×≃(_ℓ) (in the  contravariant convention)  is identified with the action of  geometric Frobenius Frob_λ^-1 on cohomology  where λ  corresponds to the first component in the identification H__ℓ≃_m×_m.

Fix for the rest of this  discussion a  split prime ℓ as above and a compact open subgroup  K̃  =   K  ×  C  ∈Υ_G̃_f   such that K = K^ℓK_ℓ, C = C^ℓ C_ℓ where K_ℓ = _2(_ℓ), C_ℓ = _ℓ^× and K^ℓ, C^ℓ are groups away from ℓ.    Let U : = ι^-1(K) and  similarly 
 write U = U^ℓ U_ℓ where U_ℓ = _ℓ^××_ℓ^×.  Let 

    ℌ _ℓ (X)  :  =    ℓ·ch (  K   ) -         ch ( K   σ _ ℓ  K   )  X   +   ch ( K γ  _  ℓ     K)   X ^  2 ∈ℋ_( K  \    ( _f ) / K )  [  X ]

where σ_ℓ  : =  diag(ℓ , 1 )  and   γ _ℓ :   =  diag   ( ℓ ,  ℓ ). 
Then  
    ℌ̃_ℓ    :    =     ℌ_ℓ  ( Frob_λ )   =   ℌ_ℓ ( ℓ^-1 C ) ∈𝒞_(K̃\G̃ /  K̃_ℓ )
    induces a _p-linear map  ℌ̃_ℓ, *    :    M__p  ( K̃  ) → M__p(K̃). Let D = C^ℓ D_ℓ where D_ℓ = 1 + ℓ_ℓ and let x_U = 1_Sh(U). Set L̃ = K × D.      We ask if there is a zeta element ( x_U ,  ℌ̃_ℓ ,  L̃ ).  Recall that such an element  would solve the corresponding question posed in   <ref>. It   is  also clear that this checking can be done locally at the prime ℓ and that via Theorem 
  <ref>, one can produce a compatible system of such relations for such ℓ. 


The local embedding of H_ℓ↪G̃ _ℓ is not the  diagonal one on the _2 (_ℓ ) copy. We may however conjugate this embedding by an appropriate element of K_ℓ to study the zeta element problem and conjugate everything back at the end by the inverse of the said element[See <cit.> for details.]. So say that H_ℓ↪G̃_ℓ is the diagonal embedding where the first component of H_ℓ corresponds to the top the left matrix entry in _2(_ℓ).    Define the following elements of G_ℓ: 
    σ_1 = [ ℓ  ;   1 ],    σ_2 =  [ ℓ 1;   1 ],     σ_3 =  [ 1  ;   ℓ ]    ,     σ_4     =     [ ℓ  ;   ℓ ]   ,        τ =  [    1 ℓ^-1;         1 ]

  and set σ̃_̃ĩ =  ( σ_j,  (  σ_j^-1) ).  Then 
  
    ℌ̃_ℓ =   ℓ·(U_ℓK̃) -   (  (U  σ̃_1K̃) +   (U_ℓσ̃_2K̃_ℓ)  + (U_ℓσ̃_3K̃_ℓ)  )  +   (  U_ℓσ̃_4K̃_ℓ)   .

It is then clear that 
 
    g_0 : =  (1,1)  ,        g_1  :  =  ( τ , 1 )  ,           g_2 : = ( 1,   ℓ  ^ -2 )

 form a complete system of  representatives for H_ℓ\  H_ℓ· ( ℌ̃ )  /  K̃_ℓ. 
 For i = 0, 1, 2, let H_ℓ, i  = H_ℓ∩ g_iK̃_ℓ  g_i^-1 and 𝔥_ℓ, i∈𝒞_(U_ℓ\  H_ℓ / H_ℓ, i )   
 denote the (H, g_i)-restriction of H̃_ℓ. Then  
 
    𝔥_ℓ,0     = ℓ· ( U_ℓ  )  -   (U_ℓ  (ℓ, 1)  U_ℓ )  
    𝔥_ℓ,1    =     ( U_ℓ  (ℓ, 1) H_ℓ, 1   )   
    𝔥_ℓ, 2      =  (U_ℓ (1,ℓ) U_ℓ ) -  ( U_ℓ (ℓ , ℓ )  U_ℓ )
 
from which it is easily seen that 
    (𝔥_ℓ,0,*) =  ℓ - 1 ,     (𝔥_ℓ,1,*) = 1,     (𝔥_ℓ,2,*) = 0 .
 
Finally, let  d_i : =   [ H_ℓ, i  : H_ℓ∩ g_iL̃ _ℓ  g_i^-1 ]. Then  d_0 = d_2 =  ℓ  - 1, d_1 = 1. 
Since 
    ( 𝔥_ℓ, i,*) ∈ d_i_p
 for i = 0 , 1, 2, Corollary <ref>  implies that a zeta element exists for (1_Sh_U , ℌ̃_ℓ, L̃).    
 Note that our zeta element is   supported on  g_0K ∪  g_1K, even  though H_ℓ\  H_ℓ· ( ℌ̃ )  /  K̃_ℓ has three elements.  See also Remark  <ref> for a similar observation.      
 




§ HECKE   POLYNOMIALS





 
In this section, we describe the Hecke algebra valued polynomials associated   with     representations     of the Langlands dual of a reductive group and  record  some  techniques  that can be used to compute them. On the way, we fix notations and terminology that will be  used in  carrying out the computations in Part II of this article.      
   
                   Throughout this  section,  we  let  F  denote  a local field of characteristic zero, 𝒪_F its ring of integers, ϖ a uniformizer, =𝒪_F / ϖ  its  residue field,  q=|| the cardinality of  and ord :  F →∪{∞} the additive valuation assigning 1 to ϖ. We pick once and for all    [ ]   ⊂_F  a fixed  choice  of    representatives for .  We let  F̅ denote  an  algebraic  closure  of   F  and let F^unr⊂F̅ denote the maximal unramified subextension.   For M a free abelian group of finite rank, we  will often denote by M_   the   -vector space M ⊗_.       
   








 §.§ Root data
       


Let 𝐆 be an unramified 
reductive group over F. This  means that F is quasi-split over F and split over a finite unramified extension of F.  Let  𝐀  be  a maximal F-split torus in , 𝐏⊃𝐀  a F-Borel subgroup and  𝐍 the unipotent radical of 𝐏. Let  𝐌:=𝐙(𝐀)   be      the centralizer of 𝐀 which is a maximal F-torus in 𝐆.   We will denote by  G,  A,  P, M,  N the corresponding groups of  F-points of 𝐆, 𝐀, 𝐏, 𝐌, 𝐍 respectively.    Let    X^*(𝐌) (resp., X_*(𝐌)) denote the  group of  characters (resp.,  cocharacters) of 𝐌 and let  

    ⟨-,-⟩: X_*(𝐌) × X^*(𝐌) →ℤ
 
denote the natural  integral   pairing. The natural extension of (<ref>) to X_*(𝐌)_× X^*(𝐌)_→ is also denoted  as  ⟨ - , - ⟩.     


Let 
 Φ_F̅⊂ X^*(𝐌)  denote  the set of absolute    roots        of  𝐆 with respect to 𝐌,   Φ^+_F̅⊂Φ _ F̅ the set of positive roots associated with 𝐏 and Δ_F̅ a base for Φ_F̅.      For α∈Φ _ F̅, we denote by α^∨∈ X_*   (  𝐌 ) the corresponding coroot and denote the set of coroots by Φ_F̅ ^ ∨.    Since  Φ_F̅ is reduced, Δ_F̅^∨ = {α^∨  |  α∈Δ_F̅} is a  base for  the positive  coroots in  Φ_F̅^∨.  We  let    𝐖_𝐌 =  𝐍 _(𝐌) / 𝐌 denote the absolute Weyl group  scheme of 𝐆 and set W_M  :    = 𝐖_𝐌(F̅ ).
Then left action W_M on X^*(𝐌), X_*(𝐌) induced by conjugation action  on 𝐌_F̅ identifies it with the Weyl group of the   (absolute)          root datum  ( X^*(𝐌) ,   Φ_F̅ ,  X_*( 𝐌) ,   Φ_F̅^∨ ).       Thus for α∈Φ_F̅, there is  reflection element s_α =  s_α^∨∈ W _   M that acts on  λ∈ X_*( 𝐌 ) and χ∈ X^*(𝐌) via  

    λ↦λ    - ⟨λ  ,   α⟩α  ^ ∨      χ↦χ - ⟨α^∨, χ⟩α
 
The pair    ( W_M , {s_α} _α∈Δ_F̅  ) is a Coxeter system. We let  ℓ _ F̅    : W   _  M →  the   corresponding length function.










We will also need to work with the relative root datum of .   Let X^*(𝐀), X_*(𝐀) denote  respectively    the set of characters and 
 cocharacters  of 𝐀.   As 𝐀 is split, all characters and cocharacters are defined over F.   
Let 
    res : X^* ( 𝐌  ) ↠  X ^  *  (  𝐀 )      ,      cores : X_*  (  𝐀 )   ↪ X_* ( 𝐌 )
   denote respectively the natural injection and  surjection induced by  𝐀↪𝐌. Let Γ  = (F^unr / F )  ≃ denote the unramified Galois group of F. Then Γ acts on X^*(𝐌)  via   (γ,χ)  ↦γχ(   γ^-1   x  )  where  γ∈Γ, χ∈ X^*(𝐌) and x ∈𝐌( F^unr ).   Similarly      Γ acts on X_*(𝐌) 
and the pairing  (<ref>)     is Γ-invariant under  these   actions.         
Since 𝐌 is defined over F,  the action of Γ preserves Φ_F̅, Φ_F̅^∨ (as sets).  
  Since 𝐏 is  defined over F,   Γ   also  preserves Δ_F̅, Δ_F̅ 
  ^ ∨   and the  action of Γ  on  these  bases is via  diagram   automorphisms.     We have  
    X^*(𝐌) _Γ, freeres≃  X^* ( 𝐀 ) ,             X_*(𝐀)   cores≃      X_*(𝐌)^Γ
   where X_*(𝐌)_Γ, free denotes the quotient of the group of coinvariants by torsion.  
The  pairing     
    ⟨-,-⟩: X_*(𝐀) × X^*(𝐀) →ℤ
  
is  compatible with  (<ref>) i.e.,  if λ : _m→𝐀, χ : 𝐌→_m are  homormophisms defined  over F̅, then ⟨cores λ , χ⟩ = ⟨λ, res χ⟩.   
Let   𝐖 _𝐀 : = 𝐍 _(𝐀)/𝐌      denote the Weyl group   scheme  of  with respect to 𝐀. 
Then 𝐖 _𝐀 is a constant group scheme over F and 
    𝐖 _𝐀(F) = 𝐍  _(𝐀)(F)/ 𝐌(F)  =  N_G(A)/M
 by <cit.>.    Using quasi-splitness of ,  it can also be shown that  𝐖 _𝐀(F) =   𝐖    _𝐌(F )   (<cit.>)  and that 𝐖 _𝐌(F) =   𝐍 _(𝐌)(F)/ 𝐌 (F) (<cit.>).   In   particular      𝐖_𝐀(F) is the subgroup of Γ-invariant elements in W_M.       We call W := N_G(A)/M the relative Weyl group   of . It is clear that res and cores are  equivariant under the action of W.  

 
Let Φ_F⊂  X_*(𝐀) denote the set of restrictions 
of elements of 
Φ_F̅ to 𝐀.       The elements of Φ_F (𝐀) are called the relative roots 
of 𝐆 with respect to 𝐀.  We denote by Q(Φ_F) the -span of Φ_F in X_*(𝐀).  Then Φ_F forms a (possibly non-reduced) root system  in    Q(Φ_F)_.    Since  is quasi-split, Φ_F̅ does not intersect the kernel of  restriction map.  The set of elements of Φ_F̅ that restrict to the same element in Φ_F form a single Γ-orbit.    The restrictions obtained  from the Γ-orbits of   Δ_F̅   constitute a base Δ_F for Φ_F (<cit.>)  and we denote by Φ_F^+ the corresponding positive root system.   The natural action of W on X^*(𝐀)  identifies it with the Weyl group of the root system of relative roots.  To each root α∈Φ_F, there is by definition an element α^∨ in the  vector space dual  of Q(Φ_F)_. The totality Φ_F^∨ of these elements α^∨ naturally forms a root system (<cit.>).  We refer to Φ_F^∨ as the set of  relative coroots of .   The set {α ^ ∨  |  α∈Φ_F^+} is then a system of positive (co)roots for Φ_F^∨. The subset Δ_F^∨ = {φ(α)   |  α∈Δ_F} where φ(α) = α^∨ if 2 α∉Φ_F and 1/2α^∨ if 2α∈Φ_F  is  a  base for the positive relative coroots  (<cit.>). By  <cit.>, Φ_F̅^∨   embeds naturally into X_*(𝐀). The quadruplet (X^*(𝐀),   Φ_F,   X_*(𝐀 )  , Φ_F^∨ )    thus      constitutes  a root datum  and   will be  referred to as the relative root datum of .     See    also   <cit.>.   



 §.§ Orderings
         In this subsection, we  work with    an abstract  root datum first  and then specialize the  notations to the situation of the previous subsection.  This is done to  address  the absolute and relative cases simultaneously.  The notations for abstract datum will also be used in <ref>.         

Let Ψ = (X, Φ , X ^∨ , Φ^∨ ) be a root datum. The perfect pairing X^∨× X  → given as part of this datum  will be denoted by ⟨ - , - ⟩. Given α∈Φ, β∈Φ^∨, we denote by α^∨∈Φ ^∨, β   ^ ∨∈Φ the associated elements  under the bijection ΦΦ^∨ given as part of    Ψ. We let W_Ψ denote the Weyl group of Ψ. If α is in Φ or Φ^∨, we denote by s_α∈ W_Ψ the corresponding reflection.


Let Q be the  span of Φ in X, Q^∨ the span of Φ^∨ in X^∨,  X_0 the subgroup of X orthogonal to Φ^∨ and P ⊂  Q_ =  Q ⊗_ the -dual of Q^∨.  Then Q ⊂ P are lattices in Q_.   We define X_0^∨, P^∨ in an analogous fashion. We refer to Q  (resp. P, Q^∨, P^∨) as the root (resp. weight, coroot, coweight) lattice.  The groups P / Q, P^∨/ Q^∨ are in duality and finite. It is clear that the  action of W_Ψ preserves Q, P, Q ^∨, P^∨. If χ∈ X_0,  χ -  s_αχ  = ⟨α^∨ , χ⟩α  =    0 for all α∈Φ and thus W_Ψ acts trivially on X_0. Similarly it acts trivially on X_0^∨.    
By <cit.>,  the subgroup Q + X_0 of X has finite index in X and X_0∩ Q is trivial. Thus each χ∈ X can be written uniquely as χ_0 +  χ_1 for χ_0∈ X_0,, χ_1∈ Q_. We refer to χ_0 as the central component of χ.     As ⟨λ   , χ_1⟩ = ⟨λ  ,  χ⟩ for all λ∈ Q^∨ and ⟨λ , χ⟩∈ as χ∈ X,     we see that χ_1∈ P for every χ∈ X.    There  is  thus  a  well-defined   X → P and its kernel is easily seen to be X_0.   We call the map X → P  the reduction modulo X_0 and χ_1 the reduction of χ modulo X_0.  We similarly define these notions for X^∨.   
 It is however  not true in general that  X ⊂ X_0 + P e.g.,  consider the root datum of    _2.   
   
Let Δ⊂Φ  be a  base for Φ giving a positive system Φ^+ for Φ,  Δ^∨ a base for the corresponding positive system for Φ^∨, S the set of reflections associated to Δ^∨ and ℓ : W   _ Ψ→ the resulting length function.  We say that λ∈ X^∨ is dominant (resp.,  antidominant) if for all  α∈Δ, we have  ⟨λ ,  α⟩≥ 0 (resp.,    ⟨λ   ,    α⟩≤ 0) and we  denote the set of such λ by (X^∨)^+ (resp.,  (X^∨)^-). It is clear that λ∈ (X^∨)^+ if and only if ⟨λ , β^∨⟩≥ 0 for  all   β∈Δ^∨ (since any element of Δ can be written as β^∨ or β^∨ /  2   for some   β∈Δ^∨).   We similarly define  dominant elements in P^∨ and denote their collection by (P^∨)^+. Then λ∈ X^∨ is dominant if and only if its image λ̅∈  P^∨ under reduction modulo X_0^∨   is  dominant.      

There exists a partial ordering  ≽   on X^∨ which also  depends on the choice of basis Δ^∨.  It is defined   by declaring λ≽μ  for    λ , μ∈   X^∨  if   
    λ - μ = ∑     _ β∈Δ  ^ ∨ n_ββ
 for some  non-negative  integers n_β∈. In particular, λ and μ are required to have  the  same central   component.    We say that λ is positive with respect to ≽ if λ≽ 0 and negative if λ≼  0.   We similarly define the   ordering  ≽ 
for P^∨. It is easily seen that λ≽μ 
for λ , μ∈  X^∨     iff λ, μ have the same central component and λ̅≽μ̅ 
where λ̅, μ̅∈ P^∨ denote  respectively  the reductions of λ , μ.   
         Let w ∈ W _  Ψ, β∈Δ^∨ be such   that    ℓ(w) = ℓ( w s_β) + 1. Then w β is negative.         
 
  Let V = Q^∨⊗. Then Φ ^ ∨ embeds in V and (V, Φ^∨) is a root system. Let Φ '  ⊂Φ the set of all indivisible roots. Then (V, Φ') is a reduced root system with the same Weyl group W_Ψ  and Δ^∨⊂Φ' is a base for Φ'. The result then  follows  by   <cit.>. 
    
 In general, a dominant λ∈   X^∨  need not be   positive (consider λ∈ X_0   ^ ∨) and a positive λ  need not be dominant (cf. the `dangerous bend' in <cit.>).  We however have the following result. 




    λ in X^∨ or P^∨  is dominant if and only if for all w ∈ W_Ψ, λ≽  
w λ. 
 


This is essentially  <cit.> where it is proved in the setting of root systems and where the ordering ≻ is defined by taking positive real coefficients. 
We   provide the necessary modifications.    Since both the dominance relation and ≽ on X ^∨ are compatible modulo X_0 ^  ∨
    and since the action of W_Ψ on X^∨ preserves    central  components, 
  the claim for X^∨ follows from the  corresponding   claim for P^∨. So let λ∈ P^∨.  Since  λ -  s_βλ =  ⟨λ , β^∨⟩β for any β∈Δ^∨ (see eq. (<ref>)),   we see that λ is dominant if and only if λ≽ 
  s_βλ for all β∈Δ^∨. So it suffices to show that λ≽    w λ for all w ∈ S implies  the same for all w ∈ W. This is   easily  proved by induction on ℓ(w). Write w = w' s_β where β∈Δ ^∨ and ℓ(w) = ℓ(w') + 1.  Then 
    λ 
     - wλ = λ - w'   λ  + w'( λ  - s_βλ) .
       Now λ - w' λ is positive by induction hypothesis. On the other hand, w'(λ - s_βλ   )    = w (s_βλ - λ) = - ⟨λ,  β ^∨⟩ wβ. Since -wβ∈ Q^∨ is positive  by 
    Lemma  <ref> and ⟨λ , β ^∨⟩∈_≥ 0 since λ≽ s_βλ,  we see from (<ref>) that λ≽   
 w λ. This completes the induction  step.     



 


We now specialize  back   to the notation of <ref>. If λ, μ∈ X_*( 𝐀), we write λ≽μ to denote the ordering with respect to the relative root datum.  If λ, μ∈ X_*(𝐌), we write λ≽_Mμ to emphasize that the ordering is with respect to the absolute root datum.      
The set of dominant relative (resp., absolute) cocharacters  is denoted X_*(𝐀) ^+ (resp.,  X_*(𝐌)^+).    Since res(Δ_F̅ ) = Δ_F, cores induces an inclusion   X_*(𝐀)^+↪        X_*(𝐌)^+. We denote by X_*(𝐀)_0, X_*(𝐌)_0 the groups orthogonal to Δ_F, Δ_F̅ respectively. Then X_*(𝐀)_0 = X_*(𝐌)_0 ^  Γ.     

Recall that we denote by W the relative Weyl group for . Let S := { s_α  |  α∈Δ_F} be the set of simple reflections and ℓ   =  ℓ_F : W → the  resulting  length function.  The longest Weyl element w_∘∈ W is defined to be the unique element which attains the maximum length in W. Then w_∘ is also maximal under Bruhat ordering and is the  unique element of W satisfying  w_∘·Δ_F = - Δ_F (as a set).  We have w_∘^2 = id_W.     For each λ∈    X_*(   𝐀 ), we define  λ ^ opp := w_∘λ. Then for λ∈  X_*(𝐀) ^ +, λ^opp  is the unique element in the Weyl  orbit of λ  that  lies  in    X_*(𝐀)^-. Moreover 

    λ≽μ⟺ - λ ^ opp≽ - μ ^ opp
  
for  any   λ , μ∈ X_*(𝐀 )        since - w_∘ ( λ - μ )  ≽ 0.
We will say that w_∘ =  - 1 as an element of W if w_∘ (α) = - α for all α∈Δ_F.    We can similarly define λ^opp for  any  λ∈ X_*(𝐌).  This is compatible with cores by the following.     
              w_∘  is  also the longest element in W_M.  

 Since w_∘∈  W   =    (W_M)^Γ, the action of w_∘  on Φ_F̅ is Γ-equivariant. In particular, w_∘ preserves Γ-orbits. Since restriction res: Φ_F̅→Φ_F  is W-equivariant and sends positive (resp., negative) absolute roots to positive (resp.,  negative) relative roots, we see that w_∘·Δ_F̅ =  - Δ_F̅.           
 



                 If w_∘ = - 1 as an element of W, then λ + λ^opp∈ X_*(𝐀)_0 for any λ∈ X_*(𝐀).  Moreover if λ≽μ for some μ∈ X_*(𝐀), then λ + λ^opp = μ + μ^opp.      
 
 The  first claim  follows since ⟨λ + λ^opp , α⟩ =  ⟨λ , α + w_∘α⟩ for any α∈Δ_F.   The second claim follows since λ - μ is a positive integral sum of positive coroots and applying - w_∘ acts as  identity on this sum.    
    




 §.§ Iwahori Weyl group
           
From now on, we denote X_*(𝐀) by Λ.      We fix  throughout a smooth reductive group scheme 𝒢 over   𝒪_F such that 𝐆 equals the  generic fiber 𝒢_F of 𝒢.       
Then K :=   (_F) is a hyperspecial maximal compact subgroup of G = 𝒢(F). Let A^∘ :=  A ∩ K, M^∘ :  = M ∩ K. As  is unramified, A^∘, M^∘ are the unique maximal compact open subgroups of A, M respectively. 
In  particular, these do not depend on 𝒢. Moreover W is identified with (K ∩ N_G(A)  )   /M^∘.  We have isomorphisms Λ∼→ A / A^∘∼→ M / M^∘    induced respectively   by λ↦ϖ^λ A^∘, A ↪ M (see <cit.>). We denote by 

    v : A / A^∘→Λ
  the inverse of the  negative 
   isomorphism Λ→ A/ A^∘, λ↦ϖ^-λ A^∘.   
The quotient W_I : =   N_G(A) /  M ^∘ is called the  Iwahori Weyl group of G.  It  naturally   isomorphic  to the semi-direct products M/M^∘⋊  W ≃ A/A^∘⋊ W   (<cit.>)   and we identify W_I with these groups.   The   mapping   (<ref>) induces a further 
isomorphism v :  W_I  =    A/A^∘⋊ W  Λ⋊ W where  ϖ^λ A^∘∈ W_I for λ∈Λ  is     identified with  (-λ, 1).

Let Q ^ ∨ _   F   =  Q(Φ_F^∨ ) denote the relative coroot  lattice.   The subgroup W_aff   : =     Q_  F  ^  ∨⋊ W of Λ⋊  W is   called the     (relative)  affine Weyl group.   
The group W_aff acts on the vector 
space Q^∨_F⊗ by translations and it is customary to denote the element (λ,1) ∈ W_aff by t_λ or t(λ). Similarly when the coroot lattice Q^∨_F is viewed as a subgroup of W_aff, it is written as  t ( Q ^∨ _F  ). More generally, we denote the element ( λ , 1 ) ∈Λ⋊ W  by  t(λ) and consider it as a translation of Λ⊗.   If Φ _  F is irreducible, α _ 0 ∈Φ_F  is 
 the highest root and s_α_0∈ W denotes   the reflection associated with α_0, the group   W_aff is   a  Coxeter group with  generators  S_aff : = S ⊔{ t_α_0 ^ ∨   s_α_0}.    In general,   W_aff is    a Coxeter group whose set of generators S_aff is  obtained by extending the set S by the reflections associated to the simple affine  root of  each irreducible component of Φ_F.   In particular, its rank (as a Coxeter group) is the number of irreducible components of Φ_F added to the  rank of W.  
We denote by ℓ : W_aff→ the extension of ℓ : W → and by ≥  the strong Bruhat order   on   W _ aff     induced by the set S_aff.    Via the isomorphism W _IΛ⋊  W, we  identify W_aff as a subgroup of W_I.    The  quotient Ω :=  W_I / W _aff acts  on    W_aff by automorphisms (of Coxeter groups) and one has an  isomorphism    W_I≃  W_aff⋊Ω. One extends the  length function to a function  
    ℓ :  W_I→
  by declaring the length of elements of Ω to be 0.  Similarly,  the    strong      Bruhat ordering  on W_aff  is  extended    to W_I by declaring w ρ≥ w' ρ' for w , w ' ∈  W_aff,  ρ  ,  ρ '  ∈Ω if w ≥ w' and ρ = ρ'. Each W \ W_I / W    
 has a unique minimal length representative in W_I via which we can define a partial ordering on the double cosets.     Under the identification Λ^+≃ W \ W_I / W, the ordering ≽  restricted  to  Λ^+ is identified with the ordering on  representatives in W \ W _I /  W. See <cit.>.       


   See  <cit.> and <cit.> for the role of buildings in  defining  these  groups.
Buildings will be briefly   used in <ref>.      
     


 §.§ The Satake transform
       

Fix a Haar measure μ_G on G such that μ_G(K)=1. For a ring R, let   ℋ_R(K \ G / K) be the Hecke algebra of level K with coefficients in R (Definition <ref>) and R⟨ G / K⟩ be the  set of finite R-linear combinations on cosets in G / K.  For σ∈ G, we denote by (K σ  K ) ∈ℋ_R( K  \  G  / K ) the characteristic function of K σ K which we will occasionally also write simply as (K σ K).        For λ∈Λ, denote by  e^λ the element corresponding to λ in the group algebra ℤ[Λ] and e^Wλ the  (formal)  sum ∑_μ∈ W λ  e^μ.  This allows one to convert from additive to  multiplicative notation for  cocharacters.    The half sum of positive roots   δ : =  1/2∑    _α∈Φ_F̅^+α is an element of P (Φ_F̅)    by  <cit.>.                      For λ∈Λ  = X_*(𝐀), let  ⟨λ, δ⟩ denote the  quantity ⟨cores(λ) ,  δ⟩ =  ⟨λ, res(δ ) ⟩.  



Let  ℛ=ℛ_q denote the ring ℤ[q^±1/2] ⊂ where q^1/2∈ denotes a root of x^2-q and q^-1/2 denotes its inverse.    Denote by p: G / K → K \ G / K the natural map and  p^*: ℋ_ℛ(K \ G / K) →ℛ⟨ G / K⟩ the induced map that sends the characteristic function of K σ K to the formal sum 
of left cosets γ K contained in K σ K.     Let ℐ: ℛ⟨ G / K⟩→ℛ[Λ]  denote the ℛ-linear  map defined by (   ϖ ^ λ n  K  )     ↦    q^-⟨λ,  δ⟩ e^λ for λ∈Λ, n ∈ N. 
This is well defined  by <cit.> (since   M K /  K  ≃ M / M^∘≃Λ). The  composition 
    𝒮 :    ℋ_ℛ(K \ G / K)  ℛ⟨   G / K  ⟩ℛ [Λ]
  
is then  a homomorphism of ℛ-algebras known as the  Satake transform. Its image lies in the Weyl invariants ℛ[Λ]^W.  By 
<cit.> or <cit.>), the induced map 𝒮_ over  is an isomorphism onto [ Λ ] ^W.        We note that {   (  K ϖ ^λ K )   |  λ∈Λ^+} is a basis for ℋ_ℛ(K \ G / K ) by Cartan decomposition. We are therefore interested in the Satake transform of such   functions.       For    λ∈Λ^+,  write   

    𝒮(K ϖ^λ K ) 
        = ∑_μ∈Λ q^- ⟨μ, δ⟩  a_λ(μ)  e^μ
 
where a_λ(μ) ∈_≥ 0.  By  definition,   a_λ ( μ ) is equal to the number of distinct left cosets ϖ^μ n K for n ∈ N such that ϖ ^ μ n K  ⊂  K ϖ ^ λ K.  The W-invariance of 𝒮 implies that q ^  - ⟨μ_1  , δ⟩  a_λ(μ)  =  q ^  -  ⟨μ_2 , δ⟩    a_λ(μ_2  ) 
for all  μ_1, μ_2∈Λ such that W μ_1 = W μ _    2.  Let ≽ denote the same partial ordering in <ref>.  


                For λ , μ∈Λ^+, a_λ(μ ) ≠ 0 only if λ≽μ. Moreover, a_λ(λ^opp) = 1. 
     
 Set  κ = λ ^  opp and ν := μ  ^  opp.  Then  - κ , - ν∈Λ^+.   Since the image of 𝒮 is W-invariant, a_λ ( μ ) ≠ 0 if and only if a_λ(ν) ≠ 0. By definition, this is  equivalent to  ϖ^ν N K  ∩ K ϖ^λ K ≠∅. Now ϖ^ν N =   N ϖ ^  ν as A normalizes N and Kϖ^λ K = K ϖ^κ  K as K ∩ N_G(A) surjects onto W.  Thus 
    a_λ(μ) ≠ 0    ⟺      K ϖ^κ K    ∩        N ϖ^ν K    ≠∅ .
 By <cit.> and the identification of ≽ on Λ^+ with the  Bruhat  ordering on W \  W_I  / W,  
we  get that     K ϖ^κ K  ∩   N ϖ ^ν K  ≠∅  - κ≽ - ν[the negative sign arising from  the normalization (<ref>)].      
But the last condition is  the same as λ≽μ  by    (<ref>). This   establishes the first part.  By <cit.>, K ϖ^κ K   ∩    N ϖ ^ κ K    =  ϖ^κ K i.e.,    the only coset of the form ϖ^κ     n K where n ∈ N such that ϖ^κ  n  K ⊂ K ϖ^λ K is ϖ^κ K.  The second claim  follows.     
   



   
 A weaker version  of above 
 appears in <cit.>. 
   See also <cit.>.   
    


      For λ∈Λ^+, 𝒮 ( K ϖ ^ λ K ) - q ^ ⟨λ,  δ⟩ e^W λ lies   in the  ℛ-span of { e ^  W μ  |  μ∈Λ^+ ,  μλ}.
 


 
Since w_∘δ = -  δ by Lemma   <ref>,  we see that 
   ⟨λ^opp ,  δ⟩ = ⟨λ ,  w_∘δ⟩ = - ⟨λ  ,   δ⟩. 
The second part of 
Proposition  <ref> therefore implies that 
    q^- ⟨λ^opp ,    δ⟩  a_λ(λ^opp) =  q ^ ⟨λ,  δ⟩ .
 Thus the coefficient of e^Wλ in 𝒮(K ϖ^λ K ) is q^⟨λ  , δ⟩. The claim now follows by the first part of  <ref>.     
    


 The Satake transform induces an isomorphism ℋ_ℛ (K \ G / K ) ≃ℛ  [Λ]^W of   ℛ-algebras.  
   

 Fix λ∈Λ^+. We wish to show that e^W λ lies in the image of 𝒮. Let  𝒰_0   = {μ∈Λ^+  |  μ≼Λ} and inductively  define 𝒰_k as the set 𝒰_k∖max 𝒰_k for k ≥ 1. It is clear that 𝒰_0 and hence each 𝒰_k is finite.  
By Corollary  <ref>, f_1 : =  𝒮 ( q ^- ⟨λ, δ⟩  ( K ϖ ^λ K  )     )  - e^Wλ∈ℛ [  Λ   ]  ^  W equals a sum ∑ c_λ(μ   )    e^W μ where μ runs over the set 𝒰_1   =  {μ∈Λ^+  |  μλ}  and c_λ(μ)  ∈ℛ. 
By  Corollary <ref> again,  
    f_2 :=  𝒮 (    q ^  - ⟨λ , δ⟩ ( K ϖ^λ K )  -  ∑       _μ∈max𝒰_1 q ^  -  ⟨μ   ,  δ⟩ c_λ ( μ )  ( K ϖ ^ μ K )      ) - e^W λ

is a linear combination of e ^  W μ∈ℛ[Λ]^W for  μ∈𝒰_2.     Continuing this process, we obtain a sequence of elements f_k∈ℛ[Λ]^W for k ≥ 1 that are supported on 𝒰_k and such that e^W λ + f_k lies in the image of 𝒮.   Since 𝒰_k are eventually empty, f_k are eventually zero and  we obtain the  desired   claim.    


        Suppose w_∘ = -1 as an element of W. Then the transposition  operation    ℋ_R(K  \ G / K ) corresponds under Satake transform to the negation of cochracters  on  ℛ[Λ]   ^   W.    
     
  For λ∈Λ^+, (K ϖ^λ K ) ^t   =  ( K ϖ^κ    K )  where κ := -λ^opp∈Λ^+.  By    Lemma <ref>,   κ = λ +  λ_0 for some λ_0∈ X_*(𝐀)_0. Since ϖ^λ_0 is central in G,   (Kϖ^κK  )=  (K ϖ^λK ) * (Kϖ^λ_0 K)  and  as    W λ_0 =  λ_0, 
    𝒮( K ϖ^κ K ) = 𝒮( K ϖ^λ K )  e^λ_0   .
  
Now for any μ∈Λ such that a_λ(μ) ≠ 0, we have λ≽μ  by   Proposition  <ref>   and   Lemma  <ref>.  Thus             - μ^opp  =  μ +   λ_0  by  Lemma  <ref>.  The result now follows since e^Wμ· e^λ_0  =  e^W (μ + λ_0 )   =  e^ W  ( - μ   ).           
 
                 For λ∈Λ ^  +,  we   call  the element    q^⟨λ, δ⟩ e^W λ∈ℛ[Λ]^W the leading term of the Satake  transform  of    ( K ϖ^λ K ) and the number q^⟨λ, δ⟩ its  leading coefficient.      If gK ⊂ K ϖ^λ K is a coset, we call the unique cocharacter μ∈Λ  such that gK = ϖ^μ n K for some  n ∈  N  the    shape of the coset gK. The shape μ of any g K ⊂ K ϖ^λ K for λ∈Λ^   +      satisfies λ≽μ by the  results  above.     

 
Proposition <ref> and  most of  its corollaries may be found in several places in literature,  
though the exact versions           we needed     are    harder to  locate.    We have chosen to include proofs primarily to illustrate our conventions, which  will also be useful in computations in Part II.    
Cf. <cit.>. 

 
  
One can strengthen  Proposition  <ref>  to   a_λ(μ) ≠ 0  ⟺λ≽μ. See <cit.>.   
 





 §.§ Examples
            
In this subsection, we provide a few examples  of Satake transform computations for _2 to illustrate  our  conventions     in a simple setting.    

Let 𝐆 = _2 , F, 𝐀 = _m×_m ↪𝐆 be the standard diagonal torus and K = _2 (  _F  ).    For i = 1, 2, let e _i :  𝐀→_m for i =1  , 2 be the characters given  by  diag(u_1, u_2) ↦ u_i, i = 1 , 2 and f  _i :   _m→𝐀 be the cocharacters that insert u into the i-th component. Then Φ = {± ( e_1 - e_2 ) } and Λ =  f_1⊕ f_2. We will denote λ =  a_1 f_1 + a_2 f_2∈Λ by (a_1 ,a_2 ).  We take χ  : = e_1 - e_2∈ X^*(𝐀) as the positive root, so that δ = χ/2 and  Λ^+ is the set (a_1,   a  _2 ) such that a_1≥ a_2. Let α   :    = e^f_1, β   :   = e^f_2 considered as elements of the group algebra [ Λ ].   Then   ℛ [Λ] ^W = ℛ [α^±, β^± ] ^S_2 where the non-trivial element of S_2 acts via α↔β. 

    
Let λ = f_1∈Λ^+. Then  
λ^opp = f_2. As is well-known,  
    K ϖ^λ K = [ 1  ;   ϖ ] K   ⊔  _ κ∈ []  [ ϖ κ;   1 ]   K   .

In this decomposition, there is 1 coset of shape f_2 and q cosets of shape f_1. Therefore, we  obtain     

    𝒮 ( K ϖ^λ K )  
    =  q^1/2β + q ·  q^ -1/2α =  q^1/2  ( α    +   β   )  ∈ℛ [ Λ]^W .
 
    




      Let λ = 2f_1∈Λ^+. Then   
λ^opp =  2f_2. It is easy to see that   
    K  ϖ^λ K = [   1    ;     ϖ^2 ] K   ⊔  _ κ∈ [] ∖{ 0  }[ ϖ κ;   ϖ ]  K    ⊔   _ κ_1, κ_2∈ [] [     ϖ ^  2 κ_1 + ϖκ_2;                     1 ]   K   .

In this decomposition, there is  one coset   of shape 2f_2, q-1 cosets of shape f_1 +  f_2  and q^2 of shape    2f_1. So,        

    𝒮 ( K ϖ^λ K )       = q β ^2  + (q-1)  ·αβ +    q^2·   q^-1α ^2
           = q (α ^2+  β ^2 )   +  ( q - 1 ) αβ∈ℛ [   Λ  ] ^  W     .
  
            
  One can  in fact  write an explicit formula for 𝒮 ( K ϖ^λ K) for  any  λ∈Λ. See   <cit.> for a formula   
in terms of  ℛ-basis α ^mβ ^n of ℛ[Λ].    
 



 §.§ Macdonald's formula
   

The Satake transform is not  explicit in the sense that the    coefficients of the non-leading terms are not explicit. In general, the coefficients can be quite cumbersome expressions in q.  There is however the following formula   due to I.G. Macdonald <cit.> (see also <cit.>).
   
           Suppose 𝐆 is split and Φ_F̅ = Φ_F is  irreducible. Then  for  any    λ∈Λ ^+, 


    𝒮 ( K ϖ^λ K ) =  q^⟨λ ,  δ⟩/ W_λ (q^-1 )      ∑_ w ∈ W ∏ _ α∈Φ^ +
       e ^  w λ· 1 -  q ^ - 1   e ^  -  w α ^ ∨/ 1 - e ^  -  w α^∨

   where W_λ(x) : = ∑_w ∈ W^λ x^ℓ(w) denotes the Poincaré  polynomial of the stabilizer W^λ⊂  W of λ.   

 
For arbitrary reductive groups, there is a similar but slightly more complicated expression as it takes into account divisible/multipliable  roots and different contributions of root group filtrations.  We refer the reader to  <cit.> and     <cit.> for details.  These formulas however will not be needed.        

 Retain the notations of <ref>. We have e^-χ^∨ = α^-1β and 

    1 -  q ^  - 1   e ^ - χ ^ ∨/ 1 - e^-χ^∨      =  α - q^-1β/α - β ,             1 -   q ^  - 1      e^χ^∨/ 1 - e ^χ^∨   =  β -  q ^-1α/β - α    .
 
For  λ = 2  f_1, we  compute    
    𝒮 ( K ϖ^λ K )       =  q  (    α ^  2 ·α - q^-1β/α - β  +    β ^2·β - q ^-1α/β  -   α )    
        =  q  ( α^2 + αβ + β ^2 ) - αβ
           =  q ( α^2  + β^2  ) + (q-1) αβ
 
which agrees   with    Example <ref>. 





 §.§ Representations of Langlands dual
   

    

Let 𝐆̂ denote the dual group of  𝐆 considered as a split reductive group over . Let 𝐌̂⊂𝐆̂ denote the maximal torus such that X_*(𝐌̂ ) = X^*(𝐌). We let 𝐏̂ be the Borel subgroup of 𝐆̂ corresponding to Φ̂_F̅^+  :=   (Φ_F̅ ^ ∨)^+⊂ X^*(𝐌̂) = X_*(𝐌). The action of Γ on based root datum of 𝐆 together with a choice of pinning determines an action of Γ on 𝐆̂ which is  unique up to an  inner automorphism by 𝐌̂.  We define the Langlands dual to be ^L𝐆 =   ^   L     𝐆_F   : =  𝐆̂⋊Γ considered as a disconnected locally  algebraic group over . We  refer the reader to <cit.> for a detailed treatment of this group.      See also <cit.>.     

 The subscript F in the notation ^L𝐆_F  is not meant to suggest base change of algebraic groups but rather the fixed field for the Galois group Γ.  If E / F is an unramified field extension, and ^ L𝐆_E denotes the subgroup 𝐆̂⋊Γ_E of ^L𝐆 _ F.      
 

Since the weights of algebraic  representations of 𝐆̂ are elements of  X^*(𝐌̂ ) =  X_* ( 𝐌 ), we  also refer to  elements of X_*(𝐌) as coweights. 
For each dominant coweight λ∈ X_*(𝐌)^+,  there exists a simple representation (π, V_λ  )   of     𝐆̂ unique up to isomorphism such that   λ≽_Mμ for any coweight μ appearing in V_λ (<cit.>). Since 𝐆̂ is defined over , so is the representation V_λ  (<cit.>). For μ is a coweight of  V_λ,  we  denote by V_λ^μ the  corresponding    coweight space.     


Let   φ : 𝐆̂→𝐆̂ be  an   endomorphism  that sends 𝐏̂,  𝐌̂ to themselves and preserves λ i.e.,  λ∘φ  = λ as maps 𝐌̂→_m.      
Then  the  representation  of 𝐆̂ obtained via the  composition π∘φ also has   dominant coweight λ and is therefore  isomorphic  to  V_λ.  Since End(V_λ ) ≃ (<cit.>), there is a unique  isomorphism   
    T_φ : (π,  V_λ )    ( π∘φ , V_λ     )
     of   𝐆̂-representations     such that T  _  φ is identity on  the highest  weight space V _λ   ^λ.  In other words, 
 T_φ : V_λ→  V_λ  is  determined by the conditions that    T_φ( gv) = φ(g) T_φ (v)    for all g ∈𝐆̂(), v ∈ V_λ and that  T_φ : V_λ^λ→  V_λ^λ is the  identity  map. Let us define (g, φ ) : V_λ→ V_λ to be  the mapping v ↦  g  ·     T_φ ( v ) for any g ∈( ), v ∈ V_λ.    If ψ : → is another such  automorphism, it is easily  seen by the characterizing property of these maps that 
T_ψ∘ T_φ = T_ψ∘φ, so  that           
    (h, ψ)  ( ( g ,  φ    ) (v)  )  =  ( h ψ(g) ,  ψ∘φ  ) (v)

for all h, g ∈𝐆̂, v ∈ V_λ.  
Thus if Ξ⊂Aut( 𝐆̂        ) is a subgroup of  automorphisms preserving 𝐏̂, 𝐌̂ and λ,  then the construction just  described  determines an action of 𝐆⋊Ξ on V_λ extending that of 𝐆̂.   
Now suppose that the coweight  λ  lies  in Λ^+ =  X_*(𝐀) ^+↪   X_*(𝐌)^+ i.e.,  λ is Γ-invariant.    Since the action of Γ  on 𝐆̂  preserves  𝐌̂, 𝐏̂ by definition,  one can extend the action of 𝐆̂ on V_λ  to  an action of ^L𝐆_F on V_λ by taking Ξ = Γ  in     the discussion  above. Thus for λ∈Λ^+, (π , V_λ ) is naturally a representation of ^L𝐆. 
 Note that the action of Γ on V_λ may not be trivial,  even  though  it is required to be so  on the highest weight space.     See <cit.> or <cit.> for an example.    
 


Let γ denote the Frobenius element in Γ. Recall that the trace of a finite dimensional algebraic -representation (ρ, V) of ^L𝐆̂_F is defined to be the  map 
    tr_ρ : 𝐌̂() →     (m̂, γ)  ↦tr (   ρ(m̂, γ)  )

where m̂∈𝐌̂().  By  <cit.> and its proof,  tr_ρ   is naturally an element of [Λ]^W.  Since the weight spaces V^μ of V are defined over  and (1, γ) acts on these spaces by finite order rational matrices, the trace of ρ(1,γ) on V^μ is necessarily integral. Hence   the trace of ρ(m̂,γ) = ρ(m̂,1) ρ(1,γ)  restricted to  V^μ is an integral multiple of μ(m̂) for any m̂∈𝐌̂ ( ). It follows    that     tr_ρ belongs to the sub-algebra [ Λ] ^W of [ Λ ] ^W.    In particular, 
 the trace of ⋀^i V_λ for  any  
   λ∈Λ^+ lies in [Λ]^W for all i. Cf.<cit.>.       

     Let λ∈Λ ^  +. The  Satake polynomial 𝔖_λ(X)  ∈  [  Λ ]^W   [ X   ] is defined to be the  reverse  characteristic polynomial of 𝐌̂⋊γ acting on V _ λ.     For s  ∈1/2, the  Hecke   polynomial ℌ_λ, s(X) ∈ℋ_ℛ(K \ G / K )  [X]  centered at s  is defined to be the unique polynomial  that satisfies   𝒮 ( ℌ_λ, s ( X) )  =  𝔖_λ( q ^  -  s      X)  ∈ℛ [ Λ  ]  ^  W. 
      
In other  words, 𝔖_λ(X) ∈ [ Λ ]  ^  W  [ X  ] is  the   polynomial of degree d =    _   V _ λ such that  the coefficient of X^k  in   𝔖_λ(X)   is (-1)^k times  the trace of 𝐌̂⋊γ  on ⋀^ k    V (λ ) and   ℌ_λ, s is the polynomial such that the    Satake  transform of the   coefficient of X^k in ℌ_λ, s (X) is q^- k s times the coefficient of X^k   in 𝔖_λ (  X    ).   


         The  coweight   we   are interested  in for a given Shimura variety for a   reductive    group  over  arise  out of the natural cocharacter μ_h : _m, →_ associated with the Shimura datum for .    The ()-conjugacy class of this  cocharacter is defined over a number field E, known as the reflex  field of the datum.    At a rational prime ℓ where the group  is unramified, choose a prime v of E above it. Then E_v / _ℓ is unramified and the  orbit of μ_h under the (absolute) Weyl group of _E_v is stable under the action of the unramified Galois group of Γ_E_v of  E_v. By <cit.>, we can 
pick a unique dominant cocharacter λ (with respect to a Borel defined over E_v) of the maximal split torus in _E_v whose (relative) Weyl group orbit is identified with  the Γ_E_v-stable absolute Weyl group orbit of cocharacters μ_h. This λ is the coweight  whose  associated   representation  we  are  interested  in.  
In the situation above,   F is intended to  be    E_v.    

If E_v≠_ℓ, the Satake polynomial  corresponds to a polynomial over the   Hecke algebra of (E_v) whereas the Hecke operators that act on the cohomology of Shimura variety need to be in the Hecke algebra of (_ℓ). This is remedied by considering traces of  (𝐌̂⋊γ )^[E_v: _ℓ] instead. This makes  sure that the traces on ⋀ ^k V_λ belong to [ Λ__ℓ  ] ^W__ℓ where Λ__ℓ, W__ℓ are defined relatively for  over  _ℓ.      The exponentiation by [E_v : _ℓ]  here  can then 
be interpreted as a  base change morphism from Hecke algebra of _E_v to the Hecke algebra  of __ℓ.  
The Hecke polynomial of <ref> is obtained in this manner.
    

 


 §.§ Minuscule  coweights
       
The  representations of ^L𝐆_F  that will  be interested   in    will be   associated to  certain dominant cocharacters that arise out of a Shimura  data.  Such cocharacters  satisfy the special condition of being `minuscule'. In this  subsection, we recall this notion   and  record some  results scattered over several exercises of <cit.>. 
The  reader may consult 
<cit.> and  
<cit.> 
for general reference of the material provided here. Cf.  <cit.>.     

It will also  be  convenient to record our results in terms of  abstract  root data. Fix Ψ an abstract  root datum (X, Φ, X^∨, Φ^∨ )  and retain the notations introduced in <ref>  before Lemma  <ref>.  We  assume throughout that Φ  is  reduced.    

 Let λ be an element in  X^∨ or P^∨.    We say that   λ    is minuscule if ⟨λ , α⟩∈{ 1 , 0 , - 1  } for all α∈Φ.      


A subset S of X^∨ or  P^∨   is said to be saturated or Φ-saturated if for all x  ∈ S, α∈Φ and integers i  lying between 0 and ⟨ x , α⟩, we  have     x - i α   ^  ∨∈  S.    For λ in  X^∨ (resp., P  ^  ∨), we define S(λ) to be the smallest saturated subset of X^∨ (resp.,  P ^ ∨ ) containing λ i.e.,  S ( λ ) is the intersection of all saturated subsets 
in X^∨  (resp.,  P ^∨) that contain   λ.  
   
Given λ∈ X^∨, we will  denote  its reduction modulo X_0 in P^∨ by λ̅. Similarly given a set S ⊂ X^∨, we denote the set of reductions of its  elements  by  S̅.   It  is then easy to see that λ∈ X^∨ is minuscule  iff λ̅ and S  ⊂ X is saturated only if S̅ is.  Moreover if  λ∈  X^∨,  the reduction of  S(λ) equals S( λ̅).        
If a subset S of X^∨ or P ^∨ is saturated, then s_α (x ) = x - ⟨ x , α⟩α ^∨ belongs to S for  all  x ∈ S, α∈Φ.  Thus any saturated set is W_Ψ-stable.  In particular, the orbit W _ Ψλ is contained in S(λ)  for  any  λ in X^∨ or P^∨.    




                               A  dominant   λ in P^∨ or X^∨ is minuscule if and only if S(λ ) =  W_Ψλ.    
    
 

Let λ∈ X^∨.  Then λ is minuscule if and only if λ̅ is and S(λ) equals W_Ψλ if and only if S(λ)  = S(λ̅) equals  W  _ Ψλ̅.   It  therefore suffices to establish the claim  for λ∈ (P^∨)^+.      Denote  V^∨ = P^∨⊗, V = Q ⊗.    Then P^∨⊂ V^∨,  Q ⊂ V are dual lattices under ⟨ - , - ⟩.    
Let ( - ,  -   )  :   V^∨× V ^∨→  
be a W _ Ψ-invariant pairing. Then V is identified with V^∨,  ⟨ - ,  -   ⟩ with ( - , -), Q with 
 P^∨    and α∈Φ_F̅ with 2  α ^ ∨  /  (   α ^ ∨  ,   α ^ ∨  ).  In  particular,    
    (  λ  ,  α^∨ ) =    ⟨λ, α⟩/      2 ·  ( α^∨ , α^∨   )  .
 Note that ⟨λ  ,  α⟩ and therefore  (λ , α^∨ ) are non-negative for α∈Φ as  λ is  dominant. 
   


()  Suppose S(λ) = W  _ Ψλ and  suppose  moreover  for   the    sake of contradiction that λ is not  minuscule. Then there exists  α∈Φ ^  + such that k : =  ⟨λ , α⟩ > 1.   Then (  λ  , α^∨ ) =  k/2     (α^∨, α^∨ ). Set μ :  = λ - α^∨∈   P ^ ∨. Then μ∈ S(λ) by definition. Now  
    (μ , μ )  = (λ   , λ ) -  k ( α  ^ ∨ ,  α^∨  )  +  ( α^∨ , α^∨)  < (λ, λ ) .

Since elements of W_Ψλ must have the same length with respect to ( - , - ), μ∉ W_Ψλ = S(λ), a contradiction.  Therefore k ∈{0, 1 } and  we deduce that  λ is minuscule.    
 

()  Suppose  that  λ is minuscule.  For all w ∈    W _ Ψ, ⟨   w λ ,  α⟩  = ⟨λ , w ^-1α⟩∈{  1,  0 , - 1 } which implies that  w λ -  i  
   α   ^  ∨∈{ w λ ,  s_α ( w λ  ) } for integers i lying between 0 and ⟨  w λ  ,  α⟩.   Thus W _ Ψλ is saturated and  therefore W   _ Ψλ  =  S ( λ  ).         
 

 
                             Every  non-empty  saturated subset of the coweight lattice  contains a minuscule element. 

  Retain the notations in the proof of  Proposition  <ref>. Let S ⊂  P^∨ be a saturated subset. Let λ∈ S be the shortest element i.e.,  λ := (λ, λ)^1/2 is minimal possible for λ∈   S. We claim that λ is minuscule. Suppose on the contrary that there exist α∈Φ such that ⟨λ, α⟩∉{ 1,0,-1 }. Replacing α with - α if necessary, we may assume that ⟨λ , α⟩ > 1. Then λ - α   ^ ∨∈   S by definition and the  length calculation in the proof of <ref> shows that λ - α is a shorter element.   
 
Under additional  assumptions, one  can describe the  minuscule   elements of    X^  ∨    more  explicitly. Let Δ   = {α_1 , …, α_n} and let ω̅ _1, …,  ω̅_   n∈ P  ^ ∨ denote the basis dual to the basis Δ of Q.  The   elements   ω̅_i  are referred to as   the fundamental coweights of Φ.    If Φ is irreducible,  there exists a  highest root (<cit.>) 
    α̃ = ∑_j=1^n m_α_jα_j∈Φ ^  +

where m_α_j≥  1 are integers. Let J ⊂{1, …, n } be the subset of indices j such that m_α_j = 1.  
     For  irreducible  Φ,   {ω̅_j}_j∈ J is the set of all  non-zero minuscule elements in (P^∨)^+.  These elements form a system of representatives  for non-zero classes in    P^∨  /  Q^∨. 
    
  Let λ∈  ( P   ^  ∨ )   ^   + be non-zero.  Since    ω̅ _1, …, ω̅_n is a basis of P^∨, we can write λ =  a_1ω̅_1  + …  +   a_nω̅_n uniquely. Since λ is dominant and non-zero, we have   a_1, …, a_n≥ 0 and at least one of  these  is  positive, say a_k. Now  λ is minuscule only if a_1 m_α_1 +  …  + a_n m _  α_n =   ⟨λ , α̃⟩ = 1 as both a_k , m_α_k≥ 1. But  this   can  only   occur  if   a_k = 1, k ∈ J and a_i = 0 for i ≠ j.   Thus minuscule elements of P ^∨     - { 0 } are contained in the  set  {ω̅_j}_j ∈ J.  Since α̃ is highest, any root ∑_j=1^n p_α_jα_j∈Φ_F satisfies m_α_j≥ p_α_j and one easily sees that   all  ω̅_j for j ∈ J are    minuscule.  The second  claim  follows   by     Corollary of Proposition 6 in <cit.>   
    



For  λ∈  X^∨, set  Σ ( λ )  : =  {μ∈  X^∨  |  λ≽  w  μ for all   w ∈ W _ Ψ}. Similarly define Σ(λ) ⊂ P^∨ for  λ∈ (P^∨)^+. Then λ∈Σ(λ)  by   Lemma   <ref> and Σ(λ) is  easily seen to be saturated.  Therefore       W_Ψλ⊂ S(λ )  ⊂Σ(λ).

               A dominant   λ in X^∨ or P^∨ is minuscule  if  Σ(λ ) =  W_Ψλ. The converse holds if Φ is irreducible. 
  
  It is clear that Σ(λ) = W_Ψλ   is  equivalent to Σ( λ̅ ) = W_Ψλ̅ so it suffices to prove these claims for λ∈ (  P  ^ ∨ )  ^  +.  
 

() Suppose Σ (λ)  =   W_Ψλ.     As, W  _ Ψλ⊂ S(λ)  ⊂Σ ( λ ), the equality Σ(λ)  =  W_Ψλ implies that S(λ)   =    W_Ψλ which by Proposition <ref> implies that λ is minuscule. 


()  Suppose Φ is irreducible and λ∈  ( P^∨ ) ^  +   is   minuscule. Then λ∈ Q^∨ implies λ is zero, since  the  only non-zero  dominant minuscule  elements  in   P^∨ are those fundamental coweights which by Lemma <ref> form representatives of non-zero  elements in P^∨/Q^∨. So is suffice to prove the claim for λ∉ Q^∨.   
Suppose now on the contrary  that  there exists a  μ∈Σ(λ) - W_Ψλ.   
We may assume μ is dominant since Σ(λ) - W_Ψλ is   stable under W_Ψ and W_Ψμ contains a dominant element.    Since Σ ( λ ) is saturated and contains μ,  Σ ( λ ) ⊃  S(μ ).  By Corollary   <ref>    S(μ) contains a minuscule element λ_1. Since all elements of W_Ψλ_1 are minuscule and S(μ) is W_Ψ-stable, we may  take λ_1 to   be  dominant. Since S(μ) ⊂Σ(λ), λ≽_♭λ _1. In particular, λ - λ_1∈ Q  ^ ∨. Since λ∉ Q ^ ∨, λ and λ_1 are   distinct    non-zero dominant coweights that  represent  the  same non-zero  class in P ^ ∨ / Q ^ ∨. But this 
  contradicts the second part of Lemma <ref>. Hence  Σ(λ) must equal W_Ψλ.  The final claim is immediate.   
   
Now resume the notations of <ref>. Fix λ∈Λ^+ and let V_λ be the irreducible representation of ^L of highest weight λ.    For each μ∈ X_*(𝐌)^+ with μ≼λ,     the dimension (as a vector space over ) of the coweight space V_λ^μ is called the multiplicity of μ in V_λ. Corollary <ref> implies that when λ is minuscule and Φ _F̅ is irreducible, the set of  coweights in V_λ  is  just the Weyl orbit W_Mλ. Since W_M permutes the weights spaces, the  multiplicities of all coweights  are 1. If  is split, then the action of Γ on  is trivial and so is its  action on the coweight spaces of V_λ. We therefore get the following  result.     
  Suppose  is split and Φ_ F  = Φ_F̅ is  irreducible.  Then for all  minuscule  λ∈Λ^+,    𝔖_λ(X)   =   ∏_μ∈ W λ ( 1 - e^μ X ) ∈ℛ[Λ]^W.  
  




  The content of this  subsection is developed in Exercises 23-24   of 1 and Exercise 5 of 2 in <cit.>.  While the results are well-known, the  version  we need and their 
written proofs  seem    harder to  find.      We have included proofs here  for  future  reference.      
 



 §.§ Kazhdan-Lusztig theory
 

 



  We  finish this section  by  recording  an  important property of the coefficients of Satake transform when taken  modulo   q - 1. We assume for all of this  section that  𝐆 is split   and  Φ_F̅  = Φ_F is  irreducible.   We refer the reader to <cit.>, <cit.> and   <cit.> for the material  presented  here.  
See  also  <cit.> for a   generalization to non-split case.  




The Hecke algebra  ℋ_ℛ(W_I) of W_I  is the unital associative   ℛ-algebra with ℛ-basis {  T_w }_ w ∈ W _I subject to the relations  
     T_s^2   =  (q-1) T_s  +  q T_e               for s ∈ S_aff 
 
   T_w T_w'   = T_ww'               if ℓ(w) + ℓ(w') = ℓ(ww') 
 
Each element T_w possesses an inverse in  ℋ_ℛ ( W_ I ). Explicitly,  T_s^-1 =  q^-1 T_s    - ( 1 - q ^-1 ) T_e.  The -linear map ι :  ℋ_ℛ(W_I)    →ℋ_ℛ(W_I) induced by T_w↦ (T_w^-1)^-1 and q^1/2↦  q^   -  1/2 induces a ring automorphism of order two known as the  Kazhdan-Lusztig involution.      
 For each y , w ∈ W_I such that x ≤ w in   (strong)  Bruhat ordering, the Kazhdan-Lusztig polynomial P_x,w  (q)  ∈[q] (considering q  as   an   indeterminate) are uniquely characterized by the following  three  properties:   
  1   

  * ι ( q ^  - ℓ ( w ) / 2 ∑ _  x   ≤ w  P_x,w(q) T_x ) =   q ^ ℓ ( w ) / 2 ∑  _  x   ≤  w   P_x,w ( q ) T_x,  

  * P_x,w(q) is a polynomial of degree at most (  ℓ(w) - ℓ(x) - 1  ) /  2 if x  ⪇  w, 

  * P _   w ,  w    ( q )  = 1.  
  If x ≰ w, we extend the definition of these polynomials  by  setting   P_x,w(q) = 0.  We will refer to P_x,w for any x,w ∈ W_I as KL-polynomials.
    


For any λ∈Λ, there is a unique element denoted w_λ which has the longest possible length in the  double coset W t(λ) W  ⊂ W_I. When λ∈Λ^+, this element is t(λ) w_∘ and ℓ(t(λ) w_∘ ) = ℓ ( t(λ) + ℓ ( w_∘ ) =  2 ⟨λ ,  δ⟩  +  ℓ ( w_∘  ). For  any λ,  μ∈Λ^+, we have  λ≽μ  (<ref>) iff w_λ≥  w _ μ.  


   Let λ∈Λ^+ and χ_λ∈ [ Λ ] ^ W  denote  the  trace of 𝐌̂ on V_λ.  Then 
    χ_λ =  ∑     _ μ≼λ q ^  - ⟨λ ,   δ⟩ P _  w_μ ,  w_λ (q)    𝒮  ( K  ϖ^μ K  )
 where the sum runs over μ∈Λ^+ with μ≼λ.     
   
 See <cit.>. We  also  note that the proof provided in  <cit.> carries over with minor changes.  
   

  χ_λ = ∑    _μ≼λ  P_w_μ , w_λ (1) e ^  W λ. 

 Using Macdonald's formula (Theorem <ref>)  for the expression 𝒮(K ϖ^μ K)  in the Kato-Lusztig formula  <ref>,   we obtain an expression for χ_λ as a linear combination in e^W μ which has coefficients in ℛ_q (see <cit.>). Since the χ_λ is independent of q, we can formally replace q with 1 which yields the expression above.   
 
Let   ℐ =  ℐ_q⊂ℛ _q denote the ideal generated by q ^1/2 - 1 and let 𝒮 =  𝒮_q :  = ℛ/ℐ.    For f ∈ℛ[Λ] ^W, we let [f]  ∈𝒮[Λ]^W denote the image of f.  Similarly, for ξ∈ℋ_ℛ(K \ G / K ),  we let  [ξ]  ∈ℋ_𝒮 ( K \ G / K ) denote the class of ξ. For f = ∑ _ μ∈Λ^+ c_μ e^W   μ∈ℛ[Λ]^W, let 
    ξ_f  :  = ∑    _μ∈Λ ^  +  c_μ (K ϖ^μ K  )  ∈ℋ_ℛ(K  \ G /  K ) .
 
 Let f ∈ℛ[Λ]^W and ξ =  𝒮^-1(f).  Then [ξ] = [ξ_f ].     
 

 Since χ_λ form a -basis for [ Λ]^W, it suffices to establish the claim for f =  χ_λ.  But this follows by Kato-Lusztig formula and Corollary  <ref>.    
 




§ DECOMPOSITIONS  OF  
DOUBLE  COSETS
   
    
In this section, we   derive using the elementary theory of Tits systems a  recipe   for  decomposing  certain double cosets into their  constituent  left cosets. Invoking the existence of a such a system on the universal covering of the derived group of a 
   reductive group over a local field, we   obtain a recipe   for     decomposing Hecke operators  arising out of double cosets of what are  known 
 as   parahoric subgroups  
  of unramified reductive groups.    
The method used here for decomposing such double cosets is  based on the one  introduced in <cit.> in the setting of split Chevalley groups. 
Theorem   <ref>, the main result of this section,  will be our primary tool for executing the machinery of <ref>  in concrete  situations.


 §.§ Motivation
             
    
To motivate what kind of decomposition we are looking for,  let us take a look at the case    of  decomposing K σ K where K = _n(_v) for v a rational prime and σ = diag(v,…, v,  1, …, 1 ) where there are k number of 1's. Let G denote _n(_v).    There is a natural G-equivariant bijection between G / K and the set of _v-lattices in _v^n where K is mapped to the standard lattice. Then σ K  corresponds to the  lattice generated by the basis where the first n -  k standard vectors are replaced by multiples of the uniformizer v. Thus K σ K 
/ K corresponds to the K-orbit of this lattice. It is clear that any such lattice lies between the standard lattice _v     ^n and v _v ^n.   Reducing modulo v therefore gives a bijection between  K  σ K /     K and  the   𝔽_v-points of the Grassmannian Gr(k,n) of k-dimensional subspaces in an n-dimensional vector space.     Since Gr(k,n)(𝔽_v) admits a stratification by Schubert cells, one obtains an explicit description of K  σ  K  /   K  by  taking _v lifts of their 𝔽_v points.  See Example  <ref> that illustrates   this for n = 4,   k =  2.          We would like a similar recipe for more general reductive groups and arbitrary cocharacters.      
 




 §.§ Coxeter systems
     Throughout this subsection,  (W,S)  denotes   a Coxeter system. Given X  ⊂ S,  we  let W_X⊂ W be the group generated by X. Then (W_X, X) is a  Coxeter system itself  and  W_X∩ S = X. We refer to groups  obtained  in this manner as  standard parabolic subgroups  of (W , S ).   Let ℓ : W → denote the length function. Then ℓ_| W_X is the length function on W_X. Given X  ,  Y ⊂ W and a ∈ W, consider  an element w ∈  W_X  a  W_Y of minimal  possible length. The deletion condition for Coxeter groups implies that  any w ' ∈ W _  X   a  W_Y can be written as w' = x w y for some  x ∈ W_X, y ∈  W_Y  such  that  
    ℓ ( w'  ) = ℓ ( x ) +  ℓ (  w )  +  ℓ  ( y )  .
  It follows that w ∈ W_X a W_Y is the unique element of minimal possible  length.    We refer to w as the (X,Y)-reduced element   of    W_X a W_Y and denote the set  of   (X,Y)-reduced elements in  W by [ W_X\ W /  W_Y ].   If w ∈ W is (X,   ∅ )-reduced, then we have the stronger property that   ℓ ( x w ) = ℓ ( x )  + ℓ(w) for all  elements x ∈  W_X. An arbitrary σ∈  W can be written  uniquely    as σ   =  xw for some  x ∈ W _X and w ∈ W a (X, 
  ∅)-reduced   element.      Similarly for (∅ , Y )-reduced  elements. An element in W  is (X,Y)-reduced iff  it is (X,   ∅     )-reduced and (  ∅,   Y)-reduced. 
  
  
The   stronger properties of minimal length  representatives for   one-sided cosets of parabolic subgroups can be generalized to double cosets as follows.   Let  σ∈ W be  (X, Y )-reduced. Then   W_X∩σ W_Yσ ^-1 is a standard parabolic subgroup of (W_X, X) generated by Z:= X ∩ (W_X∩σ W_Yσ^-1)  and 

    ℓ ( τσυ )  =  ℓ ( τσ) + ℓ (  υ ) =    ℓ ( τ ) + ℓ ( σ ) + ℓ (  υ )
 
for  any τ∈ [ W_X  /  W_  X∩σ W_Y σ  ^  - 1  ], υ∈    W_Y. In other words,   the equality above holds for any (X,Y)-reduced element σ∈ W,  any (∅, Z)-reduced element τ∈ W_X     and arbitrary υ∈ W_Y.         

There is a generalization of these facts to a  slightly  larger class of groups. Let Ω be a group and Ω× W → W be  a left action that restricts to an action on Ω× S  →  S.     We refer to elements of Ω as automorphisms of the system (W,S). Since such automorphisms are length preserving, we may form the extension  W̃  :  = W  ⋊Ω and extend the length function ℓ : W̃→ by declaring ℓ( σρ ) = ℓ ( σ ) for σ∈ W, ρ∈Ω. We refer to elements of Ω⊂W̃ as length zero  elements.    Given A ⊂ W, we denote by A  ^  ρ the set ρ A ρ ^-1⊂  W. Then ρ W_Xρ ^-1 = W _  X ^ρ⊂    W for   any     X ⊂ S.    Given X , Y ⊂ S, b = a  ρ∈W̃ where a ∈  W, ρ∈Ω, there is again a unique element w  ∈  W_X  b    W_Y of minimal possible length given by w =  σρ where σ is the (X, Y  ^ ρ )-reduced element in W _X a W_  Y ^ ρ.   Moreover  W_X∩   w W_Y w^-1 =  W_X∩σ (W_  Y ^ ρ  )    σ^-1 is   still   a standard  parabolic subgroup  of W_X with respect to X and  the length  formula (<ref>)    continues to hold  when σ is replaced with w  = σρ. 
We continue to call  the unique  element σρ as the (X,Y)-reduced element of W_X b W_Y and  denote the  collection obtained over all  double  cosets  by [ W_X\W̃ / W_Y ]. If  w ∈W̃ is (X, ∅)-reduced, we  again  have ℓ(xw) = ℓ(x) +     ℓ(w) for all x ∈ W_X. 
     


 The  result on (X,Y)-reduced  elements in the first paragraph above   appear in  <cit.> from which we have also borrowed its terminology.  See   also      <cit.>.  
Detailed proofs of all claims in the second paragraph can  be found in <cit.> or <cit.>. 
Groups W̃ as above  are sometimes called quasi-Coxeter groups.    
   





 §.§ Tits Systems
    
     
 A Tits system   𝒯     is  a quadruple (G, B , N,S) where G is a group, B  ,  N are  two  subgroups  of  G     and S is a subset of N / (  B     ∩ N ) such that the following  conditions  are  satisfied:       
           

  (T1) B ∪ N generates  G and T =   B    ∩ N is a normal subgroup of N 

  (T2) S generates  the  group  W =     N /   T and consists of elements of order 2 
 

  (T3) s B w  ⊂   B   w   B   ∪  B  s w   B for all s ∈ S, w  ∈ W.

  (T4) s  B   s   ≠   B for all s ∈ S
    
We call W the  Weyl group of the system and let ν : N → W denote the natural map.  
   



   For any v, w  ∈ W, the products w B, Bvw , vBw etc   are  well-defined since if, say, n_w∈ N is a representative of w, then any other is given by n_w t for t ∈  T  ⊂  B and one has   n_w t = t ' n_w for some t' ∈  T by normality of T in B ∩ N. 
    

For any such system, the pair (W,S) forms a  Coxeter  system. We denote by ℓ : W → the  corresponding length function.  The set S equals the set of non-trivial elements w ∈ W such that B ∪  B  w  B is a group. Hence S is  uniquely determined by the groups G, B , N and the axioms (T1)-(T4).  We therefore  also say that (G,B,N) is a Tits system or that (B,N) constitutes a Tits system for  G.     The axiom (T3) is equivalent to 
    BsBwB ⊂  BwB ∪ BswB.

Since   BsBwB is a union of double cosets, it must equal either BswB or BwB  ∪ Bs wB  and the  two  cases  correspond to whether ℓ(sw) equals ℓ(w) + 1 or ℓ(w)-1.   
In particular, Bs Bs B  equals  B ∪ Bs B by (T4). 
The subsets BwB ⊂ G for w ∈ W are called Bruhat cells which provide a decomposition   

    G =      
    _w ∈ W  Bw B
    
called  the Bruhat-Tits   decomposition. If w = s_ 1⋯   s_ℓ(w) is a  reduced decomposition of W, then BwB = Bs_1 B · B s_2 B ⋯ B s_ℓ(w)  B. A subgroup of G that contains B is called a standard parabolic.    There is a bijection between such subgroups of G and subsets X of S given  in one  direction  as follows: given X  ⊂   S, we let K_X : = B W_X B ⊃ B  where W_X⊂  W is the group generated by X. Then K_X is the standard parabolic subgroup associated with X.    In particular, K_∅ = B, K_S = G.    Any  standard   parabolic  subgroup  of G   equals  its  own   normalizer  in  G.      
If N ' is a subgroup of N such that ν(N') = W_X, then (K_X, B, N', X) is a Tits system  itself.   
If X, Y ⊂  S, the bijection B \ G / B   W induces a bijection 

    K_X\ G / K_Y≅ W_X\ W / W_Y
    
given by sending K_X w K_Y↦ W_X w W_Y.   For Z  a normal subgroup of G contained in B,  denote  G ' = G/Z and let B' = B/Z, N' = N /   ( N   ∩    Z ) denote the images of B , N in G '. Set  W ' = B' /  (B ' ∩ N ') and S ' the image of S under W → W '. Then (W,S) → (W ', S') is an isomorphism  of Coxeter groups and  (G', B', N', S' ) is a Tits system which is said to be  induced   by (G,B,N,S).     



  Let (G,B,N,S) be a Tits system. We say that the system is  commensurable   if B s B / B is finite for all s ∈ S. Then  Bw B/  B is finite for all w ∈ W.  We let q_w  denote  the  quantity |  B w B / B  |   = [ B : B ∩ w B w^-1 ].


                   Let (G,B,N,S) be   a  commensurable  Tits system.  For any  σ  ,  τ∈ W such that  ℓ( σ ) + ℓ (  τ  ) =  ℓ ( στ     ),   q _τσ =  q_τ  q_σ 
  


 Since B w B / B is finite for all w ∈ W, one may form the    convolution algebra ℋ_ ( B \  G  /  B ) with product ( B w B )  *    (B v   B) given as in  <ref>. The linear map  ind : ℋ_ ( B \ G  /B ) → given by ( B w B ) ↦  q_w is   then    a homomorphism of rings. If s  ∈ S, w ∈ W, we have  
    ( Bw B) *   ( B s B ) =  ∑_ u ∈ W  c_w, s ^u ( B u B )
   where c_w,s^u = | (  B w B  ∩ u BsB )/ B  | (see eq. (<ref>)). Note that  c^u_w,s≠   0 if and only    if BuB ⊂  B w B s B.   Suppose that ℓ ( w ) + ℓ ( s) =  ℓ (w s ), so that BwBsB = Bws B. This implies that   c_w,s^ u  = 0 for u ≠ ws and   that     wBsB ⊂ B ws B.  Since BsBsB = B ∪ BsB, we have   
    ws BsB ⊂ w  ( B s B s B  )  = w ( B ∪ BsB)  ⊂   wB ∪ BwsB   .
        
Using the above inclusion, we see that     
    w B  ⊂    BwB ∩ ( ws BsB )  ⊂ BwB ∩ ( wB ∪ BwsB ) = wB
 where the last equality follows  by disjointness of BwB, BwsB. It follows that  BwB ∩ ws BsB = w B and therefore  c^ w  s    _ w, s    =  1. Combining everything together, we see that   (  B w B )  *    (   B s B  )    =    ( B ws B  ). Repeating this argument by writing σ = w s  = w' s' s, we see that ( B σ B ) =   ( B s_1 B )   *  ⋯     *     ( B s_ℓ(w) B ) where σ   = s_1⋯  s_ℓ(w) is a reduced decomposition. Since ind is a homomorphism, we see that q_σ  =  q_s_1⋯  q_s_ℓ(σ    ) and similarly for q_τ, q_τσ.   The claim    follows since the product of two reduced expressions for σ, τ in that order  is  a  reduced 
 word    expression for στ.          
    
         Let (G,B,N,S) be a Tits system  and   φ : G →G̃ be a homomorphism of groups. Then φ is said to be  (B,N)-adapted  if 
   1.5    

  (i)  φ⊂ B, 

  (ii)  for all g ∈G̃, there is h ∈  G such that g φ(B) g ^-1 =   φ ( h B  h ^-1   ) and g φ(N) g^-1 = φ ( h N h^-1 ).      

For any such map, φ(G) ◃G̃ and the induced map G / (φ ) ↪G̃ is adapted with respect to the induced Tits system on G / φ. 
   
Let φ : G  →G̃ be a (B,N)-adapted  injection  and consider G as a (necessarily normal) subgroup of G̃.   Denote by T = B ∩ N, W = N / T as above  and  set  Ω =  G̃ / G.        Let B̂, N̂ denote respectively  the normalizers of B, N in G̃ and  set   Γ =  B̂∩N̂.  Since    every g ∈G̃ has a  h ∈ G such that g^-1 h ∈Γ, we  see  that     G̃ = Γ G. If g is taken to be in B̂, h is forced to lie in B as   B equals its own normalizer in G.  Therefore    B̂ = Γ   B.   That N_G (B) = B   also    implies  that Γ∩  G =   Γ∩   B from which  it follows   that    Γ / Γ∩ B and B̂/ B are   both    canonically isomorphic to Ω.
 
 
 
 Define  Ñ = N Γ  and  T̃ = Ñ∩ B.  As Γ normalizes N, Ñ = N Γ = Γ N is a group and therefore   so is T̃.      Invoking N_G(B) = B again, we see that T̃ =   N Γ∩ B = T (Γ∩ B ).  Since Γ normalizes both B and N,  it  normalizes the intersections  T = B ∩  N and Γ∩ B. Thus  Γ normalizes  the  product  T̃ = T ( Γ∩  B  ).  If n ∈ N, b ∈Γ∩ B, there exist n' ∈ N such that bn = n '  b. The decomposition  (<ref>)    implies that n', n represent the same class in W, and so n 'n^-1 = b n b ^-1 n ^-1∈ T. This implies that n b ^-1  n ^-1 lies in T̃. It  follows from this  that N  also 
 normalizes T̃.   Consequently,   T̃ is a normal subgroup  of Ñ. 
We  let 
    W̃  =  Ñ /  T̃ .
      
Since N   contains   T,       N ∩T̃  =  N ∩ T ( Γ∩ B ) = T   ( N ∩Γ∩ B ) = T .    
 Similarly Γ∩T̃ =  Γ∩  B. 
Thus      
the   inclusion of N (resp. Γ) in Ñ allows us
to identify W (resp.  Ω) as a subgroup of W̃.   Since  Γ normalizes N, Ω normalizes W.  Since Γ∩ N = Γ∩ B  ⊂T̃,  W ∩Ω is trivial in W̃. 
It follows that 
    W̃  =  W  ⋊Ω .
 Since γ (B ∩ B w B )  γ^-1 = B ∩ B γ w γ^-1 B for any w ∈ W, γ∈Γ and   since    B ∩ B u B is a group for u non-trivial   if and only if u ∈ S, we see that Ω normalizes  S.   Consequently Ω acts  on (W,S) by automorphisms and we may extend the length function from W to W̃. From the decomposition  (<ref>)    and the normalizing properties of Γ, we obtain a generalized Bruhat-Tits     decomposition  
    G̃  =    
    _ w ∈W̃  B  w    B  .
 
Similarly, if X, Y ⊂ S, K_X, K_Y⊂ G denote the corresponding groups, we obtain from  <ref>  a    decomposition     K_X\G̃  / K_Y≅  W_X\W̃ / W_Y.    



 For the general theory of Tits systems, we refer the reader to <cit.>.  
The material on (B,N)-adapted morphisms   and commensurable Tits systems is developed in Exercises 2, 8, 22, 23, 24 of op. cit. and we have  included their proofs here.   
The terminology of Definition <ref> is taken from <cit.>. This  notion      is referred to as  generalized Tits systems in  <cit.>.  
    
   
 




 §.§ Decompositions
    Assume for all  of this subsection   that  (G,B,N,S) is a commensurable Tits system and  φ : G  ↪G̃ is a (B,N)-adapted  inclusion.    Retain also the notations W, W̃, Ω and q_w for w ∈ W introduced above.  For each s ∈ S, let _s⊂ G   denote  a   set   of    representatives of B /  ( B ∩ s  B s^-1   ) (so |_s| = q_s) and  let s̃ denote a  lift of s to N under ν    (so that ν( s̃ ) = s). Define 
    g_s :  _s→   G ,   _s∋κ↦κs̃

considered as a map of sets.  
Fix a w = σρ∈W̃ where σ∈ W, ρ∈Ω and let ρ̃∈Γ denote a lift of ρ. Then ρ̃ B = B ρ̃ is independent of the choice of the lift and we may  therefore   denote ρ̃ B 
 simply  as    ρ B.     Let  m = m_w   : = ℓ  ( σ ) denote the length of σ and let    r( σ )  = (s_1,  … , s_m) denote  a  fixed  reduced word decomposition of σ. Denote by _r(σ) the product _  s_1×_s_2×⋯×_s_m  .
        B w   B    =     _ κ⃗∈_r(σ )   g_s_1 (  κ_1  ) ⋯  g_s_m ( κ_m )  ρ   B
where κ_i denotes the i-th  component of κ⃗.      
    
  We have B  w   B = B  σ  B  ρ    = B s_1 B ⋯ B s_m B   ρ. Now 

    B  σ  B           =   ⋃ _κ_1∈_1 g_s_1( κ_1 ) B s_2 B ⋯ B s_m B    
        = ⋃ _  ( κ_1, κ_2  )   
    ∈_1×_2  g_s_1 ( κ_1 ) g_s_2( κ_2 ) B s_3 B  ⋯ B s_m B   =  ⋯  = ⋃ _ κ⃗∈_r(w)  g_s_1( κ_1 ) ⋯   g_s_m ( κ _m   )    B
   
As |  B σ   B / B | =   q_σ    = q_s_1⋯   q_s_    m  by  Lemma   <ref>,    the union above  is  necessarily      disjoint.  Multiplying each coset in the decomposition above on the right by ρ and moving it inside next to σ on the left hand side,    we get the desired decomposition of B w  B.        


Retain the notations   w, σ, ρ, m.  We 
   define  𝒳_r(σ) ,  ρ   :  _    r ( σ )   →G̃       / B to be the  map κ⃗↦  g_s_1 ( κ_1 )    ⋯  g_s_m(  κ_m ) ρ̃ B.   
   (where we have suppressed the dependency on the choices of lifts). 
   Then in this notation,  
    BwB = _κ⃗∈_r(σ)𝒳_r(σ) , ρ (κ⃗) .
 
In particular, the image of 𝒳_r( σ )   ,     ρ in G̃      / B  is independent of all the choices involved. Since we will only be interested in the image  of    𝒳_r(σ),   ρ modulo subgroups of G containing B,   we will abuse our notation to  denote  this map   simply  as   𝒳_w. Moreover we will consider   𝒳    _w as taking values in G̃ as opposed to G̃/B,   if it is understood that these are representatives of left cosets for some fixed subgroup that  contains B.    
Similarly   we     denote _r(σ) by _ w.  

 
             Let X  , Y ⊂ S, W_X, W_Y be the subgroups  of W generated by X, Y  respectively and let K_X = B W_X B, K_Y = B W_Y B. For  any   w   ∈ [ W_X\W̃     /  W_Y  ],   we   have                               
    K_X  w    K_Y =    _ τ _ κ⃗∈_τ  w  𝒳_τ w     ( κ⃗ )  K_Y
     
where τ runs over [ W_X /   (  W_X ∩   w W_Y  w ^  - 1    )   ].  In particular, | K_X w K_Y / K_Y | = 
∑    _τ |  _τ w |.    
   

  First note that  K_X w K_Y  =  ⋃_ x ∈ W _X  B  x   B  w B K_Y  =  ⋃_x ∈ W_X  B x  w  K_Y where the second equality follows  
since ℓ ( x  w ) = ℓ ( x ) + ℓ ( w ) for all x ∈ W_X (see <ref>).   Since    B \G̃ / K _   Y is in bijection with    W̃ / W_Y,   we infer  that    Bx w K_Y = B x' w K_Y   for  x ,  x '  ∈   W_X     if and only if xw W_Y = x'w W_Y.  It follows that  
    K_X w K_Y   =  _ τ B τ w K_Y
   where   τ runs 
   over a set of  representatives of  W_X /  ( W_X∩ w W_Y w^-1  ) and    which we are free to take  from  the   set      A  := [W_X  / ( W_X∩  w  W_Y   w ^  -1 ) ]     ⊂ W_X. Fix  a τ∈  A.  We have   
   
    B τ w K_Y =  B  τ w B K_Y =  ⋃ _  κ∈ _ τ   w  𝒳_τ w   (  κ⃗ ) K_Y
  by  Lemma    <ref>.      Say κ⃗_1 , κ⃗_2∈_τ w are such that g_1 K _Y =  g_2  K _Y  where g_ i    : =  𝒳_τ w  ( κ⃗_i ) ∈G̃  for i = 1 , 2.    As K_Y  = B W_Y B, we have     
    g_1 K_Y =   _   y  ∈ W_Y _ κ⃗∈ _  y    g_1𝒳 _   y  (  κ⃗ )  B
  
by Lemma  <ref> again.    As g_2 B ⊂  g_2 K_Y = g_1 K_Y,  there exists   y  ∈   W _Y and  κ⃗_y∈_y  such that  g_1𝒳_y ( κ⃗_y )  B  =   g_2   B . Now observe that  
    B  g_1𝒳_y   ( κ⃗_y  )     B   ⊂ B g_1  B   𝒳_y ( κ⃗ _y  ) B   =    B τ w B   y    B
  and B τ w   B y B = B τ w y B    since ℓ ( τ w  y )  =   ℓ ( τ w ) + ℓ (  y    ) by 
  (<ref>). Therefore, g_2 B  =   g_1𝒳 _  y   ( κ _ y       ) B   ⊂   B τ w y      B.    Since g_2 B is also contained in B τ w B, we see that B τ w B = B τ w y B . This can only happen if y = 1_W_Y which in particular means that _y is a singleton and 𝒳_y ( κ⃗    _y ) B = B. We therefore  have   g_1 B = g_1𝒳_y(κ⃗ ) B = g_2  B which in turn implies  that  κ_1 =  κ_2.  The upshot is that  the right hand side of (<ref>) is a disjoint  union  for each fixed  τ∈ A.  Thus         
    K _  X  w K _  Y  =    _ τ∈  A   B τ w K_Y    =   _ τ∈  A  _ κ⃗∈ _ τ w 𝒳   _ τ w  , K_Y (   κ⃗ )

which completes the proof. 
   



  The  proof of  Theorem     <ref> is inspired by  <cit.>. 
     



 §.§ Reductive Groups


In this subsection, we recall the relevant results from the theory of Bruhat-Tits buildings. We primarily follow <cit.> in our exposition    and  refer  the reader to  book   <cit.>  for additional  details and  background.    

Retain the notations  introduced in     <ref>  and  <ref>. 
In particular,  denotes an unramified reductive group over F  and G its group of F points.    Additionally, we let 𝐆̃ be the simply  connected  covering of the derived  group  𝐆 ^ der of  𝐆 and let ψ :  𝐆̃→𝐆 denote the resulting map.  For a group 𝐇⊂𝐆, we denote by 𝐇̃⊂𝐆̃ the pre-image of 𝐇 under ψ. 


Let ℬ be the Bruhat-Tits  building  of  G̃   :  =   𝐆̃(F) and let 𝒜⊂ℬ be the   apartment   stabilized (as a subset) by   Ã : = 𝐀̃(F).  By  definition 𝒜 is an affine space under the  real   vector space Ṽ :=  X_*( 𝐀̃  ) ⊗.  Let   M̃  : =  𝐌̃ ( F). There is a unique homomorphism ν : M̃→Ṽ determined by the  condition 
    χ (  ν ( m )  ) =   -  ord  ( χ ( m)    )
 for all m  ∈M̃, χ  a F-rational cocharacter of 𝐀̃.    The kernel of ν is a maximal  compact open subgroup M̃ ^∘ of  M̃.  Set Ã^∘   :  =  Ã∩M̃^∘. Then Ã / Ã^∘ = M̃ / M̃^∘ via the inclusion Ã↪M̃ and the image ν(M̃) ⊂Ṽ  is identified   with     X_*(𝐀̃).   Let 𝒩̃ denote the stabilizer of 𝒜 (as a subset of ℬ).  The  map ν admits a unique extension 𝒩̃→Aut(𝒜 ) where Aut(𝒜) denotes the group of  affine automorphisms of 𝒜. The action of  G̃ on ℬ is then   uniquely  determined by this extension.


Fix x_0∈𝒜 a hyperspecial point via which we identify Ṽ with 𝒜.   Then ν identifies 𝒩̃ /   M̃^∘ with W_aff = Λ̃⋊ W.  Let C   ⊂𝒜  be   an alcove (affine Weyl chamber) containing x_0 such that   the  set    S_aff chosen in 
<ref> is identified with the set of reflections in the walls of C.  Let    B̃ be the (pointwise) stabilizer of C in G̃.   Then (G̃, B̃, 𝒩̃ ) 
is a Tits system with Weyl group W_aff  and  the morphism ψ : G̃→ G is (B̃ , 𝒩̃ )-adapted.  The action of G on G̃ induced by the natural map 𝐆→Aut(𝐆̃) determines an action of G on ℬ.  The  stabilizer 𝒩⊂ G of the action of G on  𝒜  equals  the   normalizer     N_G(A) of A in G.     If we denote by ν : 𝒩→Aut(𝒜) the canonical morphism, the inverse image of translations coincides with M = 𝐌(F) and 𝒩̃/ M̃ =  𝒩 / M = W.     By the discussion in <ref>,  the quotient G / ψ(G̃)  acts naturally on (W_aff    , S_aff). There is thus an induced    map ξ : G  →Aut(𝒜 ) such that each ξ(g) for g ∈ G sends C to itself.  Let   
    G ^1   :    = { g ∈ G   |  χ(g) = 1  for χ : 𝐆→_m} .
   Then M^∘ = M  ∩  G^1 and ψ(G̃)  ⊂ G ^1.      Let   B ⊂ G^1 be the set of elements that stabilizes C (as a subset of ℬ) and  K ⊂ G^1 the sub-group of elements stabilizing x_0. 
Then B is a Iwahori subgroup of G and K a hyperspecial subgroup. In particular, K =  B w B for w ∈ W.  We will  assume that the group scheme 𝒢 in <ref> is  chosen so that 𝒢 (_F ) = K. 

Finally, let G^0 = G^1∩ξ  and    let    𝒩^0 =   G^0∩𝒩.   Since G 
 = ψ(G̃) M 
 and  ψ(G̃) ⊴  G, we infer that    G^0 =  ψ(G̃) M^∘, B = ψ(B̃) M^∘, 𝒩^0 = ( ψ(G̃) ∩𝒩    ) M^∘  =  ψ(𝒩̃) M^∘.   


It is then elementary to see  (G^0 , B , 𝒩^0 ) is a Tits system with Weyl group W_aff (see   <cit.>)   
and that G^0↪ G is  a  (B,  𝒩 ^0)-adapted   whose  extended  Weyl group is the Iwahori Weyl group W_I.    
One may therefore apply  the  result of Proposition  <ref> to the inclusion G^0→ G to obtain decompositions of double cosets in K_1\ G / K_2 where K_1, K_2⊂ G^0 are subgroups containing B.   

  
If  s ∈ W_aff denotes the reflection in a wall of the alcove, B  / (  B ∩ s B s ) has cardinality q^d(s) for some d(s) ∈ and a set of representatives can be taken in the  F points of the root group U_α where α∈Φ_F is the vector part of the corresponding affine root associated with s. The precise description of d(s) is  given in terms of the root group filtrations and is recorded on the corresponding local index which is the Coxeter diagram of W_aff with additional  data.  When 𝐆 is split, d(s) = 1. We refer to <cit.>
for more details.   
    
   
In the notations of <cit.>, we have  𝐌(F)^1 = 𝐌(F)^0 = M^∘ as  𝐌 is split over an unramified  extension.    
The group G^0 
therefore coincides with <cit.>. That (G^0, B, 𝒩^0 , S_aff ) forms a Tits system is  established  in    Theorem 7.5.3 of op.cit. 
   


In the sequel,  we will  denote the Iwahori subgroup  B ⊂ G by the letter  I. 




 §.§ Decompositions    for  _2
      
Retain the notations  introduced in <ref>. Let χ^∨ = f_1 - f_2 denote the coroot associated with χ and  s = s_χ denote the unique non-trivial element in  W. Let 
    w_0 =  [     1/ϖ;   ϖ     ]  ,       w_1 = [   1; 1   ] ,    ρ  =  [   1; ϖ ]
 
Then w_0, w_1, ρ normalize A and ρ w_0ρ^-1 = w_1, ρ w_1ρ^-1 = w_0.  Under the conventions introduced, the   matrices w_0, w_1 represent the  two simple reflections S_aff = { t(χ^∨ ) s , s } of the affine Weyl group ⟨ f_1 -f_2⟩⋊ W.  The element ρ represents t(-f_2)  s_χ∈Λ⋊ W   =  W_I and is a generator of Ω = W_I / W_aff.  The action of ρ on ℬ preserves the alcove C and permutes the two walls corresponding to w_0, w_1. We say that  ρ induces an  automorphism of the   Coxeter-Dynkin diagram  
  

   [extended, labels = 0, 1, edge length = 1cm]A1   
     


given by switching the two nodes.  
Let I denote the Iwahori subgroup corresponding to the set of affine roots χ and -χ+1 (considered as functions on the space Λ⊗). Then I is the usual Iwahori subgroup of _2(_F) given by matrices that reduce to upper triangular matrices modulo ϖ. Let x_0, x_1  : _a→_2 denote the following `root group'  maps 
    x_0 : u ↦[   1    ; ϖ u   1 ],          x_1 :   u    ↦[ 1 x;   1 ]  .
 
and let [] ⊂_F denote a set of representatives of . Then x_i ( [κ] )   constitute a set of representatives for I / ( I ∩ w_i I w_i ) for i = 0 , 1.   
Let g_ w_i : [] → G be the maps  κ↦  x_i(κ) w_i. For w = s_w, 1⋯ s_w, ℓ(w)ρ_w∈ W_I a reduced word decomposition (where  s_w,i∈ S_aff ,
ρ_w∈Ω =  ρ  ^) such that w is shorter of the two elements in w W,   define    

    𝒳_w : []^ℓ(w)   → G/ K   
    
     (κ_1 ,…, κ_ℓ(w)  )      ↦  g_s_w,1   (    κ_1 ) ⋯ g_s_w, ℓ(w) (κ_ℓ(w))   ρ  _  w          K  .
 
The maps 𝒳_w may be thought of as parameterizing _F lifts of certain    Schubert cells[See <ref> that makes the connection with classical  Schubert cells of Grassmannians   more     precise.] 
and will be referred to as such.    
Proposition  <ref>    provides a decomposition of double cosets K ϖ ^ λ K for λ∈Λ  in terms of these  maps. Let us illustrate  this  decomposition  with a few simple examples.        
          Let λ = f_1. Then K ϖ ^λ K =  K ϖ^λ^opp K = K  ρ K.  Clearly ρ∈ [W \ W_I / W   ] and [ W / (W ∩ρ W ρ^-1 ) ] = W.    The decomposition therefore   reads  
    K ϖ^λ K   /    K   =  im ( 𝒳_ρ ) ⊔im ( 𝒳_w_1ρ   )   .
  
Explicitly,  we have    

    im( 𝒳_ρ ) =  * [ 1  ;   ϖ ] K    and   im(𝒳_w_1ρ )  =  * [ ϖ κ;   1 ]   K κ∈  [      ]     .
   
There are a total of q+1 left cosets contained in K ϖ^λ K.     
 


       Let λ = 2f_2. Then K ϖ ^ λ K   = K w_0ρ ^2 K and w      :     =  w_0ρ^2∈ [W \  W_I / W ] and [ W / W ∩ w W w^-1] = W.    
The decomposition therefore reads 
    K ϖ^λ   K  /   K   = im ( 𝒳_ w        )     ⊔im ( 𝒳 _  w_1  w        ) .
  
Explicitly, we have    

    im(𝒳_  w     )  =  *[   1    ;  κϖ ϖ^2 ]    K   κ∈[ ]     and   im(𝒳_w_1  w  )  = * [          ϖ^2 κ _1ϖ +  κ_2;                         1 ]   K  κ_1 , κ_2∈ [  ]   .

There are q 
( q + 1 ) cosets contained in K ϖ^λ K. 
 Cf.    Example <ref>.     

As seen from the examples, the Schubert cell maps 𝒳_w are recursive in nature and going  from    one Schubert cell to  the    `next'    amounts to applying a reflection operation on rows and adding   a  multiple of one row to another.   We also  note that the actual product of matrices in 𝒳_w in the example above may not necessarily be upper or lower triangular  as displayed e.g.,  with the choices above,  𝒳_w_0ρ^2 ( κ ) =  g_w_0(κ) ρ^2 equals 
    [       1; ϖ^2  κϖ ] .

However, since  we are only interested left K-coset representatives, we  can   replace   𝒳_w_0ρ^2 (κ ) with 𝒳_w_0ρ^2 (κ ) γ for any γ∈ K.  
In general,  multiplying by a reflection matrix on the left has the effect of `jumbling up' the diagonal entries of the matrix.  While performing these computations,  it is desirable to  keep  the `cocharacter'  entries on the diagonal and one may do so  by  applying a corresponding reflection operation on columns  using  elements of K. In the computations done in Part II,   
this will be done without any comment.           

    
In computing 𝒳_w, one can   often      establish certain `rules' specific to the group at hand  that dictate where the entries of the a particular cell are supposed to be written depending on the permutation of λ described by the word. For instance, the rule of filling a Schubert cell      

    [  ϖ^a    □; 0.9◯  ϖ^b ]

as displayed above is as   follows:     
   

  * if a ≥ b, the  0.9◯-entry is zero and the □-entry runs over a set of representatives of ϖ^a_F /  ϖ^b_F 

  * if a < b, then □-entry is zero, and the   0.9◯-entry runs over representatives of ϖ^b_F / ϖ^a+1_F.   
   
    



 §.§ Reduced words
 
Retain the notations introduced   <ref>  and  <ref>. Fix a λ∈Λ  ^  +. The recipe  of   Proposition      <ref> requires writing the reduced decomposition of the word w ∈ W_I of minimal possible length  such that K ϖ^λ K = K w K. This is of course the same for K ϖ^λ^oppK.  We may equivalently think of W_I as t(Λ) ⋊ W  via the morphism (<ref>)    and the length we seek is  the minimal possible length of  elements in  W t ( - λ^opp ) W  ⊂ t(Λ) ⋊ W.  For any μ∈Λ, we denote the  minimal possible length in W t(μ) W 
by 
ℓ_min(t(μ)).    
Let Ψ =  Φ_F^red⊂Φ_F denote the subset of  indivisible roots and let Ψ^+ = Ψ∩Φ_F^+. 
           For any λ∈Λ, the minimal possible length of elements in t(λ) W  ⊂ W_I is achieved by a unique element.  If Φ _    F is irreducible,  the length of this element is given by   
   
    ∑ _ α∈Ψ_λ  | ⟨λ , α⟩ |  +  ∑ _ α∈Ψ^λ  (  ⟨λ , α⟩    - 1  )
 
where Ψ _λ  = {α∈Ψ ^  +     |  ⟨λ , α⟩≤ 0  }, Ψ^λ = {α∈Ψ^+  |  ⟨λ,  α⟩ > 0  }.    
If  λ∈Λ^+, the minimal length in t(λ) W also equals ℓ_min ( t( λ ) ) =  ℓ_min(t (-λ^opp)).  
 
 The first claim holds generally for any  Coxeter group (<ref>). Assume Φ_F is irreducible. It is clear that P_F^∨ = P(Φ_F^∨ ) is the weight lattice associated with the irreducible reduced root system Ψ. By <cit.>,  P^∨_F⋊ W is an extension of the Coxeter group W_aff =  Q^∨_F⋊ W by Ω'  = P^∨_F / Q^∨_F which acts on W_aff by automorphisms. Thus the length function on W_aff can be extended to P^∨⋊ W. Let φ :  Λ⋊ W → P^∨_F⋊ W be the map given by (λ, w) ↦  ( λ̅ , w) where λ̅ = λX_0^∨. The φ factorizes as Λ⋊ W → ( Λ / X_0 ) ⋊ W →  P^∨_F⋊ W. As both maps in this composition are length preserving, we see that φ is length preserving. The  second  claim then follows  by   <cit.>. Since the sum is maximized for dominant λ and is the same for both λ and - λ^opp,  we obtain the last  claim.     
 


 Retain the notation of <ref>.    Let  λ = 5f_1∈Λ^+. Then 
    ℓ_min(t(λ)) = ⟨ 5f_1 , e_1 - e_2⟩ - 1 =  5 - 1 =   4 .
  Say  w ∈ W_I is of length 4 and K ϖ^λ K = K w K. Since ( ϖ^λ ) = 5, we may assume that w = v ρ^5 where v is a word on S_aff = { w_0 ,  w_1}. Now the final letter of v cannot be w_0, since ρ w_0ρ^-1 = w_1∈ K. Thus we may assume that v = v' w_1. Since we can only place w_0 next to w_1 for a reduced word, we see that   the  only  possible  choice  is         w = w_0 w_1 w_0 w_1ρ^3.    
 


 §.§ Weyl orbit diagrams
   
 
Retain the notations introduced in <ref> and   <ref>.    Besides the usual Bruhat order ≥  on the  Weyl group W, there is another partial order that will be useful to us.  We say that w ≽  x for w,x∈ W if there exists a reduced word decomposition for x which appears as a consecutive string on the left of some reduced word for w.   The pair (W, ≽ ) is then a graded lattice <cit.> and is known as the  weak (left) Bruhat order.   
 For λ∈Λ, let W^λ denote the stabilizer of λ in W.  The Weyl orbit diagram of λ is the Hasse diagram on the set of representatives of  W / W^λ of minimal possible length with respect to ≽.  As W / W^λ = W λ,  the nodes of such a diagram  can be labelled by elements of W λ.      
 


Assume  that  Φ_F   is   irreducible.     Let λ∈Λ^+ and let w_λ∈ W_I be the unique element of minimal possible length such that   K ϖ^λ K = K w_λ K.   By Proposition   <ref>,  we see that  w _ λ = ϖ^λ^oppσ_λ for a unique σ_λ∈ W and W ∩ w_λ W  w_λ^-1 is just the   stabilizer of -  λ^opp (equivalently λ^opp)  in W.   So we can make the  identification  
    [ W / ( W ∩  w_λ  W  w_λ^-1   )     ]  ≃  [ W / W ^λ^opp   ]   .

Thus the decomposition of K ϖ^λ K / K as described by   Proposition <ref> can   be viewed as a collection of Schubert cells 𝒳_μ, one for each node μ∈ W λ^opp = W λ of the Weyl orbit diagram of λ (though note that 𝒳_μ  is an abuse of notation). See the proof of Proposition <ref> which illustrates this  point.   



In the following, we adapt the convention of drawing the Weyl orbit diagrams of λ∈Λ^+ from left to right, starting from the anti-dominant cocharacter λ^opp  and ending in λ. The permutation of λ  corresponding to the node then `appears' in the matrices of the corresponding  Schubert cell.       For example, in the notations of  <ref>,  the Weyl orbit diagram of f_1  is 
    f_2  f_1
  
and the matrices in im ( 𝒳_ρ ), im  (   𝒳_w_1ρ   )  in Example  <ref>  have  `diagonal entries'  given by ϖ^f_2, ϖ^f_1   respectively.    We will often  omit the explicit cocharacters on the nodes  in these diagrams and only display the labels of the arrows. See also <cit.>.     

  Observe  that  the shape (in the sense of Definition <ref>) of the matrices in these cells  may not match the corresponding cocharacter. In Example    <ref>, the shape of the matrices that appear in the decomposition of K ϖ^2f_1 K  can be 2f_1, 2f_2  or f_1+f_2   when converted to upper  triangular   matrices.       
   




 §.§ Miscellaneous  results
      
In this subsection, we record  assortment of results that are useful in determining the structure of mixed double cosets in practice.    




    Suppose G is a group, X , Y ⊂ G are subgroups. Then for σ ,  τ∈ G,  X σ  Y  = X  τ  Y  only if X ∩σ  Y σ  ^-1 and X  ∩τ   Y   τ^-1 and X-conjugate.   

 X  σ   Y   = X  τ  Y  ⟺σ = x τ   y for x ∈ X, y  ∈  Y      X  ∩σ Y   σ   ^-1 =  x   ( X  ∩τ  Y  τ    ^-1 ) x ^-1.  
    

            Let ι :  H   ↪  G be an inclusion of groups, K ⊂ G a subgroup and U = K ∩ H. Then for any h_1 ,h_2∈ H,  g ∈ G,  Uh_1 g K = U h_2 g  K if and only if U h_1 H_g =  U h_2 H_g where H_g denotes H  ∩  g K  g^-1. Moreover  for any h ∈ H,  the index [ H_hg : U  ∩   hg   K  ( hg ) ^-1 ] is equal to [H_g : H_g∩  hU h^-1 ].      
    


  The map (of sets) H  ↠  H g K / K,  h ↦  hgK induces a H-equivariant bijection H / H_g H g K/ K where H acts by left multiplication.  Thus the orbits of U on the two coset spaces  are identified i.e.,  U  \ H /  H_g  U    \ H g K / K which proves the first claim.  For any h ∈ H, H_hg = h H_g h^-1 and H_g∩  hUh^-1 = h ( U ∩  g K g )h^-1 which proves the second  claim. 
 

The  next   result is  helpful  in  describing   the   structure of double cosets associated with  certain non-parahoric  subgroups.  It is needed in <cit.>.   
                 Let H be a group, σ∈ H an element and  U, U_1 ,X be subgroups of H such that U_1σ U / U, X U_1  / U_1 are finite sets and U_2 = XU_1 is a group.    Then  U_2σ U / U is finite and  
    e · (  U_2σ  U ) =   ∑        _δ ( δ   U_1σ U )
 
where (Y) : H → denotes the characteristic of Y ⊂   H, δ∈ X run over  representatives of X / ( X  ∩  U_1   )  and      e   =    [U_2∩σ U σ^-1 : U_1∩σ  U  σ ^-1 ]. If U_2∩σ U σ^-1 is equal to the product of    X  ∩σ  U  σ ^-1 and U_1∩σ U σ ^-1,    then e = [ X ∩σ U  σ ^-1 : X ∩  U_1∩σ U σ ^-1 ].  
    


 Let W_i   :  = U_i∩σ  U  σ  ^-1 for i = 1, 2,   Z  : =  X ∩   U_1 and let  γ_1, …, γ_m∈ U_1  be representatives of U_1 / W_1,  δ_1 ,  ⋯,   δ_n∈ X be representatives of X / Z.    We first show that δ_jγ_i  form a complete set of  distinct representatives of the coset space U_2 / W_1. Let x ∈ X, u ∈ U_1. Then there exists a z ∈ Z, w   ∈  W _1 and (necessarily unique)     integers  i, j such that xz  = δ_j, z^-1 u  w     = γ_i. In other words, x  u  W_1  =  ( x z) (z^-1 u w ) W_1 =  δ_jγ_i W_1.   Therefore, every element of U_2 / W  _     1 is of the form δ_jγ_i W   _  1 and so   
    U _2    =  ⋃_j = 1  ^ n ⋃ _  i = 1  ^  m δ_jγ_i W_1

We claim that this  union is disjoint. Suppose x, y  ∈ X, u , v ∈ U_1 are such that xu  W_1  =  y v  W_1.    Then v^-1 y^-1  x u ∈ W _1. Since U_2 is a group containing both v^-1∈  U_1 and  y^-1   x   ∈     X, v^-1 y ^-1  x ∈ U_2. Since  U_2 is equal to X ·  U_1,  there exists     x_1∈ X, u_1∈  U_1 such that v^-1 y^-1 x = x_1 u  _ 1  or  equivalently, y^-1  x  = v x_1 u _1.      Now   
    v^-1 y^-1 x u ∈ W   _  1      x_1 u_1 u ∈ W   _  1 ⊂ U_1
        x_1∈ U_1
          y^-1 x    =   v  x_1  u_1∈  U_1    x Z = y Z
    
Thus if x , y are distinct modulo Z, xu W _ 1, y v W _1 are distinct left W_1-cosets for  any u, v ∈ U_1. Thus, in the union above, different j  correspond to necessarily distinct   
W_1-cosets. It is clear that δ_jγ_i_1  W _  1   = δ_jγ_i_2  W_1 iff i_1 = i_2.   Thus the  union above is disjoint as both δ_j and γ_i vary.   

Now  we prove the first  claim.   Let  p : U_2 / W_1→ U_2 / W_2 be the natural  projection  map. Since U_2 / W_1 is finite, so is    U_2 / W_2 and therefore U_2σ U / U. Moreover, as W_2 / W_1↪ U_2 / W_1, e = [ W_2 :  W_1 ] is   finite.   Let   y =        a W_2∈ U_2 / W_2 be a W_2-coset of U_2.  Then p^-1( y) = { a w W_1 | w ∈ W_2}  and  we have  
    | p^-1(y) | = p^-1(W_2) = [  W_2 : W_1  ]    =   e .
   Thus  in     the list of  mn left  W_2-cosets given by   δ_1γ_1 W_2, δ_1γ_2 W_2  , … ,    δ_nγ_m  W_2,     each element of U_2 / W_2 appears exactly e times. Equivalently, among the mn left  U-cosets δ_1γ_1σ U,  δ_1γ_2σ U , … ,   δ _  m γ _ n σ   U,   
each element of U_2σ U/U appears exactly e times. Since U_1σ U =  _  i = 1  ^ m γ_iσ U, we see that  
    e ·(U_2σ U ) = ∑  _ i ,j  (  δ_jγ_iσ U )  = ∑ _j ( δ_j U_1σ U )
    
and the   first   claim  is  proved.  The second claim follows since W_2 = (X ∩σ U  σ^-1) W_1 implies that  W_2/ W_1 = (  X ∩σ U σ^-1 ) W_1 / W_1 = ( X ∩σ U σ ^-1 ) / (  X ∩ W_1 ).   





   
Part 2. Examples
 
    


§ ARITHMETIC CONSIDERATIONS
   

In this section, we record two embeddings of Shimura-Deligne  varieties   that are of arithmetic interest  from the perspective of Euler systems. Our goal here is only to motivate the local zeta element problems  arising from these scenarios, cast them in the axiomatic framework of <ref> and justify  various  choices of data in order to align these problems with the actual arithmetic situation.  In particular, we will  make no attempt to study the arithmetic implications of these problems.   
In the sections that follow, we solve the resulting combinatorial problems using techniques developed in Part I.  These examples are meant to test our machinery in situations where the computations are relatively straightforward in comparison to, for instance, <cit.>. For a concrete  arithmetic application of  such  combinatorial  results  to  Euler system constructions,     
 we refer the reader to <cit.>.  





 §.§ Unitary Shimura varieties
 
Let E ⊂ be an imaginary quadratic number field  and  γ∈(E/ ) denote the non-trivial automorphism. Let J = diag(
1,…, 1
, 
-1,…,-1
) be the diagonal matrix  where there number of 1's is p and the number of -1's is q.  Clearly γ(J)^t  = J i.e.,    J is E /-hermitian. Let  GU_p,q denote the algebraic group over  whose R points for a -algebra R are given by 
    GU_p,q(R)  : =  { g ∈_p,q(R)   |  γ(g)^tJ γ(g) = sim(g) J   for some sim(g) ∈ R^×} .

The  resulting map 
sim : GU_p,q→_m is a character called the similitude. 
 Let 
    h :     →𝐆_
        
    z       ↦diag(
    z,…,z
    , 
    z̅, …, z̅
    )
 
and  let 𝒳 be the 𝐆()-conjugacy class of h. Then (𝐆, X) constitutes a Shimura-Deligne data   that satisfies (SD3) if p, q ≠ 0   (see   <cit.> for terminology).        The dimension of the associated Shimura varieties    is     p q.  There is an identification 𝐆_E≃_m,E×_p+q, E induced by the isomorphism of E-algebras E ⊗ R ≃ R^×× R^×, (e,r) ↦ (er , γ(e)r) for any E-algebra R. The cocharacter μ_h : _m, →_m, ×_p+q, associated with h is given by z ↦   ( z ,  diag(
z,…,z
,
1,…,1
)   ). The reflex field is then easily seen to be E if p ≠ q and  otherwise.   


          


For m ≥ 1 an integer, let : = GU_1,2m-1. Then the so-called arithmetic middle degree[one plus the dimension of the variety] of the Shimura varieties  of  is 2m. Thus one  construct classes in this degree   by  taking   pushforwards   of      special cycles of codimension m. One such choice is given by the  fundamental cycles of Shimura varieties of  
    𝐇 : = GU_1,m-1   × _ _mGU_0,m.
 
where the fiber product is over the similitude map.    There is  
a natural embedding 𝐇↪ which constitutes a  morphism of SD data and gives an embedding of    varieties is  over E.    
We note that μ_h for  corresponds to the representation of ^L𝐆_E which is trivial on the factor 𝒲_E and  which  is the standard representation on = _m×_n. Thus at a choice of a split prime  λ of E  above ℓ, we are interested in the Hecke polynomial  of the standard representation of _n×_m.  This case is studied <ref>.  When ℓ is inert, we are interested in the base change    of the standard L-factor (Remark  <ref>).  
This setup is the studied  <ref> for the case m = 2.   As  we  are  pushing  fundamental  cycles of the  Shimura varieties of 𝐇, we are led to consider the trivial functor that models the distribution relations of these cycles. See <cit.> for a description of the relevant Galois representations which the resulting  norm relations are geared towards.   


To construct classes that go up a tower of number fields, we need to specify a choice of torus 𝐓 and a map ν : 𝐇→𝐓, so that the Shimura set associated with 𝐓 corresponds to non-trivial abelian extensions of the base field E. We can    then construct classes in towers by considering the diagonal embedding ↪×. 
One such choice is 𝐓 :=  𝐔_1, the torus of norm one elements in E. It is considered as a quotient of 𝐇 via 
    ν :  𝐇→𝐓        (h_1, h_2)  ↦ h_2 /  h_1     .

The extensions determined by the associated reciprocity law are anticyclotomic i.e.,  the natural action of (E/) on them is by inversion. The behaviour of arithmetic Frobenius Frob_λ at a prime λ of E in an unramified extensions contained in such towers is rather  special. Let ℓ be  the rational prime of  below λ. When ℓ is split, we denote by λ̅ the other prime above ℓ. Then Frob_λ is trivial if ℓ is inert and Frob_λ = Frob_λ̅^-1 if ℓ is split. If ℓ is split, the choice of λ above ℓ allows us to pick  identifications 𝐇__ℓ≃_m×_m×_m and 𝐓__ℓ≃_m, so that ν is identified with the map (c, h_1, h_2) ↦ h_2 /  h_1. With these conventions, the  induced map ν∘μ_h sends the uniformizer  at λ in ∈ E_λ ^×  to   1 ∈𝐓(_ℓ) if ℓ is inert and to ℓ^-1∈_ℓ≃𝐓(_ℓ) if ℓ is spilt.    The group 𝐓(_ℓ) has  a    compact open subgroup of index ℓ +1 (resp., ℓ-1) if ℓ is inert (resp., split). These groups provide the `layer extensions'   for our zeta element problem.  






 The choice of ν is made to match that in <cit.>. One equivalently work with ν' that sends (h_1, h_2) ↦ h_1 / h_2 in which case λ is sent to ℓ∈(_ℓ) for ℓ split. The Shimura varieties we have written also admit certain CM versions, and the local zeta element problem studied in <ref> apply to these more general versions    too.   





  That the resulting Euler system is 
non-trivial is the subject of a forthcoming work. This particular embedding of Shimura varieties is motivated by a unitary analogue of the period integral of Friedberg-Jacquet <cit.>, <cit.>.     A first step towards interpolating these  periods and the construction of a suitable p-adic L-function is taken in <cit.>, <cit.>.    
  



 The inert case of the situation above studied in <ref>  also   serves as a precursor for a   slightly 
 more involved calculation performed in <cit.> for the twisted exterior square representation.   
 




 §.§ Symplectic  threefolds
    
  
Let 𝐆 : = GSp_4 
and 𝐇 = _2×__m_2    where the fiber product is over the determinant map. We have an embedding ι : ↪ 
obtained by considering the automorphisms of the two  orthogonal  
sub-spaces of the standard symplectic vector space V spanned  by e_1 , e_3 and e_2, e_4 where e_i are the standard bases vectors.  Let    
    h :  →𝐆_     
    (a+b √(-1)) ↦[  a     b;     a     b; -b     a;    -b     a ].
   
Note that h factors through ι. Let  𝒳_ (resp., 𝒳_)    denote 
the () (resp.,()) conjugacy class of h. Then (_ ,  𝒳_),  (𝐆_, 𝒳_)  satisfies axioms SV1-SV6 of <cit.> and in particular,  constitutes a Shimura data. These  Shimura varieties are respectively the fibered product of two modular curves and the Siegel modular threefold that parametrizes abelian surfaces with polarization and  certain level structures.  The reflex fields of both of these varieties is  . The cocharacter μ_h associated to h  corresponds to the four dimensional spin representation of ^LGSp_4 = GSpin_5×𝒲_ and we are thus interested in establishing norm relations involving the Hecke polynomial associated to the spinor representation. See <cit.> for a description of the relevant four dimensional  Galois representations to which such norm relations are geared  towards.   


As the codimension of the two families of Shimura varieties is 1, one needs to push classes from H^2_ of the source variety to be able construct classes in arithmetic middle degree of the target Shimura variety.    As first proposed by Lemma in <cit.>, one can take (integral) linear combinations of the cup products of two Eisenstein classes in the H^1 _ of each modular curve for this purpose. The distribution relations of such cup products  can then be modelled via the tensor product of two CoMack functors associated to Schwartz spaces of functions on 2 × 1 adelic column vectors minus the origin. This tensor product is then itself    a   Schwartz  space over a four dimensional adelic vector space (minus two planes that avoid the origin) which then becomes  our (global) source functor. 
The local source bottom class (<ref>) is then  the such a  characteristic  function.  
  Apriori, one can only define Eisenstein classes integrally by taking integral linear combinations of torsion sections determined by the level structure of the modular curves.  The main result of <cit.> upgrades this association to all integral Schwartz functions, which justifies  our use of these function spaces as source functors for the zeta element  problem.     

To construct classes in a tower, we  can consider the torus 𝐓 = _m which admits a map ν : 𝐇→𝐓 given by sending a pair of matrices to their  common  determinant. 
As above, we consider the embedding 𝐇↪×𝐓 which in this case also factors through 𝐇↪.  With this choice, the map induced by μ_h : _m→ is identity i.e., locally at a prime ℓ, the pullback action of ℓ∈(_ℓ)  corresponds to the action of geometric Frobenius. Then   1 + ℓ_ℓ⊂(_ℓ)  provides us with a  `layer extension' of degree ℓ - 1.   Under  the  reciprocity  law  for  , these layer extensions correspond to the ray class extensions of  of degree ℓ - 1.  
  Although the zeta element problem is only of interest over _ℓ, we have chosen to work with an arbitrary local field for consistency of notation.  
  
  The question that this  construction  leads  to   a non-trivial   Euler system is  addressed  
in <cit.>. 
    


 As the arithmetic middle degree is even, one may ask if interesting classes can be constructed in this degree via special cycles. Such a setup was proposed in <cit.> which  allows one to construct  classes  over an imaginary quadratic field.  It would be interesting to see if this  construction indeed sees the behaviour of an  L-function.  
 



 
















§ STANDARD -FACTOR OF  
 



In this section,   we study the  zeta  element problem for the split case of the embedding discussed in <ref>. 
    
 The symbols F, 𝒪_F, ϖ, , q and []  have the same meaning as in   Notation <ref>. 
The letter 𝐆 will denote the group scheme 𝔾_m×GL_n over _F where n is a positive integer and is assumed to be even    from   <ref>     onwards.  We will denote G : =  𝐆  (F) and  K : =  𝐆(𝒪_F). For a ring R, we let ℋ_R  =    ℋ_R(K \ G / K) denote the Hecke algebra of G of level K with coefficients in R with respect to a Haar measure μ_G such that μ_G(K)=1.    For simplicity, we will often  denote  (K σ K )  ∈ℋ_R   simply       as (K σ K ).    
 





 §.§ Desiderata
                
Let 𝐀 = _m^n+1 and dis :   𝐀→𝐆 be the embedding given by 
    (u_0, u_1, …, u_n ) ↦ ( u_0,  diag( u_1 ,  … ,  u_n  )    ) .

Then dis
identifies 𝐀 with a maximal    torus in 𝐆. We denote A : = 𝐀(F) the F-points of A  and A ^∘  :  = A ∩  K the unique maximal compact subgroup.  For i = 0, … n, let   e_i : 𝐀→_m be the projection on the i-th component  and   f_i : _m→𝐀 be the cocharacter inserting u in the i-th component of 𝐀. We will denote by Λ the cocharacter lattice f_0⊕⋯⊕ f_n. The   element a_0 f_0  + … + a_n f_n∈Λ will  also  be  denoted as (a_0, …, a_n ).  The set Φ⊂ X^*(𝐀 ) of roots of 𝐆 are  ± (   e_i - e_j  ) for 1 ≤ i < j ≤  n      
which constitutes an irreducible root system of type A_n-1. We let Δ  =  {α_1, …, α_n-1}⊂Φ where 
    α_1 = e_1 - e_2,     α_2 = e_2 - e_3,     …,      α_n =  e_n-1 - e_n .
Then Δ  constitutes a base for Φ. We let Φ ^  + ⊂Φ denote the set of  resulting positive roots.   The  half  sum  of  positive  roots  is  then

    δ : = 1/2∑_k=1^n (n-2k+1) e_k
  
With respect to the ordering induced by Δ, the  highest root is α_0 = e_1 - e_2n. We let I = I_G  be the standard Iwahori subgroup  of G, which  corresponds to the alcove determined by the simple affine roots α_1  + 0, α_2 + 0, …,  α_n-1   +       0, - α_0 + 1.    The coroots corresponding to α_i are  
    α_0  ^ ∨ =  f_1 - f_n    ,    α_1^∨ = f_1 -  f_2,     α_2^∨ =   f_2  - f_3,      …,     α_n-1^∨ =  f_n-1 - f_n
   and their  span in Λ is denoted by Q ^∨.   An  element    λ = (a_0 , …, a_n )  ∈Λ is     dominant iff a_1≥ a_2≥…≥ a_n and anti-dominant if all these inequalities hold in reverse. We denote the set  of dominant cocharacters by Λ^+. The translation action of λ∈Λ on Λ⊗ via x ↦ x + λ is denoted  by t(λ). We denote ϖ^λ∈ A the element λ(ϖ ) for λ∈Λ and      v : A / A ^∘→Λ be the inverse of the map Λ→ A / A ^∘, λ↦ϖ^-λ A^∘.  Let s_i be the reflection associated with  α_i for i = 0, …,  n.  The action of s_i on Λ is given explicitly  as  follows:          
         

  * s_i acts by the transposition f_i↔ f_i+1 for i = 1,2, …, n   -  1 


  * s_0 acts by transposition  f_1↔  f_n. 

For λ∈Λ, we let e ^ λ∈ [ Λ ] denote the element corresponding to λ and e ^  W λ∈ [ Λ ] denote the element obtained by   taking     the formal  sum of elements in the orbit W λ.
Let  S_aff = { s_1 , s_2 , … , s_n-1     ,      t ( α _   0   ^ ∨    )    s_0} and W, W_aff, W_I be the Weyl, affine Weyl and Iwahori Weyl groups respectively determined by A.  We consider W_aff as a subgroup of affine  transformations of Λ⊗.  We have   
       

  * W  =   ⟨  s_1  , …, s_n-1⟩≅   S_n-1,

  * W_aff = t(Q^∨ ) ⋊ W

  * W_I =   N_G(A) / A^∘ =  A/A^∘⋊ W  v≃Λ⋊ W
 
where v is the map (<ref>).    
The pair (W_aff, S_aff )  forms a  Coxeter system of type Ã_n-1.   We   consider  W_aff    a   subgroup of  W_I via  W_aff≃ Q^∨⋊ W  ↪Λ⋊ W   v≃    W_I.  
The natural  action of W_aff on Λ⊗ then extends to W_I with λ∈Λ acting as a translation  t(λ).  We set    Ω : = W_I /  W_aff,   which is a free abelian group on two generators  and we  have W_I≅  W_aff⋊Ω.   
We  let   ℓ :  W_I→ denote the induced length function with respect S_aff. Given λ∈Λ, the minimal length ℓ_min(t(λ)) of elements in the coset t  ( λ ) W is  achieved by a unique element. This length can be  computed using Lemma <ref>.    We  let      
    w_1 : = 
        0.9[ 0 1      ; 1 0        ;     1      ;       ⋱    ;         1  ;           1 ],      
    w_2   :    =    0.9[ 1        ;   0 1      ;   1 0      ;       1    ;         ⋱  ;           1 ],   … ,      w_n-1  : =  0.9[ 1          ;   1        ;     ⋱      ;       1    ;         0 1;         1 0 ],
    

    w_    0      : = 0.9[       0                                 1.091/ϖ;               1                ;                       1;                               ⋱;                                       1;       ϖ                                       0 ],   ρ   =  
    0.9[ 0 1        ;   0 1      ;     0 1  ;       ⋱ ⋱  ;         0 1; ϖ         0 ]
 
which we consider as elements   of N_G(A) (the normalizer of A in G) whose component in _m is 1.  The   classes of w_0 , w_1 , …, w_n-1 in W_I   represent t(α_0 ^∨ ) s_0,   s_1 , … , s_n-1 respectively and    the class of  ρ  is  a   generator of  Ω / ⟨ t( f_0 ) ⟩.    The reflection s_0 in α_0 is  then  represented  by  w_α_0 :  =  ϖ^f_1  w_0.    We will   henceforth    use the   letters    w_i, ρ to denote both the matrices and the their classes in W_I if no confusion can arise. We note that  conjugation by ρ on W   _  I  acts by  cycling  the (classes of) generators via  w_n-1→ w_n-2→…→ w_1→ w_0→  w_n-1, thereby  inducing  an   automorphism of the extended   Coxeter-Dynkin  diagram  
 
    [extended,Coxeter,
    edge length= 1cm,
    labels=,1,2,n-2,n-1, labels*=0]
    A[1]

where the labels below the vertices correspond to the index of w_i. Note also that ρ^n = ϖ^(1,1,…,1)∈ A is central. 
For i = 0,1, …, n - 1, let 
x_i : _a→𝐆 be the root group maps defined by  

    x_1    : u  ↦0.9[ 1 u    ;   1      ;     ⋱    ;       1  ;         1 ],   x_2   : u   ↦0.9[ 1        ;   1 u    ;     1    ;       ⋱  ;         1;   ],
       …,      
    x_n-1 :  u  ↦0.9[ 1        ;   1      ;     ⋱    ;       1 u;         1;   ]
    , 
     
     x_0     :   u ↦0.9[   1                ;       1            ;           ⋱        ;               1    ; ϖ u               1;     ]

where again  the matrices are considered as elements of  with 1 in the _m component.  Let  g_w_i  : [  ] → G be the maps κ↦ x_i(κ ) w_i. Then I w_i I = _κ∈ []  g_w_i(κ) I.  For w ∈  W_I such that w is the unique minimal length element in the   coset   w W,  choose a reduced word decomposition  w = s_w,1 s_w,2⋯ s_w, ℓ(w) ρ_w  where s_w,i∈  S_aff, ρ_w∈Ω. Define  

    𝒳_w :  [   ] ^ ℓ(w)    →  G / K 
     
     (κ_1 ,   …,   κ_ℓ(w)  )   ↦    g_s_w,1 ( κ_1 )  ⋯  g_s_w, ℓ(w)  ( κ_ℓ (w)    )   ρ_w  K

where we have suppressed the dependence on the  decomposition chosen in the notation.   By   Theorem  
<ref>,  the    image of 𝒳_w is independent of the choice of   decomposition and    # im ( 𝒳_w )    =  q ^ ℓ(w). 
We  note that   ℓ(w) =    ℓ_min( t(-λ_w )  ) where λ_w∈Λ is the   unique     cocharacter   such that w K  =  ϖ^λ_w K.  
 Cf. the matrices in  <cit.>. 






 §.§ Standard Hecke  polynomial
 
     Let ℛ=ℛ_q denote the ring ℤ [ q^±1/2  ] and let y_i:=e^f_i∈ℛ[Λ] the element corresponding to f_i. Then ℛ[Λ]=ℛ[y_0^±, ⋯, y_n^±].   We are interested in the characteristic polynomial of the standard representation of the dual group 𝐆_F =   𝔾_m×GL_n whose highest  coweights are μ_std  =f_0+f_1. Note that μ_std  is the cocharacter obtained from the  Shimura     data in <ref>.  
Since μ_std is minuscule, 
the (co)weights of the associated representation are the elements in the Weyl orbit of μ_std.  These are f_0+f_1, f_0+f_2 , … , f_0 + f_n.     
The Satake polynomial (see Definition <ref>) for μ_std is therefore

    𝔖_std(X)=(1-y_0 y_1 X)(1-y_0 y_2 X) ⋯ (  1  -   y_0  y_n X        )    ∈ [ Λ ] ^  W  [X]
 
As in <ref>, we let 𝒮: ℋ_ℛ→ℛ[Λ]^W denote the Satake isomorphism.
 The polynomial ℌ_std, c(X) ∈ℋ_ℛ[ X] is defined so that 𝒮(ℌ_std, c(X))=𝔖_std(q^-c/2 X)
for any c ∈ℤ. 





        Let  ϱ = ϖ^f_0ρ∈ N_G(A).     Then  


    ℌ_std, c(X)= ∑_k=0^n(-1)^k q^-k(n-k+c)/2(K ϱ^k K) X^k .

In particular if n is even and c is odd, ℌ_std,c(X) ∈ℋ_[q^-1][X].  



 
 Let  p_k =  p_k(y_1, …, y_n) ∈[Λ]^W denote the k-th elementary symmetric polynomial in y_1 , …, y _n. 
Then 𝔖_std(X) = ∑_k=0^n (-1)^k  x_0^k  p_k   X^k. So it suffices to  establish that 
    𝒮 ( K ϱ^k  K ) =  q^k(n-k)/2 x_0^k  p_k  .

For k ≥ 1,  set μ _k : = f_0 + f_1 + … + f_k∈Λ   ^  +. Then,  K   ϱ ^k    K     = K ϖ^μ_ k K as double cosets. But  μ  _k are  themselves minuscule. Therefore,   Corollary     <ref>   and the second part of Corollary <ref>  together  imply   that  𝒮 ( K ϖ^μ _  k     K ) is supported on x_0^k p_k and  that  the coefficient of x_0 p_k is q^⟨μ _  k    ,   δ⟩ where δ is as in (<ref>).  One  easily calculates that   ⟨μ _k, δ⟩ = k(n-k)/2.      
    


        
 
 The formula for ℌ_std,c  was first obtained by Tamagawa <cit.> and the case n = 2 is due to Hecke <cit.>,   hence the  terminology `Hecke polynomial' – see the  note at the bottom of <cit.> and  the historical commentary in 4, 8 of  <cit.>.   Cf.  <cit.>. 
  



 An alternate proof of Proposition  <ref>   that does not use  Corollary      <ref> may be obtained using the decomposition of K ρ ^  k   K  described in  Proposition  <ref> which is closer in  spirit to the proof by  Tamagawa.     
  






 §.§ Decomposition of minuscule operators
               In this section, we  study the decomposition of Hecke operators K   ϱ   ^   k        
K for k ∈{ 1, …, n } into individual  left cosets. Here  ϱ  = ϖ^f_0ρ as above.   Since  (ϖ^k,1) ∈ G is central, it suffices to describe the decomposition K  ρ ^k    K, so that  the left coset  representatives γ  will have 1 in the _m-component.  




    

 Let k be an integer satisfying 1  ≤  k ≤ n. A Schubert symbol of length k   is a k-element subset 𝐣 of [n] : = { 1 , …, n }. We write the elements of 𝐣 = { j_1, …, j_k} such that j_1 < ⋯ < j_n.   
The   dimension of 𝐣 is defined to be 𝐣 = j_1 +  … + j_k - k+12.    The set of Schubert symbols of length k is denoted by J _  k.  We have | J _k |  = nk.  
 
We define a  partial order ≼ on J_k by declaring 𝐣≼𝐣' for    symbols   𝐣 = { j_1, …, j_k}, 𝐣' = { j_1', …, j_k' }  if j_i≤ j_i' for all i = 1 , …, k.  Then (J_k, ≼) is a   lattice (in the sense of order theory). The smallest 
and the largest elements of J_k are {1, …, k} and {n-k+1, …,  n  }   respectively. We assign a grading to J_k so that the smallest element has length is 0.          
 

  For  𝐣∈ J_k,  the   Schubert cell 𝒞_𝐣     is the finite subset of Mat_n × k ( F   ) consisting of all n × k matrices C such that   
          
 
  * M has 1 in (  j_i , i )-entry, which are referred to as  pivots. 
 
  * the entries of M that are  below or to the right of a pivot are zero, 
 
  * M has entries in [] ⊂_F elsewhere. 
    
 Then | 𝒞_𝐣 | =  q ^ 𝐣.  Given C ∈𝒞_𝐣, we let φ_𝐣(C) ∈_n(_F) be the n × n matrix obtained by  inserting the i-th  column of 𝒞_𝐣 in the j_i-th column of φ_𝐣(C),  making the rest of the diagonal entries ϖ and inserting zeros elsewhere.  
 
 
 
 
We let 𝒳_𝐣⊂_n(F) denote the image of φ_𝐣(𝒞_𝐣)
and consider 𝒳_𝐣⊂ G by taking 1 in the _m-component.              
   




         Let n =   4, , k   = 2.  Then the Schubert cells are    
    𝒞_{ 1 ,  2 }  =    0.9[ 1  ;   1;    ;     ],      𝒞_ { 1,3  }  =   0.9[ 1  ;   *;   1;     ] ,       𝒞_{2,3 }  =       0.9[ * *; 1  ;   1;     ]


    𝒞  _ {  1 , 4 }     =       0.9[ 1  ;   *;   *;   1 ],       𝒞_ { 2 , 4 }  =    0.9[ * *; 1  ;   *;   1 ] ,        𝒞  _ { 3,  4   }     =   0.9[ * *;   *; 1  ;   1 ]
       
where the star entries are elements of [] and zeros are omitted.  The corresponding   collections 𝒳_ 𝐣     are   

    𝒳_{1,2}
    =0.9[ 1      ;   1    ;     ϖ  ;       ϖ ], 
       𝒳_{1,3}
    = 
      0.9[ 1      ;   ϖ *  ;     1  ;       ϖ ],  
       𝒳_{2,3 }
    =      
      0.9[ ϖ * *;   1  ;     1  ;       ϖ ],
     

    𝒳_{1,4}     
    =   
      0.9[ 1      ;   ϖ   *;     ϖ *;       1 ],   
       𝒳_{2, 4}   
    =   
      0.9[ ϖ *   *;   1  ;     ϖ *;       1 ], 
       𝒳_{3,4}     
    =      
      0.9[ ϖ   * *;   ϖ * *;     1  ;       1 ]
   We have a total of   1 +  q +  q ^  2     +  q ^   2     +   q ^   3    +  q ^    4 matrices  in  these  six  sets.     




   
     For 1 ≤ k ≤ n, K ρ^k K =   _ 𝐣 ∈ J_k _ γ∈𝒳_𝐣γ K .   
    




    Let λ _  k  =    ∑_i=1^k f_ n - k+i∈Λ ^-. We   have        ρ^k W ρ^-k =  ⟨  S_aff∖  w_n -  k⟩ and therefore 
 W ∩ρ ^k W ρ^-k =    Stab_W (  λ _  k  ).   
By Theorem     <ref>, 
    Kρ^k K =  _  w ∈ [ W  / W  ^λ_k ] im (  𝒳_w ρ   ^  k   )  .

where W^λ_k :=  Stab_W(λ_k) and [ W / W^λ_k ] denotes the set of representatives in W of W / W^λ_k of minimal  possible length.  
For λ∈ W λ_k, let 𝐣(λ )  ∈  J_n-k be the  Schubert symbol consisting of integers 1 ≤  j_1 <  … < j_n-k≤ n such that the coefficient f_j_i in λ is   0. If w ∈ [W / W ^λ_k    ] and λ = w  λ_k ∈ W  λ_k, we let 𝐣(w) : =  𝐣( w  λ_k ). We let ≼ denote the left (weak) Bruhat order on W with respect to S. Then (W,  ≼) is a graded lattice with grading given by length.     
      

Claim 1. 
The   map w ↦𝐣(w) sets up an order preserving  bijection   [W / W^λ_k ]   J_n-k. 
       
The set W / W ^λ_k is in one-to-one correspondence with the orbit W λ_k⊂Λ. The orbit consists of the nk permutations of the cocharacter λ_k = f_n-k+1 +  ⋯ + f_n. Picking a permutation of λ_k in turn is the same  thing as choosing n-k integers 1 ≤  j_1 <  … <  j_n-k≤ n such that f_j_1 , …, f_j_n-k have coefficient zero in the permutation of λ_k. This establishes the bijectivity of w ↦𝐣(w).  The identity element is mapped to {1 , …, k } and one establishes by induction on the length that the mapping preserves the orders.     




Claim 2. For all  w ∈ [W / W  ^  λ _  k  ],  
im ( 𝒳_w ρ )  =  {γ K   |  γ∈𝒳_𝐣(w)    }.   
   
We proceed by induction on the the length of w. If w is of length 0, then w is the identity element  and 𝐣 =  𝐣_λ_k  =      { 1, …, n-k }.  Now  im(𝒳_ρ^k ) = {ρ  ^  k       K } is a singleton and  𝒳_𝐣 = ϖ^λ_k K. As ρ  ^  k       K  =  ϖ ^ λ_k  K, the base case holds. Now suppose that the claim holds for all w ∈ [W / W ^ λ _k ] of length m.  Let v=  s w where s ∈{ s_1 , … , s_n-1}, w ∈ [W / W ^λ_k] such that ℓ(v)  = ℓ(w) + 1 and ℓ(w) = m.  Let 𝐣_v, 𝐣_w be the Schubert symbols corresponding to v, w respectively. By Claim 1,    
there  exists a unique  j ∈{ 1, …, n-1 } such that i ∈𝐣_w, j +1 ∈𝐣_v and 𝐣_w∖{ j + 1  } = 𝐣_v∖{ j  }. If σ K  ∈𝒳_ w ρ, then σ  K =  φ_𝐣(w)(C) K for some C ∈𝒞_𝐣_w by  by induction hypothesis.    Denote τ    := φ_𝐣(w)(C).  By definition, τ ( j , j   )    = 1, τ ( j+1,j+1 ) = ϖ and τ ( j,j_1 ) = τ ( j_2, j  ) =  τ ( j+1, j_3 )  =  0 for j_1 , j_2 > j, j_3≠ j + 1. 

    τ =   
    [name = matrix]  
    ⋱
             
            1      0    ⋯     0 
           0     ϖ
          ⋮      
            0           ⋱[decorate, decoration= calligraphic brace,transform canvas = yshift=-2em, thick] (row-6-|col-4.east) – node[midway, below=1pt]0.8j (row-6-|col-3.west)   ;
       [decorate, decoration= calligraphic brace,transform canvas = yshift=0em,xshift=2.7em, thick] (row-3-|col-6) – node[midway, right=1pt]0.8j (row-4-|col-6)   ;

    
Then g_w_j(κ)  τ K = x_j(κ) w_jτ   w_j   K i.e., the the effect of multiplying τ   K by g_w_jx_j is to switch the rows and columns in indices  j and j + 1 and then adding κ times the j+1-st row to  the j-th row.  Clearly, x_j(κ)   w_jτ  w_j∈𝒳_𝐣(v). Since g K was arbitrary,  we  see that im(𝒳_w ρ^k   )     = {γ K   |  γ∈𝒳 _ 𝐣(w)    } for w ∈ [W / W ^ λ_k ] with ℓ(w) = m + 1.  By induction, we  get the claim.         
  
  This can also  be proved  directly  by appealing  to the stratification of the Grassmannian that parametrizes  n-k-dimensional subspaces in an n-dimensional vector space over a finite field.  
 


 §.§ Mixed decompositions
    



    
From now on,  let n = 2m be even.    If g ∈_2m(F), we will denote by A_g , B_g, C_g, D_g∈Mat_m × m  (F)  so that  
    g  =    [ A_g B_g; C_g D_g ] .

If g ∈ G, then A_g, B_g, C_g, D_g denote the matrices associated with the _2m(F) component of G.  Moreover, we adapt the following      
 An element of _2m(F) is considered as an element of G via the embedding _2m(F) ↪ G in the second component.    
    

Let ι :  𝐇↪𝐆 be the subgroup generated 𝐀 and root groups of Δ∖{α_m}.     Then 𝐇≃_m×_m×_m embedding block diagonally in .  We  denote H       =  𝐇(F),  U = H ∩ K and H_1 = H_2≃_n the two components so that H = F^×× H_1× H_2.     If h ∈ H, we denote by h_1, h_2 the components of H in H_1, H_2  respectively.     We let W _  H ≃ S_m× S_m denote  the Weyl  group of W which we consider as a subgroup of W generated by s_1 , …, s_m-1, s_m+1  ,  …, s_2m.   The roots of H are denoted by Φ_H. These are ± (e_i - e_j ) for 1 ≤ i, j ≤ m and for m+1 ≤ i, j , ≤ 2m and we have a partition Φ_H =  Φ_H_1⊔Φ_H_2 into a union of two root systems isomorphic to A_m-1.    For α = e_i - e_j∈Φ_H and k ∈,  we let U_α, k denote the unipotent subgroup of H with 1's on diagonal  and    zeros elsewhere except for the (i,j) entry, which is required to have ϖ-adic  valuation less than or equal to k.     


For  k = 0, … , 2m,    let P_k  denote the set of pairs (k_1, k_2 )  of   non-negative  integers   such that k_1 + k_2 = k and k_1,k_2≤ m. For κ =  ( k_1, k_2 ) ∈ P_k,  denote l(κ) :  =   min   (k_1 , m - k _2  ) and  let      

    λ _κ : =  ∑_ i = 1 ^k_1 f_i  +  ∑_ j = m-k_2  + 1  ^ m  f_m+j∈Λ
   

For i = 0, …, m,  let   
 t_i  :=   diag( ϖ^-1…,  ϖ^-1_ i   ,    0 , …, 0   _ m - i   ) ∈Mat_m × m(F) and  

    τ_i  : =  [ 1_m t_i;     1_m ]∈_2m  (F).

Set   H_τ_i :  = H ∩τ_i K τ_i^-1.  For g ∈ G, let U ϖ^Λ g K denote the set of all double cosets U ϖ ^ λ g K for λ∈Λ.      





























            For i = 0, … , m, the collections  U  ϖ ^ Λτ_i K are disjoint.      

  It suffices to show that H τ_i K are distinct double cosets. Suppose for the sake of contradiction that that τ_i∈  H τ_j K for i ≠ j. Then τ_i ^  -1 h τ_j∈ K.  Say h = (u, h_1 , h_2 ).  Now      
    τ_i ^-1  (h_1, h_2 ) τ_j    =    [                  h_1 h_1 t_j   -  t_i h_2;                                       h_2 ]

and therefore τ_i ^-1  h τ_j∈ K implies that h_1, h_2∈_m (    _F ) and h_1 t_j  - t_i     h_2∈  Mat   _m × m  ( _F ).  But the second condition implies that the reduction  modulo   ϖ     of one of h_1, h_2 is singular (the determinant vanishes modulo ϖ), which contradicts  the  first  condition.  






               For each k = 0, 1, …, 2m,  
(K ρ ^k K)  =     ∑_κ∈  P_k∑ _  i = 0  ^  l  (κ)   ( U ϖ ^ λ_κτ_i  K)       .    



 We first claim that for each  k = 0 , 1 …, 2m, the double  cosets  U ϖ^λ_κτ_i    K for distinct choices of κ∈ P_k and i = 0 , 1, …, l (κ).   By Lemma   <ref>, two such cosets are disjoint for distinct i, so it suffices to distinguish the cosets for different κ but  fixed i.    By  Lemma <ref>, it suffices to show that U ϖ^λ_κ H_τ_i are  pairwise disjoint  for κ∈ P_k.   Since H_τ_i⊂ U, it in turn suffices to show that U ϖ^λ_κ      U are  pairwise disjoint  for  κ∈  P _  k.     But this follows by Cartan    decomposition for  H.  

Fix a k.  For κ = (k_1 , k_2 ) ∈  P _  k,  let 𝐣 = {1,…, k_1}∪{ m -  k_2  + 1, …, 2m }. From the description  of  the Schubert cell 𝒳_𝐣 and Proposition <ref>, it is easy to see that   ϖ^λ_κτ_i K ⊂ K   ρ ^  k      
K (and therefore ϖ^λ_κτ_i K ⊂τ_i U ϖ^λ_κτ_i K)   for all  κ∈  P_k, 0≤  i  ≤ l(κ). So to prove the claim at hand,  it  suffices to show that for any γ∈ G such that γ K ⊂ K ρ^k K, there exist κ and i such that U γ  K   =     U ϖ^λ_κτ_i K.  By Proposition <ref>, it suffices to restrict attention to γ∈𝒳_𝐣 for some Schubert symbol 𝐣∈ J_k.  Furthermore, since any  γ∈𝒳_𝐣  has non-zero   non-diagonal     entries  only   above a pivot and these entries are in _F, we can replace γ by an element γ ' such that A_γ', D_γ' are diagonal matrices and U γ K =  U γ'    K.  Let us define a  set 𝒴_𝐣⊂_n(_F ) that contains all such γ' as follows. An element g ∈ G lies in 𝒴_𝐣  if        
         

  * the diagonal of g has 1 (referred to as pivots) in positions (j,j) for j ∈𝐣 and ϖ if j  ∉𝐣,

  * A_g, D_g are diagonal matrices  and C_g = 0, 

  * B_g has non-zero entries only in  columns of H   that contain a pivot and rows that  do  not.    
 
For any 𝐣∈  J_k, let 𝐣_1 (resp., 𝐣_2     ) denote the   subset of  elements not  greater   than m (resp.,   strictly   greater than m) and let κ(  𝐣 )  :  =  ( |  𝐣_1  | ,  |  𝐣_2 | )  ∈  P_k. It suffices to establish the following.   
    



Claim.  For any Schubert symbol  𝐣∈  J_k and  any γ∈𝒴_𝐣 there exists an integer   i  ∈{0, 1, …,   l(κ (  𝐣 )     )  }  such that U γ K = U ϖ^λ_κτ_i K. 
       
We prove this by induction on m. The case m = 1 is   straightforward.  Assume the truth of the claim for some positive integer m -  1 ≥ 1. If 𝐣_1 = ∅, then A_γ = I_m, B_γ = C_γ = 0 and D_γ  is diagonal.     Since w_m+1, …, w_2m lie in both U and K,   one can put all the k ≤ m pivots in the top diagonal entries of D_γ and we are done.  We can similarly rule out the case 𝐣_2 = { m+1, …, 2m }.  Finally, if B_γ = 0, we can again  use  reflections in H   to  rearrange  the   A_γ and D_γ diagonal  entries to match ϖ^λ_l,k.    


So suppose that k  _  1    :  =    |  𝐣_1| > 0,     
k_2 : =  | 𝐣  _2    | < m and B_γ≠ 0.  Pick j_1∈𝐣_1 such that the j_1-th  row of  B_γ is non-zero    and let j_2∉𝐣_2, m+1 ≤ j_2≤ 2m be such that the (j_1, j_2 ) entry of γ in B_γ is not 0.    If j_1≠ 1, then using row and columns operations, one can switch the first and j  _       1-th row and columns to obtain a  new matrix γ '.  Clearly,  γ '  is an  element of 𝒴_𝐣 ' for some new 𝐣', U γ K = U γ ' K and the (1,j_2) entry of  γ ' is non-zero.      Similarly if j_2≠ m+1, we can produce  a  matrix using row and columns operations so that (m+1, m+1) diagonal entry of the new matrix is 1 and the class of this matrix in U \ G / K is the same as γ.  The upshot is  that we may  safely  assume  that j_1 = 1, j_2   =    m+1 (so in particular,   1 ∈𝐣,  m+1 ∉𝐣).           


    γ  =       
    cccc|cccc[margin]
                            
       ⋱               
          ϖ               *         
             ⋱         
                □      
                   ⋱
                       1     
                         ⋱[decorate, decoration= calligraphic brace,transform canvas = yshift=-2em, thick] (row-8-|col-4.south east) – node[midway, below=1pt]0.9j_1 (row-8-|col-3.south west)   ;
       [decorate, decoration= calligraphic brace,transform canvas = yshift=-2em, thick] (row-8-|col-8.east) – node[midway, below=1pt]0.9j_2 (row-8-|col-7.west)   ;
       [decorate, decoration= calligraphic brace,transform canvas = yshift=0em,xshift= 3 em, thick] (row-3-|col-8) – node[midway, right=1pt]0.9j_1 (row-4-|col-8)   ; 
        ⇝    γ'   =   cccc|cccc[margin]
    ϖ              *
       ⋱
                            
             ⋱         
                1        
                   ⋱
                      □   
                         ⋱[decorate, decoration= calligraphic brace,transform canvas = yshift=-2em, thick] (row-8-|col-4.south east) – node[midway, below=1pt]0.9j_1 (row-8-|col-3.south west)   ;
       [decorate, decoration= calligraphic brace,transform canvas = yshift=-2em, thick] (row-8-|col-8.east) – node[midway, below=1pt]0.9j_2 (row-8-|col-7.west)   ;
        [decorate, decoration= calligraphic brace,transform canvas = yshift=0em,xshift= 3 em, thick] (row-3-|col-8) – node[midway, right=1pt]0.9j_1 (row-4-|col-8)   ;

  
Since the top  left     diagonal entry of B_γ is non-zero,   we  can  use   elementary   operations for   rows and  columns  with labels in 𝐣_2[the   non-zero  columns of B_γ are above a pivot of γ]   to make all the other entries of the first row of B_γ zero and keep D_γ a diagonal matrix.    The  column  operations  may   change the other rows of B_γ but the new matrix still  belongs  to    𝒴_𝐣  and
has same class in U \ G  / K.  Similarly, we can use  elementary operations for rows and columns with labels in {1, …, m }∖𝐣_1 to make all the  entries below (1,m+1) in B_γ equal to zero,   while keeping A_γ a diagonal  matrix.      
Finally, conjugating by an   appropriate        element of  the compact diagonal   A^∘⊂ U, we can  also  assume that the top left entry of B_γ is  1.    



In summary,  we have arrived at a matrix    that has the same class in U \ G / K as the original γ  and   has    zeros in rows and columns  labeled 1, m + 1 except for the diagonal entries  in  positions  (1,1), (m+1, m+1), (1,m+1) which are ϖ, 1,   1 respectively.  The  submatrix  obtained by deleting  the first and (m+1)-th  rows  and columns is a  (2m-2) × (2m-2) matrix in 𝒴_𝐣' for some 𝐣' of cardinality k-1.  By induction, this matrix can be put into the desired  form  using the groups U and K associated with _m×_2m  - 2. The  possible value of i  that can  appear  from this   submatrix have to be at most  max(k_1  -     1, m-1+k_2)  by  induction  hypothesis  and therefore the bound for possible   i holds for m as well.    This  completes  the    proof.          
 




 Suppose 
  m = 2 and k = 2, so that P_k = {  (2,0), (1,1) , (0,2)  }.  Proposition <ref> says that 
  

    K [ ϖ  ;   ϖ;     1;       1 ]  K          =      U [ ϖ;   ϖ;     1;       1 ]   K  
       +      U 
     [ ϖ   1;   ϖ;     1;       1 ]  K      +      U  [ ϖ   1;   ϖ   1;     1;       1 ]   K 
         +        U  [ ϖ;   1;     1;       ϖ ] K  
      +      U  [ ϖ   1;   1;     1  ;         ϖ ]  K       +       U  [ 1  ;   1;     ϖ;       ϖ ]   K
 

 
 




 §.§ Mixed degrees
                
For 1 ≤  r   ≤ m, let 𝒳_r : = _r(F). We have inclusions 𝒳_1↪𝒳_2↪…↪𝒳_m obtained by a considering a matrix σ∈𝒳_r as a  (r+1) × (r+1) matrix whose top left r × r   submatrix is σ, has 1 in last  diagonal entry and zeros elsewhere.  For each r, let  
let

    _  r      :  𝒳_ r → G      σ↦ι( σ , σ )   =   [ σ  ;   σ ]∈ G
 
where σ is considered as an element of H_1, H_2  as above, so that j_ r factorizes as 𝒳_ r ↪𝒳_m   G.     We henceforth consider all 𝒳_ r as  subgroups of G and omit _r unless necessary.  We  denote 𝒳_r  ^∘  =  𝒳∩ K  ≃_ r (_F ).   

For α = e_i - e_j∈Φ_H, k ∈, let U_α, k be the unipotent subgroup of matrices  h ∈ H such that the diagonal entries  of h are 1, the (i,j) entry of  h has valuation at least k   and all other   entries are 0.  For each r ≥ 1, let ψ_r  : Φ_H→   be the function 
    ψ_s (α ) =    1      if α∈{  e_i  - e_j∈Φ_H   |    either  1 ≤ j ≤   r  
     or  m + 1  ≤ i  ≤  m +  r }
    
    0       otherwise

and let H _ ψ_ r  be the subgroup generated by U_α, ψ_ r (α) and A ∩τ_r  K τ_ r ^-1   
More explicitly, H_ψ_ r 
is the subgroup of  elements (v,h_1, h_2) ∈ U satisfying the three   
conditions below:   
         

  * all the non-diagonal entries in  the first r columns of h_1 are divisible by ϖ,

  * all non-diagonal  entries in the  first r rows of h_2 are divisible by ϖ,

  * the difference of the j and j+ m -th  diagonal entries of h = ( h_1 , h_2 ) ∈ G is divisible by ϖ for all j = 1, …, r.  
 


            H_τ_ r   =    𝒳_ r    ^ ∘     H_ψ_ r     =  H_ψ_ r  𝒳_  r    ^ ∘  for  r   = 1, … ,  m.        



  The _m component on both sides are _F^× and we may therefore ignore  it.    Let h  = ( h_1 , h_2 ) ∈ H.   Then   h ∈ H_τ_ r and if and only if h ∈ U and 
    h_1   t _ r   -  t _ r    h_2∈Mat_ m ×  m       (   ϖ_F )
  
(see  the calculation in  Lemma  <ref>). It is then clear that H_τ_r ⊃𝒳_r^∘· H_ψ_ r.  Let h   =  (h_1 ,  h_2 )  ∈ H_τ_ r. From the description of H_τ_r, we see that  the   r ×  r  submatrix σ  formed by first r rows and columns of h_1 must be invertible (and similarly for h_2). Then _r (σ^-1) · h has the top r  ×  r block equal the identity matrix. Since this matrix lies in H_τ_ r, we see  again from the 
description of elements of H_τ_ r   that _  r  ( σ ^-1 ) h  ∈  H_ψ_r. This    implies the reverse inclusion H_τ_r ⊂𝒳^∘ _  r   H_ψ_ r. Since the product of 𝒳_r ^∘ and H_ψ_ r is a group, 𝒳_r ^∘  H_ψ_r   =  H_ψ_r 𝒳_i^∘.    
  
Recall that Φ_H = Φ_H_1⊔Φ_H_2.  
Declare   α_1 , … , α_m∈Φ_H_1 and - α_m+1 , …,  -  α_2m∈Φ_H_2 to be the set of positive roots of Φ_H.    Then α_1,0 : = e_1 - e_m∈Φ_H_1, α_2, 0   =  e_2m   - e_m+1∈Φ_H_2 are the highest roots.   Let s_1, 0, s_2, 0∈ W_H denote the reflections associated with α_1 ,   0, α_2,   0  respectively. Then the  affine Weyl group W_H, aff (as a subgroup of W_aff)  is  generated by 
    S_H, aff  = { t( α_1,   0  ^ ∨  )  s_1,0 ,  s_1, …, s_m-1}⊔{ t(  α_2, 0  ^ ∨  )    s_2, 0 ,   s_m+1, …,  s_2m-1}
     and (W_H_aff , S_H, aff ) is a Coxeter  system of type Ã_m-1×Ã_m-1.         We denote by ℓ_H  : W_H→ the resulting  length    function. The  extended Coxeter-Dynkin   has two components

    [extended,Coxeter,
    edge length= 1cm,
    labels=,1,2,m-2,m-1, labels*=0_1]
    A[1]      [extended , Coxeter,
    edge length= 1cm,
    labels=,m+1,m+2,2m-2,2m-1, labels*=0_2]
    A[1]

where the labels 0_1, 0_2 correspond to the two affine reflections corresponding to α_0,1,  α_0,2.   

Now let   I _  H_1  (resp.,  I_H_2)   be the Iwahori subgroup of H  _  1 (resp.,   H_2 )  consisting    of   integral  matrices  that reduce modulo ϖ to upper triangular (resp.,  lower triangular)  matrices and set   I_H :  = _F^× ×     I_H_1× I_H_2. Then I_H is the Iwahori subgroup associated with alcove determined by S_H,   aff. We    let    

    ρ_1 : =     0.9[ 0 1        ;   0 1      ;     0 1  ;       ⋱ ⋱  ;         0 1; ϖ         0 ]∈ H_1    ρ_2   : =    0.9[ 0         ϖ; 1 0        ;   1 0;     1 0    ;       ⋱ ⋱  ;         1 0 ]∈   H_2

(so we have    ρ_1 = ρ_2^t).  Both   ρ_1, ρ_2  normalize I_H and the effect of conjugation w ↦ρ_1 w ρ_1^-1 (resp.,  w ↦ρ_2 w ρ_2^-1) is by cycling in  clockwise (resp.,   counterclockwise)  direction     the left (resp.,  right) component of the diagram      displayed   in          (<ref>).  We   set ρ_H :  = (ρ_1, ρ_2 ) ∈ H and  for   κ =  (k_1 , k_2 )  ∈ ^  2, we denote by  ρ_H^κ the element ( ρ_1 ^k_1 , ρ^k_2   _  2         )   ∈ H.  We will denote   by    - κ the pair (-k_1,  - k _2 ).             

                 
For r = 0 , … ,     m, let    I_H, r denote the subgroup of H which contains I_H and whose Weyl group W_ H ,  r   ⊂ W_H  is generated by   S_H,  r :   =   {     s_r … , s_m - 1   ,    s_m+r , … ,  s_2m  - 1  }.   More  explicitly,  I_H,r is the subgroup of U consisting of all   matrices  as  below  
 

    cccc|cccc[margin] 
                                               
                         
                         
                         
                         
                         
                         
                         
     (row-1-|col-1) – (row-4-|col-4) ;
         
         (row-1-|col-1) – (row-4-|col-1) ;
     
     (row-4-|col-1) – (row-4-|col-4) ;
    
       
      (row-5-|col-5) – (row-5-|col-8);
        
        (row-5-|col-5) – (row-8-|col-8);
            
          (row-5-|col-8) – (row-8-|col-8); 
    [decorate, decoration= calligraphic brace,transform canvas = yshift=0.2em, thick] (row-1-|col-1) – node[midway, above=1pt]0.9r     (row-1-|col-4);
    [decorate, decoration= calligraphic brace,transform canvas = yshift=-0.3  em,   xshift =0.1  em, thick] (row-8-|col-8) – node[midway,
    below=1pt]0.8r     (row-8-|col-5);
    [decorate, decoration= calligraphic brace,transform canvas = yshift=-1.3em, xshift = 1.0 em, thick] (row-7-|col-4) – node[midway, left =1pt]0.8r     (row-4-|col-4);    
    [decorate, decoration= calligraphic brace,transform canvas = yshift  =  0 em,    xshift =  -1.1    em, thick] (row-1-|col-5) – node[midway,  right    =1pt]0.8r     (row-4-|col-5);

such that the   non-diagonal  entries inside the  two  triangles are divisible by ϖ.     
     
  For any k = 0, …, 2m,  κ∈  P_k and r   = 0,…,  l ( κ),   we have    H _ τ_ r ϖ^-λ_κ U =  I _  H ,  r   ρ_H ^  -  κ    U.       

  Since r  ≤ l (κ), ϖ^-λ_κ commutes with 𝒳_r ^∘ and therefore H_τ_rϖ^-λ_k U =  H_ψ_rϖ^-λ_κ U. It is also easily seen that 
    I_H , r    =    H_ψ_r·    A^∘·∏_α∈Φ_H,r^+  U_α, 0
 
where Φ_H,r^+    :  = { e_i - e_j∈Φ_H ^ +   |  either  1 ≤  j ≤   r 
 or  m + 1  ≤ i  ≤  m + r     }.  Since   ϖ^λ_κ commutes with A^∘ and     U_α, 0 for α∈   S_r, we see that H_ψ_rϖ^-λ_κ  U.  Since ϖ^λ_κ U =  ( ρ_1^k_1, ρ_2^-k_2   )     U =    ρ_H^-κ  U,      the  claim  follows.   
  



For  κ = (k_1 , k_2 )   ∈  P_k, r = 0, …, l (κ), let  W_κ, r⊂ W_H,r denote the subgroup generated by S_H,r∖{ s_k_1 ,  s_2m - k_2}. Then W_κ, r is a Coxeter subgroup of W_H, r.  Let     
    P_κ  ,  r           : =  ∑_  w ∈ [ W_H,r / W_κ, r ]   q ^ ℓ  _  H       (w)

denote the Poincaré polynomial of [ W_H,r / W_κ, r  ] ⊂ W_H.    


    For any k, κ∈  P_k and   r   = 0,…,  l ( κ),  we have   [U ϖ^λ_κτ_ r     K ]_* =  P_κ,  r (q).   
    


   We have  [U ϖ^λ_κτ_r     K ]_* =   [ H_τ_   r   ϖ^-λ_κ U ] which is by definition the cardinality of H _ τ_ r ϖ^-λ_κ U / U.  By  Lemma  <ref>,  H_τ_rϖ^-λ_κ  U / U  =  I_H,r ρ_H ^  -  κ  U / U.   Theorem    <ref>  therefore  implies that [ U  ϖ^λ_κτ_r     K ]  _  * is  the Poincaré polynomial of [ W_H,r / ( W_H, r∩ρ^-κ W_Hρ^κ  )  ].    Now ρ ^  - κ  W_Hρ^κ is the subgroup of W_I,H generated by     
    S_H, aff∖ρ_H^-κ{ s_1,0, s_2,0}ρ _  H  ^ κ   =  S_    H   ,    aff∖{  s_k_1  ,  s_2m-k_2}
 
where the equality  follows  since ρ_1^-1s_1,0ρ_1 = s_1 and  ρ_2 ^-1 s_2,0ρ_2      =    s_2m (see  above for the description of the action of ρ_1 ,ρ_2 on      (<ref>)).  Thus we have    
    W_H,r ∩ρ^-k   W_Hρ^k    =  W_κ, r
  and  the   claim   follows.     
   



    With notation as above,        [ U ϖ^λ_κτ  _    r     K ] _*≡0.9m- rm-k_1m-rk_2q-1. 
  



  | W_H, r | = ( m- r) ! · ( m- r)! since W_H,r is the product of the groups generated s_r, … , s_m-1 and s_m+r , … , s_2m-1, each of which  have   cardinality  (m-r)!.  Similarly,  W_κ,  r is the product of four groups generated by four sets of reflections labeled
 
    r + 1 , …, k_1-1,         k_1 + 1,   …,  m-1 ,        m+r +  1 , … ,   2 m - k _2 - 1 ,             2m - k _2 + 1 ,  …  ,   2m-1

which have sizes ( k_1 - r ) !, ( m - k_1 ) !, (  m -   k_2 - r ) !   and   k_2 !   respectively.    
 






















 






















 §.§ Zeta  elements
 

We now formulate the zeta element problem relevant to the situation of <ref> and show that one exists using the work done above.       Let  T : =   F  ^ ×, C  =  _F^×⊂ T the unique maximal compact subgroup,  D = 1 +  ϖ_F a subgroup of index q - 1 and  ν   :   
H → T be the map given by (h_1 , h_2 ) ↦(h_2 ) /  h_1.   Let   𝒪 be any  integral  domain  containing  [ 
q ^  - 1  ].      Set     
        

  * G̃  = G × T,   

  * ι̃ = ι×ν : H  →G̃, 

  * U ⊂ H and  K̃     :   = K × C ⊂G̃ as bottom levels

  * M_H,  = M_H,  , triv the trivial functor, 

  * x_U  = 1_∈ M_H, (U) the  source bottom class, 

  * L̃  = K × D the  layer  extension 
of degree q - 1, 

  * ℌ̃_c = ℌ_std,  c  (Frob)  ∈𝒞_  ( K̃\G̃   /  K̃   ) where Frob : = ( ϖ^-1 C).
 
 This setup generalizes the one studied in <ref>. 
  



     There  exists a   zeta  element for   
(x_U , ℌ̃ _c,  L̃  ) 
for all c ∈∖  2.         
    

  For  each k_2  =  0 , …,  m and i an integer such that 0 ≤ i ≤ m -  k_2,  let g_i,   k_2  :  = (1 , τ_i, ϖ^-2  k_2 ) ∈G̃ and J_i,  k_2  :   = {  ( k_1  ,  k_2 )     |   i ≤  k_1≤  m ,  k_1∈}.   For each i, k_2 as above, let 
    d_i,k_2 :  = [ H ∩ g_i,k_2K̃  g_i,k_2^-1 :  H ∩ g_i,k_2L̃ g_i  ,  k_2^-1 ] .
   By Lemma  <ref>(iii),  d_i,k_2 = [  H   _ τ_i∩τ_i K τ_i^-1 : ν^-1(D) ]. We therefore write d_i for d_i,k_2.   Since ν (  H ∩τ_i K τ_i^-1  ) =  C for i =0,…, m-1,   we  have      d_i = q - 1. Now if (h_1 , h_2 ) ∈ H_τ_m, then h_1 - h_2∈ϖ·Mat_m× m(_F ). Thus, ν ( H_τ_m ) ⊂ D (if fact, equal) and H_τ_m = ν^-1(D). This implies that  d_m = 1. To summarize, 
    d_0 = … = d_m-1 = q - 1,       d_m = 1 .
    
Next, for  each (i,  k_2  ) as above and  j = (k_1,k_2 ) ∈ J_i,k_2, denote  h_ j  :  =  (  ϖ^ k f_0 ,  ϖ^λ_j   )  ∈ H and σ_j =  ι_ν ( h_j  ) ·  g_i,k_2 =  ( ϖ^k f_0 ,  ϖ^λ_j  ,  ϖ^- k   )   ∈G̃    
where k in these    expressions denotes k_1 + k_2.      
Denote by J the  disjoint union of J_i,v for all possible i, v as above.     By   Proposition  <ref>, Proposition  <ref> and   Lemma <ref>(a),  
    ℌ̃_c     =   ∑ _ j ∈ J   b_j  ( U σ_jK̃   )
    
where b_j∈[q^-1] for j = (  k_1 , k _2  ) ∈ J_i,k_2 is given by ( - 1 ) ^ k  q ^  - k (2m-k +c)/2 and  k = k_1 +  k _2 as before.     
In particular, b _j≡ (-1)^kq-1. It is then clear that 
    H \ H ·(ℌ̃_c )  /  K̃ = { g_i,k_2  |   0 ≤ k_2≤ m   ,       0 ≤ i ≤ m - k_2}.
  
Let 𝔥_i,k_2 denote the ( H , g_i,k_2 )-restriction of ℌ̃_c.   By  Corollary  <ref> and  Lemma  <ref> (ii),

    (  𝔥_i,k_2^t)        = ∑_ j ∈ J_i,k_2  c_j       [  U σ_jK̃ ]  _  * 
       ≡∑_ k_1 = i ^m  (-1)^k_1+k_2m  - i  m - k_1m-i    k_2q-1
            =  (-1)^k_2m-i k_2· (-1)^i  (1-1)^m-i =   0
    
for all i, k_2 as above   such that i < m. 
Since d_m = 1,  the  criteria of  Corollary <ref> is satisfied.      
    

     
For m = 2, the coefficients  ∑_j ∈ J_i,k_2    c_j      [U σK̃ ] _ * as follows
         

  * 1-q^-c+3/2(q+1)+q^-(c+2) for g_0,0, 

  * q^-(c+2)-q^-3 c+1/2 for g_1,0,

  * q^-(c+2) for g_2,0

  * (q+1)(q^-(c+2)(q+1)-q^-3/2(c+1)-q^-(c+3)/2) for g_0,1,

  * q^-(c+2)-q^-3/2(c+1)(q+1)+q^-2 c for g_1,2,

  * q^-(c+2)-q^-3/2(c+1) for g_1,1.
   When  c=1, the  sets   g_0,1K̃, g_1,2K̃, g_1,1K̃ do not contribute to the  support of the  zeta element, since their corresponding coefficients all vanish.  An induction argument shows that for c=  1, the zeta element is only supported on g_i,0K̃.   
    

The normalization at c= 1 is relevant for the setting   <cit.> (corresponding to the L-value at s = 1/2), and the  coefficients of the zeta element we obtain match exactly with those of  test vector specified  Theorem 7.1  of  loc.cit.  
More precisely, the  coefficient  denoted `b_i' in Theorem 7.1 (2) of  loc.cit.  is the coefficient for g_i,0 computed in the proof  above multiplied with q/q -1·μ_H(U) /μ_H(V_i) (after replacing ℓ in loc.cit.    with q).  Note also that what we denote by V_i here  is denoted `V_1,i' in loc.cit.   One of the chief advantages of the approach here is that one does not need to compute the measures μ_H(V_i) in Definition <ref> which seem to have far more complicated  formulas.     
               
  

        Notice that the   g_m,0 (equivalently, τ_m in the decomposition  Proposition <ref>) only arises from a single Hecke operator K ϱ^m K. By  Corollary  <ref>, we see that a zeta element exists only if the degree d_m is 1. 
In Theorem <ref>, this was guaranteed by the choice of ν and T.  If   say, ν is replaced by the product of determinants of H_1, H_2, then no zeta elements exist.  So in a sense, one  can only hope to make `anticyclotomic'  zeta elements in this  setting.        
 




§ BASE  CHANGE -FACTOR OF  
     

In this  section,  
we study the inert  case of the embedding discussed in <ref>.  We first collect some generalities on the unitary group GU_4. Let E / F be  separable extension of of degree 2, Γ : =Gal(E / F),  γ∈Γ the    non-trivial element. Let 

    J=[     1_2; 1_2     ]
 
where 1_2 denotes the the 2 × 2 identity matrix. Then J  =  γ(J)^t is Hermitian.  We let 𝐆   =   GU_4 be the reductive group over F given whose R points for a F-algebra R are given by  
    𝐆 (R) ={g ∈GL_4(E ⊗ R) |γ(^tg)   J g =   sim (g) J  where sim (g)   ∈ R ^ ×}.
   Then 𝐆 is the unique quasi-split unitary similitude group of split rank 3 (see <cit.>).  It's derived group is a special unitary group whose  Tits index 
is  ^2 A_3,2^(1)  (see <cit.>). The mapping 𝐆→_m, g ↦sim (g) is  referred to as the   similitude.  The   determinant  map : 𝐆→Res_E/F_m then satisfies γ∘ ·  =  sim^4. For R an  E-algebra, we let 
    γ_R: E ⊗ R → E ⊗ R, x ⊗ r ↦γ(x) ⊗ r
 the map induced by γ and 
    i_R: E ⊗ R →  R × R
 the isomorphism x ⊗ r ↦(x r, γ(x) r), where x ∈ E, r ∈ R. We let π_1, π_2: E ⊗ R → R the projections of i_R to the first and second component respectively. We have an induced action γ_R: GL_4(E ⊗ R) → GL_4(E ⊗ R) and an induced isomorphism i_R: GL_4(E ⊗ R) →GL_4(R) ×GL_4(R) given  by  (g_i, j) ↦(π_1(g_i, j), π_2(g_i, j). 
Under the identification i_R, the group 𝐆(R) ⊂GL_4(E ⊗ R) is identified with the subgroup of elements (g, h) ∈GL_4(R) ×GL_4(R) such that 
    (^t h,^t g) ·(J, J) ·(g, h)=(r J, r J).
  We thus have functorial isomorphisms
ψ_R:(c_R: pr_1∘ i_R): 𝐆(R) ∼⟶𝔾_m×GL_4(R) via which we identify 𝐆_E𝔾_m×GL_4 (as group schemes
over E) canonically.   
 The symbols F,  _F , ϖ ,  = _F  , q = q_F have the same meaning as in <ref>. We let E / F denote an unramified quadratic extension  and  set  q_E = |  |_E =  q^2 where _E is the residue field of E.  We denote by [_F ], [_E]   a fixed choice of representatives in _F, _E of elements of _F, _E   respectively.  We let   𝐆 be the group defined above and denote  
    G=𝐆(F) ,        G_E=𝐆(E) ψ=   E^××GL_n(E),        K_Eψ=_E ^ ××_4 ( _E ),         K =  K_E∩𝐆(F).
  For a ring R, we let ℋ_R,
ℋ_R, E denote the Hecke algebras ℋ_R(K \ G / K), ℋ_R(K_E\ G_E / K_E) over R  respectively.   For  simplicity, we will denote
(K σ K ) ∈ℋ_R simply  by (K σ K ).  Similarly for ℋ_R,E.          






 §.§ Desiderata
         Let 𝐀 =𝔾_m^3, dis : 𝐀→𝐆 be the map

    (u_0, u_1, u_2) ↦[     u_1                        ;             u_2                ;                 u_0/u_1        ;                         u_0/u_2 ]

which identifies 𝐀 with the maximal split torus of 𝐆. Let 𝐌 be the normalizer of 𝐀.  Then ψ : 𝐌_E𝔾_m, E^5 and we consider _m,E^5 as a maximal torus  of 𝐆_Eψ=𝔾_m, E×GL_4, E via (u_0, …, u_4) ↦(u_0, diag(u_1, …, u_4)). We will denote A : = 𝐀(F), M : = 𝐌(F).      We have X^*(𝐌)=ℤ e_0⊕⋯⊕ℤ e_4, X_*(𝐌)=
ℤ f_0⊕⋯⊕ℤ f_4, where f_i, e_i are as in <ref>. The Galois action Γ on X_*(𝐌), X^*(𝐌), is as follows:

    γ· e_i=        e _  0     if  i = 0 
     e _  0  - e _  i + 2     if  i = 1 , … , 4     γ·   f_i   =  
    f_0+⋯+f_4    if  i=0 
    
    -f_i+2    if  i=1, …, 4

where e_i=e_i-4, f_i=f_i-4 if i>4. For i=0,1,2, let
       

  * ϕ_i: 𝔾_m→𝐀 by sending u to the j-th component,

  * ε_i: 𝔾_m→𝐀, dis(u_0, u_1, u_2) ↦ u_i
 
Then X^*(𝐀)=ℤε_0⊕ℤε_1⊕ℤε_2, X_*(𝐀)=ℤϕ_0⊕ℤϕ_1⊕ℤϕ_2.   Let  res  :   X^*(𝐌) → X^*(𝐀), cores :  X_*(𝐀) → X_*(𝐌) be the maps obtained by restriction and inclusion respectively. Then

    res(e_i)= ε_i    if  i=0, 1,2 
    ε_0-ε_i-2    if  i=3,4
        cores(ϕ_i)= f_0+f_3+f_4    if  j=0 
    
    f_i-f_i+2    if  j=1,2

We let Φ_E denote the set of absolute roots of 𝐆_E as in <ref> for n = 4 and Φ_F denote the set of relative roots obtained as restrictions of Φ_E to 𝐀.  Then Φ_F={±(ε_1-ε_2), ±(ε_1+ε_2-ε_0), ±(2 ε_1-ε_0), ±(2 ε_2-ε_0)},  which   constitutes a
root system of type C_2. 
We choose β_1=e_1-e_2. β_2=e_2-e_4 and β_3=e_4-e_3
as simple roots and let Δ_E={β_1, β_2, β_3}. In this ordering, the half sum of positive roots is  
    δ =  1/2( 3 e_1 + e_2 - e_4 - 3 e_3 )
  and 
β_0=e_1-e_3 is the highest root. The set Δ_E and β_0 are invariant under Γ, and the labeling is chosen so that (absolute) local Dynkin diagram (with the bar showing the Galois orbits) is the diagram on the left 

    [fold, scale  =  1.7, extended,    edge length  = 0.7,  labels=0,1,2,3]Aooo      [scale = 1.7,  labels*=1,2,1, extended,    edge length = 0.7,     labels=0,1,2]Coo
 
The set of corresponding relative simple roots is therefore Δ_F={α_1, α_2} where   α_1=ε_1-ε_2, α_2 =  2  ε_2 -  ε_0. 
With this ordering, the highest root is α_0=2 ε_1-ε_0.  The  associated simple coroots are  α_0 ^ ∨ = ϕ_1, α_1^∨  =   ϕ_1  -  ϕ_2, α_2^∨   =  ϕ_2 and we denote by Q^∨ their  span in Λ.  Executing the recipe provided in 1.11  of  <cit.> on the absolute diagram above, we find that the  local index or relative local Dynkin diagram (see 4 of op.cit.) is the diagram on the right above.    Here,  the  indices below the diagram correspond to the affine roots -α_0+1, α_1, α_2 and the indices above the diagram are half the number of  roots of a  semi-simple  group of  relative  rank 1 whose  absolute Dynkin-diagram is the corresponding Galois orbit in the diagram on the left.   The endpoints of the diagram on the right, and in particular the one labelled 0, are hyperspecial and hence so is the subgroup K by construction. The diagrams above can be found in the fourth row of the  table on p. 62 of op. cit. 
 For λ =  a_0ϕ_0 + a_1ϕ_1 + a_2ϕ_2∈ X_*(𝐀),   ⟨λ , δ⟩ can be computed by pairing λ with  res(δ) = -2  ε _0  + 3 ε_1 +  ε_2 and equals - 2 a_0 + 3 a_1 + a_2. Note also that 
    2 ·res(δ) = 2 (ε_1  -   ε_2    )  +  2  (  ε_1  +  ε_2   -   ε_0  )  +  ( 2   ε _  1      -   ε_0 )    +   ( 2  ε_2  -   ε_0  )

is a weighted  sum  of  the positive roots in Δ_F, with the weights given by   the  degree of  the splitting field of the  corresponding  root. 
 

From now on, we denote by Λ the cocharacter lattice X^*(𝐀) and denote by t the translation action of Λ on Λ⊗.      An  element λ =  a_0ϕ_0 + a_1ϕ_1 + a_2ϕ_2∈Λ will be denoted by (a_0 ,a_1 , a_2 ) and ϖ ^ λ denotes the element λ(ϖ)  ∈  A. Let s_i, i = 0,1,2 denote the simple reflections associated α_i. 
The action of s_i on Λ is given explicitly as follows: 
          

  * s_1 acts as a transposition ϕ_1↔ϕ_2,

  * s_2 acts by sending ϕ_0↦ϕ_0+ϕ_2, ϕ_1↦ϕ_1, ϕ_2↦ - ϕ_2

  * s_0=s_1 s_2 s_1 acts by sending ϕ_0↦ϕ_0 + ϕ_1, ϕ_1↦ -  ϕ_1, ϕ_2↦ϕ_2. 
   
As before, we let e^λ (resp.,  e^W λ ) denote the element in the group algebra [ Λ ] corresponding to λ (resp.,  the formal sum over W λ ).  Let  S_aff =  { s_1 , s_2 ,  t( α_0 ^∨ )  s_0} and  W, W_aff and W_I denote the Weyl, affine Weyl, Iwahori Weyl groups respectively. We  consider    W_aff as a group of affine transformations of Λ⊗. We have   
           

  * W  ≅  (  / 2  ) ^2⋊  S_2, 

  * W_aff=  t ( Q  )     ^∨⋊     W the affine Weyl group

  * W_I = A / A^∘⋊  W  Λ⋊  W,     
 
The pair (W_aff, S_aff) is a Coxeter system of type C̃_2 and we consider   W_aff⊂  W_I via v.  Then W_I  =  W_aff⋊Ω.  Given λ∈Λ, the   minimal possible length of elements in t(λ) W is obtained by a unique element. This length is given by 

    ℓ_min(λ) = ∑_λ∈Φ_λ^1 |  ⟨λ , α⟩   |     +  ∑_α∈Φ_λ^2  ( ⟨λ  ,   α⟩ - 1  )
   
where Φ_λ^1 = {α∈Φ_F^+  |  ⟨λ , α⟩≤ 0  },  Φ_λ ^2  =  {α⟩∈Φ_F^+  |  ⟨λ , α  >  0  }.  When λ is dominant, the first sum is zero, and the length is then also minimal among  elements of W t(λ) W.  Consider the following elements in the normalizer N_G(A):  

    w_0=  0.9[         1/ϖ    ;       1        ;   ϖ            ;               1 ],        w_1=  0.9[   1    ; 1      ;       1;     1   ],     
      w_2=  0.9[ 1      ;       1;     1  ;   1     ],   ρ =  0.9[       1;     1  ;   ϖ    ; ϖ       ] .
 
The  classes of w_0, w_1, w_2 represent t(α_0^∨) s_0, s_1, s_2 in W_I and ρ represents t(-ϕ_0) s_2 s_1s_2, which is a generator of Ω≅ℤ.   The  conjugation action of ρ switches w_0, w_2 and  keeps w_1 fixed, inducing an automorphism of the extended Coxeter diagram [Coxeter,   labels = 0,1,2   ,  extended,]C2.  

Let ξ∈_E ^ × be an element of trace 0 i.e.,  ξ + γ (ξ ) =  0.
Let  x_1 : Res_E/F_a→𝐆 and x_i  : _a→𝐆 for i = 0 , 2 be the root group maps 

    x_0: u ↦0.9[    1               ;         1          ; ϖξ u         1     ;                   1 ],          x_1: u ↦0.9[     1     u            ;           1            ;                 1      ;             -  u̅     1 ],            x_2: u ↦0.9[   1            ;       1     ξ u;           1    ;               1 ],

where u̅ : = γ(u). We let _w_i = _w_2 :=  _F, _w_1 :  = _E  and for i = 0,1,2,  we denote by  g_w_i : [ _w_i ]  →     G the map u ↦  x_i (  u )  w_i. If I denotes the Iwahori subgroup[note that K is hyperspecial, i.e., its Weyl group equals W] of K whose reduction modulo ϖ lies in the Borel of () determined by Δ_F,  then I w_i I  = _κ∈ [_w_i]  g_w_i ( κ )  .     
For  w ∈ W_I such that w is the unique  minimal length element in w W,  choose a reduced word decomposition   w  =   s_w,1 s_w,2⋯  s_w,ℓ(w)ρ _w where s_w  , i ∈  S_aff and  ρ_w∈Ω.  Define  
    𝒳_w :   ∏_ i  = 1  ^ ℓ(w)   [ _w_i]      →   G 
    
       (κ_1, …,   κ_ℓ(w) )      ↦   g_s_w,1   ( κ_1 )  ⋯  g_s_w,ℓ(w)    ( κ_ℓ(w) )   ρ_w
   
where we have suppressed the dependence on the the  decomposition of w  in the notation.   By   Theorem  
<ref>,  the    image of 𝒳_w is independent of the choice of   decomposition.     



 §.§ Base change Hecke  polynomial
  Let  y_i :  = e^ϕ_i∈ [ Λ ], so that [ Λ ] =  [ y_0 ^ ± , y_1 ^ ± ,  y_2   ^  ±  ] and  let  ℛ_q : =  [ q ^±1/2], ℛ_q^2   :     =  [ q^-1 ]. The abelian group homomorphism 1+γ: X_*(𝐌) → X_*(𝐌) given by f ↦ f+γ· f has image in Λ=X_*(𝐌)^Γ and hence induces a  map e^1+γ on ℛ_q^2-algebras

 

ℋ_ℛ_q^2(  G_E )  [r, "𝒮_E"]   [d, "BC", swap]        ℛ_q^2[X_*(𝐌)]^W_E   [d, "e^1+γ"]        

ℋ_ℛ_q(    G   ) [r, "𝒮_F"]       ℛ_q[Λ]^W_F
  

 
corresponding to which we have what is called the base change map    
    BC   :  ℋ_ℛ_q^2 ( G_E )  →ℋ_ℛ_q(G) .
     
The Satake polynomial that we need to  consider here is the base change of the Satake polynomial of 𝐆_E associated with the standard representation considered in <ref>.
This polynomial is

    𝔖_bc(X)=(1-y y_1 X)(1-y y_1^-1 X)(1-y y_2 X)(1-y y_2^-1 X) ∈ℤ[Λ]^W_F[X] .
       
where y =  y_0   ^    2   y _  1    y _   2.    
 We have a componentwise embedding ^  L 𝐆 _E↪ ^  L 𝐆_F.     
Given  an unramified L-parameter φ : 𝒲_F→ ^  L 𝐆_F, let  t̂⋊Frob_F ^-1 : =  φ( Frob_F ^ -1  ), where Frob_F∈𝒲_F denotes  a lift of the arithmetic  Frobenius, we have  
    φ(Frob_E ^-1 ) = (  t̂⋊Frob_F ^-1     )   ^2  =  t̂γ(t̂)   ⋊Frob_E ^-1∈ ^  L 𝐆_E .
    If   we think of t̂ as the Satake parameters of an unramified representation π_F  of 𝐆(F), then  t̂γ( t̂  ) are the Satake parameters of an unramified representation π_E of 𝐆(E) which is called the base change of π_F. The base change map BC above can then   also    be characterized as in <cit.>.       
    


      
 
 We define     ℌ_bc, c(X) ∈ℋ_ℛ[X] to be the image of ℌ_std, c(X) under the map BC for c any integer.   Equivalently, ℌ_bc,c(X) is the unqiue polynomial such  that 𝒮_F   ( ℌ_bc,c(X)    ) = 𝔖_bc (q^-c X ).  
 







         We have         
       

  * 𝒮_F(K ϖ^(2,2,1) K) =   q  ^3 e ^W(2,2,1) +  ( q -1  ) ( q^2 + 1 )  e ^(2,1,1)  , 

  * 𝒮_F(K ϖ^(4,3,3)K) =  q^4  e^W(4,3,3) +  q ^ 3 ( q - 1 )    e ^W(4,3,2)  +  q ( q - 1 ) ( 1 + q + 2q^2)    e ^(4,2,2) .    

  
 
  Note that since (δ)  ∈ X^*(𝐀), the Satake transform of (K ϖ^λ K ) for any λ∈Λ  all have coefficients in [q^-1][Λ]^W. The leading coefficients are obtained by Corollary 
<ref> which also shows that the support of these transforms is on Weyl orbits of cocharacters that are succeeded by λ under ≽.  
 

(a) Since (2,2,1) - (2,1,1) = α_1^∨ + α_2 ^ ∨, (2,2,1) ≽ (2,1,1) and it is easily seen that  (2,1,1) is the only dominant cocharacter which (2,2,1) succeeds. 
Thus  
    𝒮( K ϖ^(2,2,1) K )  = q^3 e^W(2,2,1) + b    e ^(2,1,1)
  
for some b ∈[q^-1].  To obtain the value b, we use the decomposition recipe of Theorem <ref>. Note that ℓ_min(2,2,1) = 1   and that  K w K =  K ϖ ^ (2,2,1) K where w=  w_0ρ^2. So we see  from the the Weyl orbit diagram   

    (2,0,1)[r, "s_1"]    (2,1,0)[r, "s_2"]    (2,1,2)[r, "s_1"]    (2,2,1)
 
that  | K w_0ρ^2 K / K| =   q   + q ^3 + q ^4 + q^6. Of these, the number of cosets of shape a permutation of (2,2,1) is ∑ _ μ∈  W (2,2,1)    q ^ ⟨λ + μ , δ⟩  =  1   +  q^2 + q^4 + q ^ 6  by W-invariance of 𝒮 (see Corollary <ref>). Thus the number of cosets of shape (2,1,1)  is 
    q + q^3 + q^4  + q^6  - ( 1 + q^2 + q^4 + q^6 )  = ( q -1  ) ( q^2 + 1 ) .
   
Since ⟨ (2,1,1) , δ⟩ = 0, the claim follows. 



(b)  Arguing as in part (a), we have  

    𝒮(K ϖ^(4,3,3)K) = q^4 e^W(4,3,3) + b_1   e ^W (4,3,2) + b_2    e ^(4,2,2)
 for some b_1 , b_2∈[q^-1].  Here we need a more  explicit  description of the Schubert cells in order to find b_1, b_2.    Observe that ℓ_min(4,3,3) = 3  and that K ϖ^(4,3,3) K = Kw K where w  = w_0 w_1 w_0ρ^4.   The Weyl orbit diagram of (4,3,3) is  
     


  (4,1,1)  [r,"s_2"]     (4,1,3)  [r,"s_1"]     (4,3,1)  [r,"s_2"]      (4,3,3) .


 


By Theorem <ref>,  K ϖ^(4,3,3)  K /  K = _ i = 1 ^3im ( 𝒳_σ_i) where σ_0 = w, σ _1 = w_2σ_0, σ _2 = w_1σ_1, σ _3 = w_2σ_2.  
Explicitly,   

    im ( _σ_0 )      =   0.85*([      ϖ       ;             ϖ              ; x_1ϖ^2 a  ϖ^2    ϖ^3       ; -a̅ϖ^2 x  ϖ^2           ϖ^3 ]) K 
         a   ∈ [_E]   ,  
           x, x_1∈ξ [_F]  , 
    
    im ( _σ_1 )      =   0.85*([           ϖ                                    ;     -ϖ^2  a         ϖ^3             x ϖ^2 + y ϖ;     x_1 ϖ^2                     ϖ^3       a ϖ^2;                                               ϖ ]) K       a   ∈ [_E]   ,  
           x, x_1 , y   ∈ξ [_F]
    im ( _σ_2 )       =   0.85*([             ϖ^3 a_1  ϖ -ϖ^2  a̅   x ϖ^2 +   y ϖ                ;                               ϖ                                ;                                               ϖ                ;                        ϖ^2  x_1   a ϖ^2 -ϖ a̅_1             ϖ^3 ]) K 
         a ,  a_1∈ [_E]   ,  
           x, x_1,   y   ∈ξ [_F],    
    im ( _σ_3 )       =   0.85*([               ϖ^3                         x ϖ^2 + y ϖ    a_1  ϖ - a ϖ^2;                                 ϖ^3    a ϖ^2 - a_1  ϖ x_1  ϖ^2 + y_1  ϖ;                                                     ϖ                  ;                                                                       ϖ ]) K      a ,  a_1∈ [_E]   ,  
           x, x_1,  y , y_1∈ξ [_F].
 

From the cells above, it is not hard  to see that the shape of any  coset in
 

  * im(𝒳_σ_0) is  
  * (4,1,1) if x_1 = x = a =  0, 

  * (4,1,2) if x_1 = a = 0, x ≠ 0, 

  * (4,2,1) if x_1≠ 0 and  a a̅ + x x_1ξ^2∈ϖ_F, 

  * (4,2,2) if either x_1 = 0, a ≠ 0 or x_1≠ 0 , a a̅ + x x_1ξ^2∉ϖ_F  
  

  * im(𝒳_σ_1) is (4,1,3) if x_1 = a = 0,  
(4,2,2) if x_1 = 0, a ≠ 0 and (4,2,3) 
if x_1≠ 0,   

  * im( 𝒳_σ_2) is (4,3,1) if x_1 = 0 and  (4,3,2) if x_1≠ 0,   

  * im(𝒳_σ_3 ) is (4,3,3).  

So in K ϖ^(4,3,3) K / K,  there are exactly   
q^6(q-1) cosets of shape (4,3,2). Since ℐ(ϖ^(4,3,2)K) = q^-3 e^(4,3,2), 

    b_1  = q^-3· q^6(q-1) = q^3(q-1)
 by W-invariance of 𝒮.  Thus the number of cosets in K  ϖ^(4,3,3) K / K whose shape is in W (4,3,2) is ∑_μ∈ W (4,3,2) q^⟨μ  , δ⟩  q^3(q-1)  =   (q-1)( 1 + q^2  + q^4 + q^6 ).  
Since the nubmer of cosets of shape in W( 4,3,3) is ∑_μ∈ W (4,3,3)  q ^⟨μ , δ⟩ = 1 + q^2 + q^6 + q^8 and | K ϖ^(4,3,3) K / K | = q^4 + q^5 + q ^7 + q^8, we see that 
    b_2      =    q^4 + q^5 + q ^7 + q^8  - ( 1 + q^2 + q^6 + q^8 ) -  (q-1)( 1 + q^2  + q^4 + q^6 )  
        =     q ( q - 1 ) ( 1 + q + 2q ^2     )
  
 


       We have

    ℌ_bc,c(X)    =( K ) 
       -q^-(c+3)((K w_0ρ^2 K ) + (q^2+1)(1-q)(K ρ^2 K)) X 
       +q^-( 2c+4)( (K w_0 w_1 w_0ρ^4 K ) + ( 1 -q ) ( K w_0ρ ^4  K )     + (q^2+1)(1 - q + q^2) ( K ρ^4 K ) ) X^2
       -q^-(3c+3)((K w_0ρ^6 K  )  + ( q ^2 + 1 ) (  1  - q  )   ( K  ρ    ^6 K   )    ) X^3
       +q^-4 c   ( K ρ^8 K  )   X^4∈ℋ_[q^-1][X]
 
where the words appearing in each Hecke operator are of minimal possible length. 
   
   Since  

    𝔖  _ bc , c   (  X )  =   1 - e^W (2,2,1  )     X       +  ( e ^W(4,3,3)   + 2  e ^   W ( 4,2,2)   ) X ^2  -   e ^    W    ( 6,4,3)   X^3  +  e ^  (8,4,4)   X^4  ,

the result follows from  Proposition  <ref>.   
   



 §.§ Mixed   coset   structures

Let 𝐇 be the subgroup of 𝐆 generated by the maximal torus 𝐌, and the root groups corresponding to ±α_0, ±α_2. Then 𝐇  =  GU_2×_μGU_2. Here GU_2 is the 
reductive group over F whose R points for a F-algebra R are given by 
    GU_2(R) = {  g ∈_2(E ⊗ R)   |  γ (  ^  t  g ) J_2 g = μ(g)  J_2  , μ(g) ∈   R ^ ×}
   where J_2 =         [   1; 1   ] and the fiber product in 𝐇 is over the similitude character of the two copies of GU_2. Explicitly, we get the embedding 

    ι :  →       (  (  [ a b; c d ] )   ,   (   [ a_   1    b_1;  c _ 1    d_1 ] )  )  
    ↦ (    [   a       b;     a_1     b_1;   c       d;     c_1     d_1 ] )
   
We let  H  =   𝐇 (F ),   U = H ∩ K = (_F). The Weyl group of W_H of H can be identified with the subgroup of W generated by s_0, s_2 and is isomorphic to S_2× S_2. 
 
This embedding is isomorphic to the one obtained by localizing the global one in <ref> by a local change of variables that sends J in  (<ref>) to diag(1,-1,-1,-1), which can be explicitly written by the formula given in <cit.>.  
 
To describe the twisted restrictions arising from the Hecke polynomial ℌ_bc,c(X), we define the  elements τ_0 = 1_G and 
    τ_1  =  
      0.9[   ϖ         - 1;       ϖ   1;           1    ;               1 ],     τ_2 =   
      0.9[ ϖ^2         - 1;     ϖ^2   1;           1    ;               1 ],      τ_3 =    0.9[ ϖ^2   ϖ   1 - ϖ;       ϖ   1    ;           1    ;         - 1   ϖ ]   .
 

     If 2 ∈_F^×, then   Hτ_i K are pairwise disjoint  for i = 0 , 1, 2, 3.  
  
  If Hτ_i K = H τ_j K, there exists an h ∈ H such that τ_i^-1 h τ_j∈ K. Writing h as  in  (<ref>), 
the matrices  h τ_1, h τ_2, h τ_3, τ_1^-1 h τ_2 respectively have the form     
   
0.9 [   aϖ         * -  a;         *    *    *;   cϖ         *   -c;         *    *    * ],         0.9[ a ϖ ^  2                 *      - a;                 *        *        *;   c ϖ ^2                 *       -c;                 *        *        * ],          0.9[  a  ϖ^2       *       *   - a ϖ;               *       *       *;   c ϖ^2       *       * -  c  ϖ;               *       *       * ],     0.9[         a  ϖ            *            *    d_1 - a ϖ;        - c ϖ            *            *            *; c  ϖ   ^   2                         *         -  c;                         *            *          d_1 ]   
 
where * denotes an expression in the entries of h and an empty space means zero. It is then easily seen that first column in each of these matrices becomes an integral multiple of ϖ if we require it to lie in K, which is a contradiction. Moreover 

    τ ^-1_1 h  τ  _3   =   
    0.9([                  a ϖ                a+c_1   a + b + c_1 - d_1ϖ               d_1 -a;                 -c ϖ               a_1 -c a_1 - b_1 - c - d  ϖ               b_1 +c;                c ϖ^2                  c ϖ                  c+d                 -c ϖ;                                    c_1  ϖ             c_1 -d_1               d_1  ϖ ])   and   τ_2^-1   h  τ  _3     =   
    0.9([        a   a+c_1ϖ        * d_1 - aϖ;       -c        *        *        *;    c ϖ^2      c ϖ        *     -c ϖ;            c_1  ϖ        *   d_1  ϖ ]) .

  
If τ_1 h τ_3∈ K, then  a_1  - c, a_1 - b_1 - c - d, b_1 + c ∈_F and this implies that c - d ∈_F. Since c + d ∈_F as well and 2 ∈_F^×,  we have c, d ∈_F. Similarly  we can deduce that c_1, d_1∈_F. This forces all entries of h to be integral.   But then the first column is an integral multiple of ϖ, a contradiction. 
Finally, note that if τ_2 h τ_3^-1∈ K, then a , c , c_1, d_1∈_F and column expansion along the fourth row forces ( τ_2^-1 h τ_3) ∈ϖ_F, a contradiction.  
 
For w ∈ W_I, let ℛ (w) denote U \ K w K / K.  When writing elements of ℛ(w), we will only write the corresponding representative elements in G and it will be understood that these form a complete system of representatives.  For g ∈ G, we denote H ∩ g K g^-1 simply by H_g. Observe that 
    [              1;                   1;  - 1;      -  1 ]∈     H _τ_1
   
is a lift of s_0 s_2∈ W_H. Therefore 
 U ϖ^λτ_1 K = U ϖ^s_0s_2(λ)τ_1K.  
            If 2 ∈_F^×, then  
          

  * ℛ (w_0ρ^2) = {ϖ^(2,2,1),  ϖ^(2,1,2),  ϖ^(1,1,0)τ_1,  τ_3} ,   

  * ℛ ( w_0 w_1 w_0ρ^4 ) =   {ϖ^(4,3,3),  ϖ ^  (3,2,2) τ_1,   ϖ^(2,1,1)τ_2 }.  
   

   
 Note that  Lemma <ref> implies that U ϖ^λτ_i K ≠ U  ϖ^μτ_j K for any λ , μ∈Λ if  i ≠ j. Lemma <ref> implies that Uϖ^Λ K is in one-to-one correspondence with U ϖ^Λ U, and Cartan decomposition for  distinguishes ϖ^(2,2,1) and ϖ^(2,1,2) in ℛ(w_0ρ^2). Thus all the listed  elements represent distinct classes.  It remains to show that these also exhaust all of the classes.      
 

∙ w = w_0ρ^2. From the Weyl orbit diagram drawn in the proof of Proposition <ref> (and Theorem <ref>), we see that Kw K / K   =  im(𝒳_w)  ⊔im (_w_1 w   )   ⊔im (   𝒳_w_2 w_1 w   )  ⊔im  (  _w_1 w_2 w_1 w  ).   
Thus to describe ℛ(w), it suffices to study the orbits of U on Schubert cells corresponding to the words σ _0 : = w_0ρ^2, σ _1  : =  w_1σ _0 and σ _2   :  = w_1 w_2σ _1. These cells are 

    im(𝒳_σ_0)  =  0.9* ([   1;       ϖ        ; x ϖ     ϖ^2    ;               ϖ ]) K    x ∈ξ  [_F],    im(  𝒳_σ_1)  =    0.9* ([     ϖ     a;           1            ;                 ϖ      ;        x  ϖ - a̅ϖ  ϖ ^2 ]) K      a ∈ [_E]  ,  
          x   ∈ξ   [_F],
 

    im ( 𝒳_σ_2  )  = 
    0.9* ([              ϖ^2           a_1  ϖ a a_1 +  y+ x  ϖ            -ϖ a̅;                                 ϖ                a                 ;                                                  1                 ;                                             -a̅ _1                ϖ ]) K       a ,  a_1∈ [_E]   ,  
           x,  y   ∈ξ   [_F]   .

For the σ _0-cell, one eliminates the entry x ϖ by a row operation and conjugates by w_α_0  : = ϖ^(0,1,0)w_0 to arrive at the representative  ϖ^(2,2,1).  For the  σ_1-cell, one eliminates x  ϖ and conjugate by w_2 to arrive at   
    0.9[     ϖ                 a;         ϖ^2 - a̅ϖ      ;                 ϖ      ;                       1 ]
 
If a = 0, we get the representative ϖ^(2,1,2). If not, then conjugating by diag(-a^-1,1,-a̅,1) ∈ M^∘ leads us to ϖ^(1,0,1)τ_1 and we have U ϖ^(1,0,1)τ_1 K = U ϖ^(1,1,0)τ_1 K.  



As for the  σ _2-cell, begin by eliminating y + ϖ x in the third column using a row operation. 
If a_1 = 0 , a = 0, then we obtain the representatives  ϖ^(2,2,1). If a_1 = 0, a ≠ 0, we can conjugate diag( a̅^-1 , 1, a , 1 ) ∈ M ^ ∘ to obtain the representative   ϖ^(1,1,0)τ_1.  Finally, if a_1≠ 0, we can conjugate by diag(a_1 ^-1 , 1,  a̅_1 , 1 ) to arrive at the matrix  
    0.9[   ϖ^2     ϖ     u - ϖu̅;           ϖ     u      ;                 1      ;               - 1     ϖ ]
 
where u =  a / a̅_1∈_E. We can assume u ∈_F by applying row and column operations. If u = 0 at this juncture, we can conjugate by w_2 and diag(1,1,-1,-1) to obtain the reprsentative ϖ^(1,1,0) τ_1, and if u ≠ 0, then conjugating by diag(1,1,u,u) gives us the representative  τ_3. So altogether, we have 
    K w_0ρ^2 K  =  U ϖ^(2,2,1)  K ⊔ U  ϖ^(2,1,2) K ⊔ U  ϖ^(1,1,0)τ_1 K  ⊔ U τ_3 K.
 
 
∙ w =  w_0 w_1 w_0ρ^4. The Schubert cells for this word were all written in Proposition <ref>(b).  Here we have to analyze the U-orbits cells corresponding to words σ_0 and σ_2-
in the notation used there. We record the reduction steps for the σ_2-cell, leaving the other  case  for the reader. 

Begin by eliminating the entries x_1 ϖ^2 and x  
 ϖ^2 + y ϖ using row operations. Conjugating by  w_2 makes the diagonal ϖ^(4,3,3) and puts the entry a_1ϖ - ϖ^2a̅ and its conjugate on the top  right anti-diagonal. A case analysis of whether a, a_1 are zero or not   gives us ϖ^(4,3,3), ϖ^(3,2,2)τ_1 and ϖ^(2,1,1)τ_2 as possibilities.  
 


 §.§ Zeta  elements

Let U_1 be the F-torus whose R points over a F-algebra R are  given by U_1 (R ) =  { z ∈  E ^ ×  |    z γ(z) = 1 }. Then U_1(F)  ⊂_E^×  is  compact.    There is a homomorphism of F-tori given 𝒩 : Res_E/ F_m→U_1 given by z ↦  z / γ(z) with kernel _m. An application  Hilbert's Theorem 90 gives us that 𝒩 is surjective, inducing isomorphism  _E  ^ × /  _F ^ ×   =  E ^ × / F ^ ×U    _1(F).      
Denote  T = C :  =    U  _1(F), D =  𝒩 (  _F ^ ×  +   ϖ_E ), and define  
    ν :  H → T,    (h_1 , h_  2  )  ↦ h_2 /  h_1   .
   Fix  an  integral domain containing [ q ^-1 ].   For  the  zeta  element  problem, we take  
          

  * G̃ : = G × T the target group,

  * ι̃  : = ι×ν : H →G̃,  


  * M_H, =  M_H, 𝒪, triv the trivial functor, 

  * U and K̃ : = K × C as bottom levels,   

  * x_ U   = 1 _∈  M_H,  ( U ) as the source bottom class, 

  * L̃  =  K × D as the layer extension degree d = q + 1, 

  * ℌ̃  _c =  ℌ_bc,c(Frob)  ∈𝒞_ (   K̃\G̃ /  K̃  )  the Hecke polynomial where Frob= (C).   
 




              If 2 ∈_F^×,    there  exists a zeta element for (x_U ,  ℌ̃_c,  L̃  ) for all c ∈∖   2.                     
   

 For i = 0 , 1, 2 , 3, let g_i =  ( τ_i ,1_T ) ∈G̃  span a zeta element. Using  centrality of ρ^2 and that c is odd,  we see that  
    ℌ̃_c≡ (1 - ρ^2)^4 (K̃) - (1  -  ρ^2) ^2 ( K̃w_0ρ^2K̃) +  (K̃ w_0 w_1 w_0ρ^4K̃)  q+1

where we view w_i , ρ etc., as elements of G̃ with 1 in the T-component. It follows from  Proposition <ref> that 
    H \ H ·(ℌ̃_c) / K̃= { H g_iK̃  |   i = 0,1,2,3 } .
  For i = 0 ,1 , 2 ,3,  let 𝔥_i∈𝒞_[q^-1](U\ H / H_g_i) denote the (H, g_i)-restriction for  ℌ̃_c (where H_g_i = H ∩ g_iK̃  g_i^-1) and d_i = [H_g_i : H ∩ g_i L g_i^-1 ]. Then 

    𝔥_0   ≡ (1 - ρ^2)^4 ( U ) - ( 1  -  ρ^2)^2  ( U ϖ^(2,2,1) U )  + ( U ϖ^(2,1,2) U )  )  +  ( U ϖ^(4,3,3) U )  ,  
    𝔥_1   ≡ - ( 1-  ρ^2) ^2 ( Ũϖ^(1,1,0) H_g_1) +  ( U ϖ^(3,2,2) H_g_1  )  ,   
    𝔥_2   ≡  (U ϖ^(2,1,1) H_g_2)  ,   
    𝔥_3   ≡  - ( 1 - ρ^2)  ^   2   ( U H_g_3)   .
 
modulo  q + 1. Since ρ^2 is central, we see that 
    (𝔥_0^t)     ≡   [ U ϖ^(4,3,3) U]_*  = (q+1) ^2≡ 0  q+1
    (𝔥_3^t)      ≡  0   q+1 .
  
Now since d_i | (q+1) for all i,  we see that d_0 |   (  𝔥_0^t ) and d_3  |   (  𝔥_3^t  ).  Next observe that  H_g_i = H ∩τ_i K τ_i^-1 for all i. If we write h  ∈ H as in (<ref>), we see that for i = 1 , 2,  
    τ_i h τ_i^-1 =  
    0.9([            a          c_1   b + c_1ϖ^i d_1  - a ϖ^i;           -c          a_1 a_1  - d ϖ^i  b_1 + c ϖ^i;          c ϖ                         d           -c;                    c_1  ϖ          c_1          d_1 ])

If  now  h ∈ H_τ_i, then the matrix above lies in K and thus all its entries must be in _F.   It is then easily seen that H_τ_1, H_τ_2⊂ U and that ν(h) ∈  1 + ϖ_E⊂  D. So d_1 = d_2 = 1 and   d_1|(𝔥_1^t), d_2 |    (𝔥_2^t)  holds  trivially.  We have therefore established that 
    d_i | (𝔥_i^t)     for     i = 0 , 1 , 2 , 3
 and the claim follows by  Corollary    <ref>.   
  

    The value of c in our normalization that is relevant to the setting of <cit.> is 1 since (q_E)^1/2  = q. Note that for even c, no zeta element exists in this setup. 
 




§ SPINOR   -FACTOR OF 
   
In the  final  section, we study the zeta element  problem for the embedding discussed  in <ref>.  


 
The symbols F, 𝒪_F, ϖ, , q and []  have the same meaning as in   Notation <ref>.     Let 𝐆 be the reductive over F whose R points for a F-algebra R are {  g ∈_4 ( R )   |  ^t g J_4 g =  sim(g) J_4 for sim(g) ∈ R ^ ×} where J_4 =  ( [         1_  2; - 1 _2        ] ) 
is the standard  symplectic matrix. The map g ↦(g) is referred to as the similitude character. We let 
    G = 𝐆(F) 
    ,    K = G ∩_4 (  _F ) .
 
For a ring R, we let ℋ_R  =    ℋ_R(K \ G / K) denote the Hecke algebra of G of level K with coefficients in R with respect to a Haar measure μ_G such that μ_G(K)=1.    For convenience, we will sometimes denote  (K σ K )  ∈ℋ_R  simply   as (K σ K ).  
   





 §.§ Desiderata
  


Let 𝐀 = _m^3, dis : 𝐀→𝐆 be the map (u_0 , u_1 , u_2 ) ↦diag ( u_1 , u_2 ,  u_0 u_1 ^-1 ,  u_0 u_2 ^-1 ). Then dis identifies 𝐀 with the maximal torus in 𝐆.   We  let A = 𝐀  ( F ) and A ^ ∘ = A ∩ K denote the unique maximal compact  open   subgroup. For i = 0 , 1 , 2,  let ϕ _ i, ε_i be the maps defined in <ref>. As before, we let  
    Λ =   ϕ_0⊕ϕ_1⊕ϕ_2
   denote the cocharacter lattice. The conventions for writing elements of Λ as introduced in <ref> are     maintained. 
The set Φ of roots of  relative to 𝐀 is  the set denoted Φ_F in <ref>.   The half sum of positive roots is  

    δ =   2 ε_1 + ε_2 -  32ε_0
  
We let α_1 = ε_1 - ε_2, α_2 = 2 ε_2 - ε_0 as our  choice of simple roots. Then  α_0 =  2 ε_1 - ε_0 the highest root.  The groups W, W_aff, W_I,  Ω,  the  set S _ aff  are analogous to the ones defined in <ref> 
We let ℓ :  W_I→ denote   the length function on W_I  The minimal length of elements in t(λ) W  ⊂Λ⋊ W ≃ W_I  can be computed   using the formula    
(<ref>).  Set 
    w_0 =   0.9[           1 /ϖ;         1;    ϖ;                 - 1 ],       w_1 =      0.9[   1    ; 1  ;       1;     1 ] ,       w_2 =      0.9[   1            ;               1;         - 1    ;       1         ],     ρ  =     
    0.9[       1;     1  ;   ϖ    ; ϖ       ]     .

These represent the elements t(α_0^∨ ) s_0, s_1, s_2, t(-ϕ_0) s_2 s_1  s_2  in W_I.    
We  let w_α_0  : =  w_1 w_2 w_1  = ϖ^ϕ_1 w_0∈ N_G(A), which is a  matrix representing  the  reflection s_α_0.        For i = 0 , 1 ,2, let  x_i : _a→𝐆  be the root group maps 

    x_0: u ↦0.9[   1            ;       1        ; ϖ u       1    ;               1 ],         x_1: u ↦0.9[    1    u          ;         1          ;              1     ;           -  u    1 ],             x_2: u ↦0.9[ 1      ;   1   u;     1  ;       1 ]
    
and let g_i :  [     ]   → G be the map κ↦ x_i ( κ ) w_i.   If I denotes the Iwahori subgroup of K whose reduction modulo ϖ lies in the Borel of () determined by Δ = {ε_1 - ε_2, 2 ε_2 - ε_0},  then I w_i I  = _κ∈ [_w_i]  g_w_i ( κ ) .      For w ∈ W_I such that w is the unique minimal length element in w W, choose a reduced word decomposition  w  =  s_w, 1 ⋯ s_w, ℓ(w) ρ_w, where s_w,i∈  S_aff, ρ_w∈Ω,  a reduced word  decomposition. As usual, define  
         𝒳_w  :   [ ] ^  ℓ(w)          →G / K    

 ( κ_1 ,   …,  κ_ℓ(w)   )         ↦g_s_w, 1      ( κ_1 )  ⋯g_s_w , ℓ(w)    ( κ_ℓ(w) )   ρ_w K      
Then im(𝒳_w ) is independent of the choice of decomposition
of w by   Theorem <ref>.    




 §.§ Spinor  Hecke  polynomial
    The dual  group of  𝐆 is GSpin_5 which has a four dimensional representation called the spin representation. The highest coweight of this representation is ϕ_0  + ϕ_1 + ϕ_2 (see <ref> for arithmetic motivation) which is minuscule. By Corollary <ref>, the coweights are 2ϕ_0 + ϕ_1 + ϕ_2/2±ϕ_1/2±ϕ_2/2. The Satake polynomial is therefore 
    𝔖_spin  ( X  ) =  ( 1 - y_0 X ) ( 1 - y_0 y_1 X ) ( 1- y_0 y_2 X ) ( 1 - y_0 y_1  y_2  X )   ∈ [ Λ ] ^ W  [X]

where y_i = e ^ ϕ_i∈ [ Λ]. Let ℛ =   [ q ^ ±1/2 ], and  𝒮  : ℋ_ℛ  ( K \ G / K ) →ℛ [ Λ] ^W denote the Satake isomorphism (<ref>).  For c ∈  - 2,  the  polynomial    ℌ_spin, c (X)  is defined so that 𝒮  ( ℌ_spin,c ( X )    ) =  𝔖_spin(q^-c/2 X ).
                              For  c ∈∖ 2,  

    ℌ_spin,c(X)      =   (K) -  q ^   - c+3/2 (K ρ K ) X   
        +  q ^-(c+2)  (  ( K w_0ρ ^2 K )  + ( q^2 + 1 ) (   K ρ^2 K )     ) X^2
        - q ^  - 3c+ 3/2  ( K ρ^3K ) X^3  +   q^-2c  ( K ρ ^ 4 K )  X^4∈ℋ_[q^-1]( K \ G    /  K  ) [X] .

where the words appearing in each Hecke operator are of minimal possible length. 

 


  We have 
    𝔖_spin(X) =  1 - e^W(1,1,1) X +   (  e ^  W(2,2,1) + 2 e ^W(2,1,1) ) X ^ 2    -  e ^  W(3,2,2) X^3 +  e ^  (4,2,2)   X^4 .

The lengths of the cocharacters appearing as  exponents in the coefficients of 𝔖_spin(X)  is computed using the formula (<ref>)  and  the  corresponding  words   are  easily  found.    The leading coefficient  (see Definition <ref>) of K ϖ^λ K for λ∈Λ^+ is   q ^  -  ⟨λ , δ⟩ (Corollary <ref>) shifted by an appropriate power of q^-c/2, which are  easily  computed.      The  coefficient of the non-leading term (K ρ^2 K)  in the monomial X^2 is computed as follows. Consider the Weyl orbit diagram  

    (2,0,1)[r, "s_1"]    (2,1,0)[r, "s_2"]    (2,1,2)[r, "s_1"]    (2,2,1)

of (2,2,1).    
From (<ref>)
and Theorem <ref>,  we see that  | K w_0ρ^2 K / K | =  q + q^2 + q^3 + q^4. 
Since the leading coefficient of the Satake transform   of  ( K w_0ρ^2 K ) is q^⟨ (2,2,1), δ⟩ = q^2, the number of cosets in K w_0ρ^2 K / K whose shape  lies in the W orbit of (2,2,1) is 
    ∑  _μ∈ W (2,2,1) q ^ ⟨     (2,2,1) + μ , δ⟩  =  1 + q + q ^3 + q ^4 .
 Thus the required coefficient is q^-c multiplied with  2 - q^-2( q + q ^2 + q^3 + q^4  -  ( 1  + q + q^3 + q^4 ) ) = q^-2 (  q^2 + 1  ).
   
 The formula for ℌ_spin,c is again  well known, e.g.,  see <cit.> or <cit.> where c is taken to be - 3.  We  have however included a proof for  completeness and to provide a check on our computations.       
The  dual  group  of   𝐆 also has a 5 dimensional representation called the standard representation. Its highest   coweight is ϕ_1 and it's  Satake polynomial is 
    𝔖_std(X) =  ( 1-  X )( 1 -  y_1^-1 X ) (   1 -   y_1 X ) ( 1 - y_2  ^-1  X  ) (  1-  y_2 X ) .

Cf.  the   polynomial   𝔖_bc(X) of <ref>.  See   <cit.> for a discussion of this L-factor.   
 



 §.§ Mixed coset decompositions
    

Let 𝐇 be the subgroup of 𝐆 generated by 𝐀 and the root groups of ±α_0, ±α_2.  Then 𝐇≅_2×__2, the fiber product being over the determinant map. 
Explicitly, we get an embedding   

    ι :  →       (  (  [ a b; c d ] )   ,   (   [ a_   1    b_1;  c _ 1    d_1 ] )  )  
    ↦ (    [   a       b;     a_1     b_1;   c       d;     c_1     d_1 ] )
    
Set  H =  𝐇(F), U = H ∩ K,  W_H = ⟨ s_0 , s_2⟩≅ S_2× S_2 the Weyl group of H and Φ_H := {±α_0,  ±α_2} the set of roots of 𝐇. 
For convenience in referring to the components of H, we let H_1, H_2 denote _2(F) (so that H = H_1×_F^× H_2) and _i : H → H_i for i =1,2 denote the natural projections  onto the two component groups of H.   To describe the twisted H-restrictions of the spinor Hecke polynomial, we introduce the following elements in G: 
    τ_0 = 0.9[ 1      ;   1  ;     1  ;       1 ],       τ_1    =   
    0.9[ ϖ     1;   ϖ 1;     1  ;       1 ],
 
As  in  <ref>, we will need to know the strucuture of H_τ_1 = H ∩τ_1 K τ_1^-1,  
Let =   [   1; 1   ]∈_2(F) and define 

    : _2(F)     ↪   H 
    
      h 
         ↦ ( h,  h    
    ) .
  
Let 𝒳^∘ = (_2(_F), 𝒳 = (_2(F) ) and 
J   be the compact open subgroup of H_τ_1 whose reduction modulo ϖ lies in the diagonal torus of 𝐇().      

      H _ τ_1  =  𝒳 ^ ∘     J  ⊊ U. In particular, H K and H τ_1 K are disjoint. 
  
  Let  h =  ( h_1 , h_2  )    ∈ H and say h_i : =    (   [ a_i b_i; c_i d_i ] )
where a_i, b_i, c_i , d_i∈ F. Then h ∈ H_τ_1 implies that    

    a_1, a_2 , c_1, c_2 , d_1 , d_2∈_F     and      a_1 - d_2   ,   
    a_2 - d_1   , b_1 - c_2,  b_2 - c_1∈ϖ_F .
  
It follows that 𝒳 ,    J ⊂   H_τ_1⊂  U. 
In particular,  
H_τ_1⊃𝒳 J. 
For the reverse inclusion, say   h = (h_1, h_2 ) ∈ H_τ_1. Since (h_1) ∈𝒳⊂ H_τ_1, we see that  (h_1', h_2') :=  (h_1^-1) ·  h lies in H_τ_1. By construction, we  have h_1 ' = 1_H_1. The conditions of the membership (1_H_1 , h_2 ' ) ∈ H_τ_1 force  are 
(1_H_1 , h_2' ) ∈  J. For the second claim, note that H_τ_1≠ U since A^∘⊄H_τ_1 and  invoke  Lemma <ref>.  
 
For w ∈ W_I,  let ℛ(w) denote the double coset U \ K w K / K.  As before,   we l only write the representative elements when describing ℛ(w) and these  representatives are understood to be distinct.   




        We have                     
         

  * ℛ ( ρ )  =   {ϖ^(1,1,1) ,   τ_1}, 

  * ℛ  (   w _0ρ  ^  2   )   =   {ϖ ^(2,2,1) ,  ϖ^(2,1,2)  ,   ϖ^ (1,1,0)τ_1}. 
 


    

  Since H K and H τ_1 K are disjoint, H_τ_1⊂ U and  U \ H / U ≃ W_H\Λ,  the listed elements represent distinct classes in their respective  double coset spaces. To show that they represent all classes, we study the orbits on K w K / K  using   Theorem <ref>.  



∙ Let w =  ρ. We have K w K  / K =     _ σim ( 𝒳_σ ) for σ∈{ w, w_2 w, w_1 w_2 w, w_2 w_1 w }.  To obtain the mixed  representatives, we need to analyze the U-action on the cells corresponding to the words σ_0 =  ρ and σ_1 =  w_1 w_2ρ.  The first is a singleton and gives ϖ^(1,1,1) (after conjugating by w_α_0  w_2). As for σ_1, we have    
    im(  
    𝒳_σ_1     ) =        0.9*  [  ϖ  a  y   ;     1      ;        1   ;       -a  ϖ ]     K      a , y ∈ []
 We can eliminate y by a row operation from U, and conjugating by w_2 gives us a matrix with diagonal ϖ ^(1,1,1). If a = 0, we obtain ϖ^(1,1,1) and  if   a ≠ 0, we conjugate by diag(1, 1, a , a) to obtain τ_1. 

 



∙   Let w =        w_0ρ  ^2. From diagram (<ref>), we have K w K  / K = _σim ( 𝒳_w ) for σ∈{ w, w_1 w, w_2w_1 w, w_1 w_2 w_1w } and it suffices to analyze the cells corresponding to σ_0 = w, σ_1 = w_1 w, σ_2 = w_1 w_2 w_1 w. These cells are as follows:  

    im(𝒳_σ_0)  =  0.9* ([   1;       ϖ        ; x ϖ     ϖ^2    ;               ϖ ]) K    x ∈,    im(  𝒳_σ_1)  =    0.9* ([        ϖ        a;                 1                  ;                          ϖ         ;              x  ϖ - 
     a  ϖ     ϖ ^2 ]) K   
    a, x  ∈ []  ,,
 

    im ( 𝒳_σ_2  )  = 
    0.9* ([              ϖ^2           a_1  ϖ a a_1 +  y+ x  ϖ             a  ϖ;                                 ϖ                a                 ;                                                  1                 ;                                              - a_1                ϖ ]) K    a , a_1, x , y ∈ []   .

The σ_0-cell obviously leads to ϖ^(2,2,1). For the σ_1-cell we can eliminate x ϖ, conjugate by w_2. If a = 0, we have ϖ^(2,1,2) at our hands and if not, then conjugating by diag(1,1,a,a) gives us ϖ^(1,0,1)τ_1. Now observe that since  () ∈  H_τ_1 is a lift of s_0 s_2, we have 
    U ϖ^(1,0,1)τ_1 K = U ϖ^(1,1,0)τ_1 K .
    
Finally for the σ_2-cell, begin by eliminating aa_1 + y  +    ϖ x. Next note that conjugation by w_2 swaps a_1 and a. Using row and column operations, we can assume that a_1 = 0. If a = 0, we end up with ϖ^(2,2,1) and if not, then  conjugation by  diag(1,1,a,a) gives us ϖ^(1,1,0)τ_1.   
 









 §.§ Schwartz space   computations
    
Let   X : = F^2× F^2 considered as a totally disonnected topological spaces.  We view elements of X as pairs of 2 × 1  column   vectors. We  let  H_1× H_2 act on X on the right  via 
    ( u⃗  , v⃗   ) · (h_1, h_2)  ↦  (  h_1 ^  - 1  u⃗  ,   h _ 2   ^  - 1   v)      u⃗  ,   v⃗∈ F^2,    h_1∈ H_1,      h_2∈ H_2 .

Via the natural embedding H ↪ H_1× H_2, we obtain an action of H on X.  
Let   be an integral domain that contains [ q ^-1 ] and let 𝒮_X  =  𝒮_X,𝒪 be 𝒪-module of 
all functions  ξ : X → which are locally constant and   compactly supported  on X. Then 𝒮_X has an induced  left action 𝒮× H →𝒮 via (h,  ξ    ) ↦ξ ( ( - ) h  ),  which makes 𝒮 a smooth representation of H.  Let Υ_H be the set of all compact open subgroups of H and 
    M_H, :  𝒫(H, Υ _    H  
      )  →𝒪-Mod
 denote the functor V ↦𝒮_X^V associated with 𝒮  (see Definition <ref>).  For u, v, w , x  ∈, let Y_u  := ϖ^u _F⊂ F,  
Y_u,v = Y_u× Y_v⊂ F^2 and   X  _u,v,w,x  := Y_u,v× Y_w, x⊂ X. We denote  
    ϕ_(u,v,w,x) :=  ( Y_u,v,w,x ) ,       ϕ̅_(u,v,w,x) = ϕ_(-u,-v,-w,-x)
 where ch(Y) denotes the  characteristic     function of Y ⊂ X.  These belong to 𝒮.  We  also denote   ϕ : =  ϕ_(0,0,0,0) for simplicity.     The element ϕ will serve as the source bottom class of the zeta element.   

             We have 
              
    
  * [ U ϖ^(1,1,1) U ]_* ( ϕ 
    ) =   ϕ̅_(1,1,1,1) + q ( ϕ̅_(1,1,0,0) +  ϕ̅_(0,0,1,1) )  + q^2ϕ, 
    
  * [ U ϖ^(2,2,1)  U ]_*(  ϕ 
    )  =   ϕ̅_(2,2,1,1) +  ( q - 1 )  ϕ̅_(1,1,1,1)  +  q^2ϕ̅_(0,0,1,1), 
    
  * [ U ϖ^(2,1,2)  U ] _*(  ϕ 
    )  =  ϕ̅  _(1,1, 2,2)  +  (  q - 1 )  ϕ̅_(1,1,1,1)   +  q ^2ϕ̅_(1,1,0,0). 
   
 

 
   
                
    




     If we denote U_1 = U_2 := _2( _F ) and pick any λ = (a_0, a_1, a_2 ) ∈Λ, we have  

    [ U ϖ ^λ U ]_* (ϕ_(u,v,w,x) ) = [U_1 t_1 U_1 ]_*   (ϕ_(u,v)  )   ⊗   [U_1 t_2 U_2]_* ( ϕ_(w,x)     )
 where t_i = diag (ϖ^ a_i,   ϖ ^  a_0 - a_i     ) for i = 1, 2  and  ϕ_(a,b) : F^2→𝒪 denote   the characteristic function of Y_a× Y_b for a,b, ∈.  The resulting functions  can be computed using the decomposition recipe of Theorem <ref>. See also 
 <cit.> for a more general  result.  
 
To facilitate checking the trace criteria  for one of the twisted restrictions, we do a preliminary calculation.  Let Mat_2 × 2 (F) be the  F-vector space 2 × 2 matrices over F. We make the identification 

    : X     Mat_2 × 2 ( F )        1.1(  ( [ u_1; u_2 ] ),   (   [ v_1; v_2 ]   )  )  
    ↦ ( [ u_1 v_2; u_2 v_1 ] )  .
 
and define a right action 

    Mat_2 × 2(F) ×_2(F)  →Mat_2 × 2(F)       (h, M) ↦ h^-1 M
  
Then for all  h ∈_2(F) and (u⃗, v⃗ ) ∈ X, 
    ( (u⃗ , v⃗ ) · ( h )  )  = ( u⃗ ,  v⃗  ) ·   h

where  is as in  (<ref>) and the action on the right hand side is (<ref>).     Let ψ∈𝒮_X denote the function such that ψ∘^-1 : Mat_2 × 2 (F) →𝒪 is the characteristic function of diag(ϖ, ϖ)^-1·_2 ( _F ). 
Let

    𝔥_1'  : = q ( U  H_τ_1  ) 
    - ( U 
    ϖ^(1,1,0)  H_τ_1 
    )  +   (   U ϖ ^ ( 2, 1,1)   H_τ_1   )      ∈𝒞_(U \ H / H_τ_1   )
 
            𝔥_1,*'(ϕ) = ψ.    
      
 By  Lemma <ref>,  U H_τ_1 = U 𝒳^∘, U ϖ^(2,1,1) H_τ_1 = U ϖ^(2,1,1)𝒳^∘ and U ϖ^(1,1,0)  
H_τ_1 
=    U ϖ^(1,1,0)𝒳 ^ ∘,    
where  we used that ϖ^(1,1,0) J ϖ^-(1,1,0)⊂ U  in the last equality.  
Moreover  ϖ^(1,1,0) =  (diag(ϖ, 1)    ) and ϖ^(2,1,1) =   ( diag(ϖ , ϖ )  ) and U ∩𝒳 = 𝒳^∘. A straightforward analogue of  Lemma <ref> implies that we have a bijection 

    𝒳^∘\𝒳^∘  h 𝒳^∘     U \ U (h) 𝒳^∘
    𝒳^∘γ   ↦  U (γ)
 
Therefore   𝔥_1,* '( ϕ ) = 
 ( q  (ϕ) - T_ϖ ^t·(ϕ) +  S_ϖ^t·(ϕ)  ) ∘ where T_ϖ, S_ϖ are the Hecke operators of _2(F) given by the characteristic functions of  _2 ( _F )-double cosets of diag(1,  ϖ ), diag(ϖ, ϖ  )  respectively, T_ϖ^t, S_ϖ^t denotes their transposes and the action of these operators is via  (<ref>)    
Now (ϕ) is just the characteristic function of Mat_2× 2 ( _F  ). A 
straightforward  computation shows that the function   
    q  (ϕ) - T_ϖ ^t·(ϕ) + S_ϖ^t·(ϕ)
 on Mat_2 × 2(F)  vanishes on any  matrix whose entries are not in ϖ^-1_F or whose determinant is not in ϖ^-2_F^×. The claim  follows.     
    
 A very closely related computation  appears in <cit.> in the context of Kato's Euler system, which is what inspired the choice of 𝔥_1 above. 
  





 §.§ Zeta   elements
 Following the discusion in  <ref>, we introduce    T =   F  ^ ×, C  =  _F^×  and D = 1 +  ϖ_F⊂ C. We let ν =  sim∘ι  : H → T be the map that sends (h_1 , h_2 ) to the common determinant of h_1 , h_2. For  the zeta  element problem,  we set   
        

  * G̃ = G × T, 

  * ι_ν = ι×ν : H  →G̃, 

  * U and K̃     :   = K × C as bottom levels

  * x_U  =  ϕ = ϕ_(0,0,0,0)∈ M_H,(U) as the  the source bottom class

  * L̃  = K × D as the  layer extension  
of degree q - 1, 

  * ℌ̃_c = ℌ_spin, c  (Frob )  ∈𝒞_  (   
K̃\G̃   /  K̃ ) as the Hecke  polynomial.    
 
            There  exists  a     zeta  element     for (x_U ,   ℌ̃ _c ,  L̃ 
) for all c ∈ - 2.          
 
 Denote ϱ   = (ρ , ϖ) ∈G̃.   By Proposition <ref>, we see that  
    ℌ̃_c≡  ( 1 + 2 ϱ^2 +  ϱ^4 )  ( K̃ )  - ( 1 + ϱ^2 ) ( K̃ϱK̃  )  + (K̃ w_0ϱ^2K̃)  q-1

where we view w_0∈G̃ via 1_G×ν.  
For i = 0, 1, let g_i = (τ_i, 1_T). By  Proposition <ref>, we see that H \  H · (  ℌ̃  _c ) / K̃  =  { H g_0K̃ ,  H  g_1K̃}   .
So it suffices to consider restrictions with respect to g_0 and g_1. 
Let 𝔥_i denote the (H,g_i)-restriction of ℌ̃_c.   Observe that 
    H_g_i = H ∩ g_iK̃ g_i^-1 = H ∩τ_i K τ_i^-1 ,
  so that 𝔥_i∈𝒞_𝒪( U \ H / H _τ_i ).     Let z =  (  ( [ ϖ;   ϖ ] ) ,  (  [ ϖ;   ϖ ] )   )  ∈ H.  Invoking  Proposition   <ref> again, we see that 

    𝔥_0   ≡   ( 1 + 2 z + z^2 ) ( U ) -  ( 1 + z ) ( U ϖ^(1,1,1) U)  + ( U ϖ^(2,2,1) U )  + (  U ϖ^(2,1,2)  U  )      
    𝔥_1   ≡𝔥_1'
  
modulo q- 1.  Note that the   action of z on ϕ in the covariant convention is by its inverse. To avoid writing minus signs, let us denote z_0 = z^-1.  Then by Lemma <ref>,  

    𝔥_0,*(ϕ)    ≡  ( 1 + 2z_0 +  z_0^2) ϕ  - ( 1 +  z_0  )  ( z_0·ϕ +  ϕ̅_(1,1,0,0) + ϕ̅_(0,0,1,1) + ϕ )    +  
          (  
     z_0·ϕ̅_(1,1,0,0) + ϕ̅_(0,0,1,1))  +  (   z_0·ϕ̅ _(0,0,1,1)   +  ϕ̅_(1,1,0,0) )  
         = 0   q-1
 
On the other hand, 𝔥_1,*(ϕ) ≡𝔥_1,*'(ϕ) = ψ. It is easily seen that the stablizer of every point in supp(ψ) in H_τ_1 reduces to identity modulo ϖ.  In particular, these stabilizers are  contained in the subgroup H ∩ g_iL̃  g_i^-1 of H_τ_1.  So by  Theorem <ref>, ψ is in the image of the trace map 
    _* : M_H, 𝒪 (H ∩  g_iL̃ g_i   ^-1  ) 
    → M_H, 𝒪(H_τ_1) .
  We now invoke Corollary  <ref>.   
    












 

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