Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$
Syed Waqar Ali Shah

§ ABSTRACT  We establish  abstract  horizontal 
norm relations involving the unramified Hecke-Frobenius  polynomials  that    correspond under the Satake isomorhpism to the degree  eight  spinor L-factors  of  GSp_6. These relations    apply  to    classes     in   the  degree seven motivic  cohomology of the Siegel modular sixfold obtained 
via   Gysin   pushforwards of Beilinson's  Eisenstein symbol 
pulled back on one  copy in a triple   product of modular curves. The proof is based on   a novel    approach  that circumvents the failure of the so-called   multiplicity one hypothesis in our setting, which precludes the applicability  of an  existing technique.  In a sequel, we combine our result with  the     previously established vertical  norm  relations for these classes to  obtain   new    Euler systems    for  the    eight dimensional  Galois representations associated with    certain     non-endoscopic  cohomological  cuspidal automorphic representations of GSp_6.   
    
Better bounds on Grothendieck constants of finite orders
    Sebastian Pokutta
    5th September 2024
========================================================



§ INTRODUCTION



  
Ever since the pioneering work of Kolyvagin,  the machinery of Euler systems has become  a standard tool for probing the structure of Selmer groups of global Galois representations  and for    establishing  specific  instances of Bloch-Kato and Iwasawa main conjectures.  Recently, there has been an interest in constructing Euler systems  for Galois representations found in the cohomology of Siegel modular varieties. 
In  <cit.>, the authors constructed an Euler system for certain four dimensional  Galois representations found in the middle degree cohomology of the GSp_4 Siegel modular variety.      They also introduced a  new  technique of  using local zeta integrals   that  has been applied with great success in many other settings (<cit.>, <cit.>, <cit.>, <cit.>).   

The natural successor  of  GSp_4 in  Euler system based investigations is  the Siegel modular variety attached to  GSp_6. This is a sixfold 
whose middle degree  cohomology realizes the composition of the spin representation with the GSpin_7-valued Galois representation associated under  Langlands correspondence with certain 
cohomological  
cuspidal automorphic representations of GSp_6  <cit.>,   <cit.>.     
A standard paradigm for constructing  Euler systems for such geometric  Galois representations is via pushforwards of  a   special family  of  motivic cohomology classes known as  Eisenstein symbols. 
A natural candidate  class 
in the  GSp_6  setting    is the  pushforward of the Eisenstein symbol pulled 
back  on one copy in a triple product of modular curves. Besides having the correct numerology, this particular choice of pushforward is  motivated   by a period integral of  Pollack and Shah <cit.>,  who showed that integrating certain cusp forms of GSp_6 against an Eisenstein series on one copy in a  triple product of _2  retrieves the degree eight (partial)   spinor  L-function  for  that     cusp  form. In <cit.>, the authors  use  this   period integral to relate the regulator of our candidate class in Deligne-Beilinson cohomology to non-critical special values of the spinor L-function, thereby providing  
 evidence that it  sits   at the bottom of a non-trivial  
 Euler system  whose behaviour can be explicitly    tied to special  L-values.     

To construct an Euler system above this  class,  one needs to produce  classes going up the abelian tower over    that   satisfy among themselves  two kinds of norm 
relations. One of these is the vertical relations that see variation along the _p-extension and are Iwasawa theoretic in nature.  These have already been verified  in   <cit.> using a general method later  axiomatized in <cit.>.  
 The other and typically more challenging kind is  the horizontal relations that see variation along ray class extensions and involve local L-factors   of the Galois representation.  These present an even  greater  challenge in the GSp_6 case,  since one is dealing with a non-spherical pair of groups and  the  so-called multiplicity one hypothesis on a local space of linear functionals fails to hold. In particular,  the  technique of   local zeta  integrals of <cit.> and  its variants cannot be applied in this situation to establish horizontal norm compatibility. 
 
The purpose  of  this article is to establish the ideal version of this compatibility   using  a    fairly     general  method developed by us  in   a  companion article   <cit.>, thereby completing the Euler system construction envisioned in <cit.>.    
For convenience and to free up notations that play no role outside the proof of our norm relations, we have chosen to cast our result in the framework of abstract cohomological Mackey (CoMack) functors[the more relaxed notion of “Mackey functor" is referred to as a “cohomology functor" in <cit.>]. 
The application to p-adic étale   cohomology and the actual Euler system construction  is recorded in a sequel <cit.>. In future, we also expect to establish an explicit reciprocity law relating this Euler system to special values of the spinor L-function by means of a p-adic L-function, thereby making progress on the Bloch-Kato and Iwasawa main conjectures in this setting.      



 §.§ Main result
 
Let = GSp_6, = ×_m  and  = _2×__m_2×__m_2 where the products in 𝐇 are  fibered  over the   determinant map.  There is a natural embedding ι : ↪ and if sim : →_m denotes the  similitude map, then post composing ι with  1_×sim : → gives us an embedding 
    ι̃ :  ↪
  via  which we view  as a subgroup of .  For ℓ a rational prime,  let G_ℓ    denote the groups of _ℓ-points of  and  
let  ℋ_R  denote the spherical Hecke algebra of G_ℓ with coefficients in a ring R.  For c an integer, let ℌ_ℓ,c(X)  ∈ℋ_[ℓ^-1][X] 
denote the unique polynomial in X such that for any (irreducible)  unramified representation π_ℓ of G_ℓ and any spherical vector φ_ℓ∈π_ℓ, 
    ℌ_ℓ,c(ℓ^-s)  ·φ_ℓ  = L(s+c, π_ℓ  ,  Spin) ^-1·φ_ℓ
  
for all s ∈ℂ. Here  L(s,  π_ℓ, Spin) denotes the spinor L-factor  of π_ℓ normalized as in  <cit.>. 
Fix  any  finite set S of rational  primes and let  G,  G̃, H denote the group of _S·_f^S-points of ,  ,  
respectively. 
Fix  also   a neat compact open subgroup K ⊂ G such that K is unramified at primes away from S.    Let 𝒩 denote the set of all square free  products of primes outside  S  (where the empty product  means  1) and for n ∈𝒩, denote  
    K[n]  = K ×∏_ℓ∤ n_ℓ^×∏_ℓ| n  ( 1+ ℓ_ℓ)      ⊂G̃  .
 
Let 𝒪 be a characteristic zero integral domain such that ℓ∈𝒪^× for all ℓ∉ S. Denote by 𝒮 =  𝒮_𝒪 the  𝒪-module  of  all locally constant compactly supported  functions  χ  :Mat_2 × 1 (_f) ∖{ 0 }→𝒪 such that χ  = 
f_S⊗χ  ^S where f_S is a fixed  function on Mat_2 × 1 (_S) that is invariant under (_S ) under the natural left action of H on such functions.   We  view the association  V ↦𝒮(V) that sends a compact open subgroup V of H to the V-invariants of 𝒮 as a CoMack functor for H.   
Let U = H ∩ K[1] and let 

    ϕ = 
    f_S⊗(^S) ∈𝒮_(U)
 where ^S = ∏_ℓ∉ S _ℓ denotes integral adeles away from S. Finally, let  Frob_ℓ denote (ℓ_ℓ^×).  
  

[Theorem <ref>]   For any 𝒪-Mod valued cohomological Mackey functor  M_G̃ for  G̃, 
any Mackey pushforward   ι̃_* :   𝒮→   M_G̃ and any integer c,   there exists a collection of classes y_n∈ M_G̃(K[n]) indexed by n ∈𝒩 such that y_1 = ι̃ _U, K[1],*(ϕ) and 
    [ℌ_ℓ,c(Frob_ℓ)]_*( y_n  )   = pr_K[nℓ],  K [n],*(y_nℓ)

for all n , ℓ∈𝒩 such that ℓ is a prime and ℓ∤  n.   
 

 
Here for a  locally constant compactly supported function f : G̃→𝒪, [ f  ]_* denotes the covariant action of f  and _* denotes the trace 
map of the functor M_G̃.   For sufficiently negative c, the Hecke polynomial  ℌ_ℓ, c(X) has coefficients in ℋ_. For such  c, the condition on invertibility of primes outside S  in  𝒪   can be dropped.     






In the intended application, the   functor 𝒮 over  parametrizes weight-k Eisenstein classes in the first motivic cohomology of the modular curve. Its  composition with   the étale regulator admits a _p-valued  version by   <cit.>,  which ensures integrality of    classes  in Galois cohomology corresponding to all choices of integral Schwartz functions.    
The set S corresponds to the set of “bad primes" where the behaviour of   Eisenstein classes is pathological and the function f_S is therefore not perturbed for Euler system purposes. The  functor for G̃ is the degree seven  absolute étale   cohomology on which (ℓ_ℓ^× ) acts covariantly  as  
 arithmetic Frobenius.   Moreover the 
 pushforward ι̃ _* is    obtained via  the Gysin  triangle in Ekedahl's “derived" category of lisse étale p-adic sheaves along with certain branching laws of coefficient sheaves on the underlying Shimura varieties.   The abstract formalism of functors used above applies to this cohomology theory  by various   results established in  
 <cit.>. 
  The   bottom class y_1 in our Euler system is meant to be a geometric incarnation of the Rankin-Selberg period integral of Pollack-Shah <cit.>[This integral is denoted by  I(ϕ, s) in loc. cit.] and is  expected to  be related to certain special values of the degree  eight   spinor L-function via this period.  See also  
 <cit.>.    
   










 §.§ Our  approach
  While Theorem <ref> is the key relation required for an Euler system,  its proof relies on  a  far more fundamental and purely local relation  that lies   at the heart of our approach. In a nutshell, our approach posits that if the convolutions of all ‘twisted’ restrictions to H_ℓ =  ( _ℓ ) of the Hecke-Frobenius polynomial with the unramified Schwartz function ϕ_ℓ =  ( [ _ℓ; _ℓ ] ) fall in the image of certain trace maps, then Theorem <ref> follows. This local relation is also exactly what is needed in <cit.>, as it allows us to synthesize the results of <cit.> with our own.

We state this relation precisely.    In analogy with the global situation, let 𝒮_ℓ denote the set of all 𝒪-valued locally constant compactly supported functions on  Mat_2 × 1   ( _ℓ  ). Again, this is a smooth H_ℓ-representation which we view as a  CoMack functor for H_ℓ.  Denote G̃_ℓ = (_ℓ) and   K̃_ℓ =  (_ℓ). For a  compactly supported function ℌ̃ : G̃_ℓ→ and g ∈G̃_ℓ, 
the 
(H_ℓ,g)-restriction   of    ℌ̃ is the function   
    𝔥_g : H_ℓ→𝒪      h  ↦ℌ̃(hg) .
   If ℌ̃ is K̃_ℓ-biinvariant,  then 𝔥_g is left invariant under U_ℓ = H_ℓ∩K̃_ℓ and right invariant under H_ℓ, g = H_ℓ∩ g K̃_ℓ 
 g^-1. It therefore induces   
an 𝒪-linear map 𝔥_g,*  :   𝒮_ℓ  (U_ℓ)   →𝒮_ℓ 
 (H_ℓ , g ). 
 Let V_ℓ , g denote the subgroup of all elements in H_ℓ, g whose similitude lies in 1 + ℓ_ℓ. 

[Theorem <ref>]     Suppose in the notation above,   ℌ̃ = ℌ_ℓ,c(Frob_ℓ) where c is any integer.  Then 𝔥_g,*(ϕ_ℓ) 
lies in the image of the trace map _* :   𝒮_ℓ  (V_ℓ,g) →𝒮_ℓ (H_ℓ, g) for every g ∈G̃_ℓ. 
 





Results analogous  to Theorem <ref> were obtained in <cit.>, which   strengthen the norm relations of <cit.> and <cit.> to their ideal (motivic)  versions.     The machinery of <cit.>  takes Theorem <ref> as input and gives Theorem <ref> as output,  and can also easily incorporate vertical norm compatibility once a local result has been established, say,   in the style of <cit.>.   
Our approach 
has also been  successfully  applied  in   forthcoming works   to obtain new  Euler systems for certain exterior square motives  in the cohomology of GU_2,2 Shimura varieties  <cit.> and for certain rank seven motives of type G_2 <cit.>.   All these results taken together point towards an intrinsic “trace-imbuing" property of Hecke polynomials attached to Langlands L-factors that seems to be  preserved under   twisted restrictions on suitable reductive  
 subgroups.  We 
hope to explain this property   more conceptually at a future  point.   
















   



 §.§ Outline
 We prove Theorem <ref> by explicitly computing the convolutions of twisted  restrictions of ℌ̃  = ℌ_ℓ,c(Frob_ℓ) with ϕ_ℓ. As this is rather involved,  we have divided the article into two parts, the first containing mainly statements and the second  their proofs.  Below we provide an  outline  of  the key steps.   

Note first of all that if 𝔥_g,*(ϕ_ℓ ) lies in the image of the trace map, so does 𝔥_η g γ,*(ϕ_ℓ  ) for any η∈ H_ℓ and γ∈K̃_ℓ. Thus it suffices to compute 𝔥_g,*(ϕ_ℓ ) for g running over  a choice of representatives for 
 H_ℓ\ H_ℓ·(ℌ̃  ) / K̃_ℓ.   
Since multiplies of ℓ - 1 obviously lie in the images   of trace maps that concern us,   it also suffices to compute these functions modulo ℓ - 1. This allows us to completely bypass the computation of ℌ_ℓ,c(X) by a property of Kazhdan-Lusztig polynomials. It is also straightforward to restrict attention to  ℌ : = ℌ_ℓ,c(1) ℓ - 1 by first restricting ℌ̃ to G_ℓ. The problem is then reduced to computing U_ℓ-orbits on certain double coset spaces  K_ℓ g K_ℓ/ K_ℓ  where K_ℓ = (_ℓ ) and   (K_ℓ g K_ℓ) is a Hecke operator  in  ℌ.    
The key technique that allows us to compute  these orbits  is a recipe of decomposing parahoric double cosets  proved in <cit.>. It is originally due to Lansky  <cit.> in the setting of  Chevalley groups.     


However even with the full force of this recipe, directly computing  the U_ℓ-orbits on  all the relevant double coset spaces  is a  rather  formidable  task, particularly because the  pair ( , ) is not spherical. See also Remark <ref>. 
What makes  this computation much more tractable is  the introduction of an intermediate group that allows us to  compute the twisted  restrictions in two steps. In the first step, we compute the restrictions   of ℌ with respect to the group H' _ℓ   = '(_ℓ) where ' = _2× __mGSp_4 .   
The pair (', ) is spherical, and a relatively straightforward 
computation shows that there are three H'_ℓ-restrictions corresponding to the representative elements 

    τ_0 =  ([ 1        ;   1    ;     1    ;       1;         1;           1 ] ), 
        τ_1 = ([ ℓ       1;   ℓ   1;     ℓ    ;       1;         1;           1 ] ), 
        τ_2 = ([    ℓ                ℓ^-1;         ℓ      ℓ^-1;              1     ;                ℓ^-1;                     ℓ^-1;                             1 ])

in G_ℓ. This is expected since a general “Schröder type" decomposition holds for the quotient    H'_ℓ\  G_ℓ / K_ℓ by a result of  Weissauer <cit.>. We  denote the (H'_ℓ, τ_i)-restrictions of ℌ by 𝔥_i. This step is recorded in <ref> and justifications  are   provided in <ref>.  

The second step is to compute the H_ℓ-restrictions of 𝔥_i for i = 0,1,2.  This essentially turns out to be a study of _2×__m_2-orbits on GSp_4-double cosets. Since (_2× __m , _2 , GSp_4 ) is also a spherical pair, this is again  straightforward for i = 0 and even for i =1 as the projection of H'_ℓ∩τ_1 K_ℓτ_1^-1 to  the   GSp_4(_ℓ)-component  turns out to be a non-special maximal compact open subgroup of  GSp_4(_ℓ). The more challenging case of i = 2 is handled by  comparing the  double cosets with a subgroup of GSp_4(_ℓ) deeper than the Iwahori  subgroup that sits in the projection of the twisted intersection.  For 𝔥_0 (resp., 𝔥_1), there turn out to be three (resp., four) restrictions  indexed again  by certain  “Schröder type" representatives. 
For 𝔥_2 however, there turn out to be  ℓ + 3 restrictions. 
We use the symbols ϱ, ς, ϑ for the set of distinct representatives of H_ℓ\ H_ℓ·(ℌ)  /K_ℓ which correspond to the H_ℓ-restrictions of 𝔥_0, 𝔥_1, 𝔥_2 respectively. The diagram below organizes these restrictions in a  tree.  


  

                                                                                                                     ℌ [dd] [rrrd] [lld]                                                                                                                                                                                                                                      

                               𝔥_0 [rd] [d] [ld]                                                                                                                                                                         𝔥_2 [lld] [ld] [d] [rd] [rrd]                                                                           

𝔥_ϱ_0     𝔥_ϱ_1                           𝔥_ϱ_2       𝔥_1 [lld] [ld] [d] [rd]     𝔥_ϑ_0     𝔥_ϑ_1     𝔥_ϑ_2                                                 𝔥_ϑ_3     𝔥_ϑ̃_k 

                               𝔥_ς_0                         𝔥_ς_1     𝔥_ς_2                                     𝔥_ς_3                                                                                                                                                                                        


 
Here the branch indexed by ϑ̃_k actually designates  ℓ - 1 branches, one for each value of k ∈{0,1, 2,3,…, ℓ-2}. Thus H_ℓ\  H_ℓ·(ℌ) / K_ℓ consists of  3 + 4 + (4 + ℓ -1 ) =  ℓ + 10 elements.  The corresponding  ℓ + 10 restrictions are recorded  in  <ref> and proofs of various claims are provided  in <ref>.   Once these restrictions are obtained, the final step is to compute their covariant  convolution   with  ϕ_ℓ. We show in <ref> that all resulting convolutions vanish modulo ℓ  - 1 except for 𝔥_ϑ_3,*(ϕ_ℓ). 
A necessary and sufficient criteria established in <cit.> allows us to easily determine that 𝔥_ϑ_3,*(ϕ_ℓ) lies  in the image of the appropriate trace map and thus deduce the truth of Theorem <ref>. 

   For comparison, the GSp_4 setting studied in <cit.> involved  only 2 restrictions, which explains why the test vector of <cit.> only required two terms to produce the  L-factor.  
 

  The mysterious vanishing of all but one of the convolutions modulo ℓ - 1 and 
 the simplicity of 𝔥_ϑ_3,*(ϕ_ℓ)  strongly suggest that a more conceptual proof of our result is possible. 
 




 §.§ Acknowledgements
   I would like to express my gratitude to Antonio Cauchi and Joaquín Rodrigues  Jacinto, whose work on Beilinson conjectures and vertical norm relations in the GSp_6 setting served as the inspiration for this article.  I am especially indebted to Antonio Cauchi for his  careful  explanation of the unfeasibility of a related construction    and for his unwavering support 
throughout the course of this project. In addition, I  thank  Aaron Pollack, Andrew Graham,  Christophe Cornut, Barry Mazur, Daniel Disegni,  David Loeffler and  Wei Zhang  for   several    valuable conversations in relation to the broader  aspects of this work. I am also grateful to  Francesc Castella,  Naomi Sweeting and  Raúl Alonso Rodríguez for some  useful  comments and  suggestions.   At various stages, the software  MATLAB® was used for performing  and organizing  symbolic matrix manipulations, which proved 
 invaluable in composing many of the  proofs.    







 



PART: 
Statements of results


§ GENERAL NOTATION
         The notations introduced here are used throughout this article except  for  <ref>.  For aesthetic reasons, we work  with an arbitrary local field of characteristic zero, though we only need the results over _ℓ.  

Let  F denote  a local field of characteristic zero, _F its ring of integers, ϖ a uniformizer, = _F / ϖ_F its residue field and q = |  |.   For a ≥ 0 an integer, we let [_a] ⊂_F denote a fixed set of representatives for _a = _F / ϖ^a_F and we omit the subscript a when a = 1. We let 0,1,-1 ∈ [] denote the elements that represent 0,1,-1 ∈  respectively.   For n an integer,  let 1_n denote the n × n identity matrix and J_2n =  ( [         1_n; - 1_n ] ) denote the standard 2n × 2n 
symplectic matrix. We define GSp_2n to be the group scheme over   whose R-points for a ring R  are given by 
    GSp_2n(R) =  { (g ,c )∈_2n(R) × R^×  |   g^t J_2n  g = c   J_2n} .

Note that GSp_2 is the general linear group _2. We let sim :  GSp_2n→_m, (g,c) ↦ c  denote the similitude map and refer to an element (g, c) ∈GSp_2n(R) simply by g.  The following group schemes will be used throughout:  

2 
 

  * = _2×__m_2×__m_2,

  * _1 = _2,  

  * _2 = _2×__m_2,

  * '  = _2×__mGSp_4,

  * '_2 =  GSp_4,

  * 𝐆    =  GSp_6
 
 
where all the products are  fibered over similitude maps. We define H, H_1, H_2,  H ',  H_2', G  to be respectively the group of F-points of the algebraic groups above and U, U_1, U_2,  U',  U_2 ', K to be the group of _F-points. 
We define projections   
   
_1:    ⟶_1                   _2 :      ⟶_2                  _1': '     ⟶_1                    _2': '     ⟶'_2 

 
(h_1,h_2,h_3) 
    ⟼h_1                 (h_1, h_2, h_3 )     ⟼(h_2,h_3)                 (h_1, h_2 )  
    ⟼h_1     
        (h_1, h_2)     ⟼h_2 







and  embeddings 
  _2 :  _2        ⟶'_2                     :        ⟶'                  ι' :  '         ⟶

( ( 
[ a b; c d ] ) , 
( 
[ a' b'; c' d' ] )   )       ⟼(   [   a       b;     a '      b';   c       d;      c'     d ' ] )                      (h_1, h_2, h_3)      ⟼(h_1,  _2(h_2, h_3)      )           ( [ a b; c d ] ,  [ A B; C D ]   )      ⟼[ a   b;   A   B; c   d;   C   D ]      

via which we consider U_2, H_2, U, H, U', H' to be subgroups of U_2 ', H_2', U', H', K, G   respectively. We let 
    ι : →
 
denote the composition ι ' 
∘ via which we view U, H as subgroups of K, G respectively. If R is a commutative ring with identity  and L_1, L_2 are compact open subgroups of G, we write 𝒞_R(L_1\ G / L_2) for the set of R-valued    compactly  supported functions f : G → R that are left L_1-invariant and right L_2-invariant.   Similar notations will be used for functions on H and H '.    
 Given a function 𝔉 : G → R and an element g ∈ G, we define the (H', g)-restriction of 𝔉 to be the function 𝔣_g : H'  →  R given by 𝔣_g(h) = 𝔉(hg) for all  h ∈ H '. We similarly define (H,g)-restriction  of 𝔉 and (H,η)-restrictions  of functions on H'   and   η∈ H'.  
  
It is easy to see that if 𝔉∈𝒞_R(K\ G /K), then 𝔣_g∈𝒞_R(U' \  H' / H_g' ) where  H'_g = H '∩ g K g^-1. If η∈ H', then the (H,η)-restriction of 𝔣_g  coincides with  the  (H, η g)-restriction of 𝔉 and  lies in  𝒞_R(U\ G / H_η g) where H_η g  =   H ∩η H_g ' η^-1  =  H ∩η g K g^-1η^-1.    






§ SPINOR  HECKE POLYNOMIAL
    



 §.§ Root datum of 
 Let  𝐀 =  _m^4  and  dis :   𝐀→𝐆 to be the embedding given by 
    (u_0, u_1,u_2,  u_3 ) ↦diag( u_1 , u_2 , u_3  ,  u_0 u_1 ^-1, u_0  u _2  ^-1 ,  u_0 u _3 ^-1  ) .

Then dis
identifies 𝐀 with a maximal (split) torus in 𝐆. We let A, A^∘ = A ∩ K denote respectively  the group of F, _F-points of 𝐀.   Let e_i : 𝐀→_m be the projection onto the i-th component, f_i : _m→𝐀 be the cocharacter inserting u into the i-th component with 1 in the remaining components. We will let 
    Λ =   f_0⊕⋯⊕ f_3
 
denote the cocharacter lattice.  An   element a_0 f_0  + … + a_3 f_3∈Λ will  also  be  denoted by (a_0, …, a_3 ). The set Φ⊂ X^*(𝐀 ) of roots of 𝐆 are
        

  * ± ( e_i - e_j  ) for 1 ≤ i < j  ≤  3, 

  * ± (   e_i + e_j  - e_0 ) for 1 ≤ i  < j ≤ 3

  * ± (  2 e_i - e_0  ) for i = 1 , 2 , 3

which makes an irreducible root system of type C_3. We choose  
    α_1 = e_1 - e_2,     α_2 = e_2 - e_3,     α_3 = 2 e_3 - e_0
 as our simple roots and let  Δ = {α_1 , α_2 ,  α_3}. This determines a subset Φ ^  + ⊂Φ of positive roots. The  
 resulting  half sum of positive roots is 
 
    δ =   -3 e_0 + 3e_1 + 2 e_2 +   e_3∈ X^*( 𝐀)
  
and  the  highest root is α_0  =  2 e_1 - e_0.  
The simple  coroots corresponding to α_i for i = 0, 1,  2, 3 are   
    α_0  ^ ∨ =  f_1,     α_1^∨ = f_1 -  f_2,      α_2^∨ = f_2  - f_3,      α_3^∨ = f_3
  and their  span in Λ is denoted by Q ^∨. The set Δ determines a dominance order on Λ. Explicitly, an  element    λ = (a_0 , …, a_3 )  ∈Λ is  dominant iff 

    a_1≥ a_2≥ a_3  and     2 a_3 - a_0≥ 0 .
 It is anti-dominant if all these inequalities hold in reverse. We denote the set  of dominant cocharacters by Λ^+.  
Let W denote the Weyl group of (, 𝐀) and s_i be the reflection associated with  α_i, i = 0, …,  3.  The action of s_i on Λ is given  as  follows:  
  
   

  * s_i acts by switching f_i↔ f_i+1 for i = 1,2,

  * s_3 acts by sending f_0↦ f_0 + f_3, f_3↦   - f_3, 

  * s_0 = s_1 s_2 s_3 s_2 s_1 acts by sending f_0↦ f_0 +                f_1, f_1↦  -  f_1.  

We have W = ⟨ s_1, s_2, s_3⟩≃  ( /2  )^3⋊ S_3 where S_3 denotes the group of permutations of three elements that acts on (  / 2  ) ^3 in the obvious manner.    


 §.§ Iwahori Weyl group
   

Let I denote  the  Iwahori subgroup  of G  corresponding to (the alcove determined by) the simple affine roots Δ_aff  =  {α_1, α_2, - α_0 + 1   }.    Explicitly, I is the compact open subgroup of K whose reduction modulo ϖ is the Borel subgroup of 𝐆() determined by Δ.   Let W_aff and W_I denote respectively the  affine Weyl and  Iwahori Weyl groups of the pair (𝐆, 𝐀).   We  view   W_aff as a subgroup of the group of affine transformations of Λ⊗. Given λ∈Λ, we let t(λ) denote translation by λ map on Λ⊗  and  write    ϖ^λ for the element  λ(ϖ) ∈ A.   Let    v : A / A ^∘→Λ be the inverse of the isomorphism  Λ→ A / A ^∘ given by  λ↦ϖ^-λ A^∘. Then 
      


  * W_aff = t(Q^∨ ) ⋊ W

  * W_I =   N_G(A) / A^∘ =  A/A^∘⋊ W  v≃Λ⋊ W,
 
where N_G(A) denotes the normalizer  of A in G. The set  S_aff = { s_1 , s_2 ,s_3 ,  t ( α _   0   ^ ∨    )    s_0}  is a generating  set  for W_aff and  the pair (W_aff , S_aff) forms   a Coxeter system of type C̃_3.  Identifying W_I with Λ⋊ W as above,  we can consider  W_aff    a   subgroup of  W_I via  W_aff =  t(Q^∨)   ⋊ W  ↪ t(Λ) ⋊ W.     The quotient   
    Ω : = W_I /  W_aff
 is  then  an infinite cyclic group and  we  have a canonical isomorphism 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 of elements in t  ( λ ) W is  achieved by a unique element. This length is given by 
    ℓ_min ( t(λ)   )    :    =      ∑_α∈Φ_λ  | ⟨λ , α⟩ |    +   ∑ _ α∈Φ^λ  (  ⟨λ , α⟩   -  1   )
    where Φ_ λ    = {α∈Φ   ^        +   |  ⟨λ ,  α⟩≤ 0  } and  Φ ^    λ    =  {α∈Φ ^  +  , |  ⟨λ ,  α⟩   > 0  }.     When λ is dominant, this is also the minimal length of elements in W t(λ) W.  Consider the following elements in N_G (A):   


    w_1 : =  1.1( [ 0 1      ; 1 0        ;     1      ;       0 1  ;       1 0  ;           1 ] ),        
    w_2   :    =  ( 1.1[ 1        ;   0 1      ;   1 0      ;       1    ;         0 1;         1 0 ]) ,         w_3  : =   1.1( [  1               ;     1            ;        0        1;          -1      ;             -1   ;        1        0 ] )
 


    w_    0      : =  1.1(  [   0         11ϖ        ;       1        ;           1    ;   ϖ           0;                  -1;                      -1 ] ),    ρ   =   0.95( [           1;         1  ;       1  ;     ϖ      ;   ϖ        ; ϖ           ] )   .
 
The   classes of w_0 , w_1 ,w_2, w_3 in W_I   represent t(α_0 ^∨ ) s_0,   s_1 , s_2, s_3 respectively and the  reflection s_0   is  represented  by  w_α_0 :  =  ϖ^f_1  w_0     =      w_1 w_2 w_3 w_2 w_1. The class of  ρ represents ω : = t(-f_0 ) s_3 s_2 s_3 s_1 s_2 s_3 which is a generator of Ω and the conjugation by ω acts by switching s_0↔  s_3, s_1↔ s_2. That is,   it induces    an  automorphism of the extended   Coxeter-Dynkin  diagram  
 
    0.9[extended,Coxeter,
    edge length=1cm,
    labels=0,1,2,3]
    C3

where the labels below the vertices correspond to w_i. Note also that ρ^2 = ϖ^(2,1,1,1)∈ A is central.   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. When referring to action of simple reflections in W on Λ however, we will  stick to the letters s_i.     


 §.§ The  Hecke polynomial
 Let  [ Λ ] denote the group algebra of Λ.  For λ∈Λ, we let e ^ λ∈ [ Λ ] denote[this is done to distinguish the addition in Λ from addition in the group algebra]  the element corresponding to λ 
and e ^  W λ∈ [ Λ ] denote the the (formal)  sum    of elements in the orbit W λ.
 We will denote y_i : =  e^f_i∈ [ Λ ] for i = 0 , … 3, so that 
    [ Λ ] =  [ y_0 ^± , …, y_3 ^ ± ] .

Let ℛ  =  ℛ_q denote the ring [ q ^ ±1/2 ]. The dual group of 𝐆 has an 8-dimensional representation called the  spin representation.  Its    highest (co)weight is f_0 + f_1 + f_2 + f_3 which is minuscule. Thus its (co)weights are 1/2 ( 2 f_0 + f_1 + f_2 + f_3 )   +  1/2 ( ± f_1±  f_2±  f_3 ) and its characteristic (Satake) polynomial is

    𝔖 _spin ( X )  =        ( 1-  y_0 X ) ( 1- y_0 y_1 X) ( 1-  y_0 y_2 X ) ( 1 - y_0 y_3 X) 
        
    ( 1-  y_0 y_1 y_2 X ) ( 1-  y_0 y_1 y_3 X )( 1- y_0 y_2  y_3 X ) ( 1 - y_0 y_1 y_2 y_3 X ) ∈ [ Λ ] ^  W  (X)    .

Let ℋ_ℛ(K \ G / K ) denote the spherical Hecke algebra with coefficients in ℛ that is defined with respect to a measure on G giving  K measure one.   Let 
    𝒮 :  ℋ_ℛ(K \ G / K ) →ℛ [  Λ ]  ^   W
 denote the Satake  isomorphism. If P = P(X) ∈ℋ_ℛ(K\ G / K )[X] is a polynomial, then 𝒮(P) means the polynomial in ℛ[Λ]^W[X] obtained by applying 𝒮 to the coefficients of the powers of X in P.      
  For c ∈, we define the degree 8 spinor Hecke polynomial  ℌ_spin,c(X) ∈ℋ_ℛ(G) [X] to be unique polynomial    such that 𝒮 ( ℌ_spin,  c     )    =  𝔖_spin( q^-c X).   

To work with this  Hecke polynomial and to describe the decompositions of the double coset operators appearing in it later on, it would be convenient to record the following.   



        For each  λ∈Λ ^  +  below,      the element  w =  w _λ∈  W_I specified is the unique element in W_I of minimal possible length  such that  K ϖ ^ λ K = K   w   K.

 
      

  * λ = (1,1,1,1), w = ρ, 

  * λ = (2,2,1,1), w =  w_0ρ^2,  

  * λ = (2,2,2,1), w = w_0  w_1 w_0ρ^2, 


  * λ = (3,3,2,2), w =   w_0 w_1 w_2 w_3ρ^3, 

  * λ = ( 4,3,3,3),  w =    w_0 w_1 w_0 w_2 w_1 w_0ρ^4,    

  * λ = ( 4,4,2,2 ), w =   w_0 w_1 w_2 w_3 w_2 w_1 w_0ρ^4.   

 


 We point out  that the translation component   of each w_λ above (i.e., the Λ-component in  W_I = Λ⋊ W) is t(- λ^opp) 
where λ^opp is the  anti-dominant element in the Weyl orbit W λ.     The minimal  
possible length in each case is computed using (<ref>) and   that ℓ ( w _λ ) =  ℓ_min ( t ( -  λ ^ opp   )   )    =  ℓ _ min  ( t ( λ ) ).  

   

 For convenience,  we will notate     

    υ_0 = w_0 ,   υ_1 = w_0 w_1 w_0 ,   υ _2 : =   w_0 w_1 w_2 w_3  ,   υ_3  : = w_0 w_1 w_0 w_2 w_1 w_0 ,   υ_4 =  w_0 w_1 w_2 w_3 w_2 w_1 w_0
    Given g ∈ G, we let (K gK ) denote the characteristic function (K g K ) : G → of the double coset K g K. 
 For an even integer  k,  we let ρ^k (K g K ) denotes the function  (K g ρ^k K ). We will use  similar notation for sums of such functions and for functions on H' and H.    
 
     The coefficients of ℌ_spin,c(X) lie in ℋ_[q^-1](K \ G  / K ) for all c ∈. If we define 




    ℌ (X)        =  ( K  )  -   (K ρ K ) X +   𝔄   X^2     -   𝔅 X^3   + (   ℭ    +    2 ρ^2𝔄  )X ^4 
     - ρ^2𝔅 X^5  +   ρ^4𝔄 X^6
        - ( K ρ ^  7   K )   X ^  7    +   ( K ρ ^  8   K )       X ^ 8  ∈ ℋ_ ( K  \ G / K  ) [ X ]

where 
  
   

  * 𝔄 =  ( K   υ_1ρ^2    K ) +    2 (   K υ _0ρ^2  K  )   +   4  (  K ρ^2 K    ),

  * 𝔅  = ( K υ_2ρ  ^3  K  ) +   4 (   K     ρ ^  3   K   ),

  * ℭ  =    ( K   υ_3ρ^4     K   )   +   (  K  υ_4ρ^4   K  ),    
  
then  ℌ_spin,c (X) is congruent to  
ℌ (X) modulo q - 1  for all c ∈. 
 
  Since the half sum of positive roots  (<ref>)   lies in X^*(𝐀), the first claim is obvious from the discussion in <cit.>. Solving the plethysm problem for exterior powers of the  spin  representation by combining i choices of coweights ( 1 , 12 , 12  , 12  )  +   ( 0,  ±12, ±12, ±12    ) for i = 0, …, 8  or simply  by expanding 𝔖_spin(X), we see that             



    𝔖 _ spin ( X)        = 1 - e ^ W(1,1,1,1) X  
        +   ( e ^  W ( 2,2,2,1)  + 2 e ^W(2,2,1,1)   +  4  e ^ (2,1,1,1) ) X^2   -   (  e^W(3,3, 2,2)  + 4  e ^W(3,2,2,2)  )     X ^3
        +    (    e ^   W ( 4,4,2,2)    + e ^    W     ( 4,3,3,3)   + 2  e ^  W (  4,3,3,2)    +   4   e ^  W ( 4,3,2,2)  +   8  e ^   ( 4,2,2,2)   ) X^4
           -  (  e ^  W ( 5,4,3,3)   +   4   e  ^    W  (  5, 3 , 3 ,3 )   )  X ^  5      +    ( e ^  W ( 6, 4,4,3)  +   2      e ^ W (6,4,3,3)  + 4 e ^   ( 6,3,3,3)   )  X ^  6  
           -   e ^  W ( 7,4,4,4) X^7 +  e ^  ( 8  ,  4 , 4,   4 )   X ^  8
 
The claim now follows by Lemma <ref> and <cit.>.  
   



 The exact coefficients in the Hecke polynomial are polynomial expressions in q translated by (possibly negative) powers of q. They  can be found explicitly using Sage by computing appropriate Kazhdan-Lusztig polynomials P_σ,τ(q) for σ, τ∈ W_I. See <cit.> for an example.  




§ RESTRICTION TO GL2 × GSP4
 
 In what follows, we will denote

    ℌ  =  ℌ(1) 
          =  ( 1 + ρ^8 ) (K) - ( 1 + ρ^6 )  (Kρ K) + (1  +  2 ρ^2  +  ρ^4 ) 𝔄 - ( 1  + ρ^2 ) 𝔅  + ℭ
 
considered as an element of 𝒞_ ( K \ G /  K ). Note that ℌ≡ℌ_spin, c(1) modulo q - 1  for all c ∈ by Proposition <ref>. Note also that ρ^k for even k is an element of H (and H').    We wish to write the -restrictions of ℌ. To this end, let us introduce the following elements in G: 

    τ_0 = 1_G, 
        τ_1 = ([ ϖ       1;   ϖ   1;     ϖ    ;       1;         1;           1 ] ), 
        τ_2 = ([  ϖ          1ϖ;     ϖ    1ϖ;        1   ;          1ϖ;             1ϖ;                 1 ]).

For w ∈ W_I, we denote  ℛ(w) =    U' \ K w K / K. When listing elements of ℛ(w), we will only write the representative element and it will be understood that no two elements represent the same double coset. Similar convention will be used for other double coset spaces.    
    
 With notations and conventions as above,    
    

  * ℛ(ρ)= 
{ϖ^(1,1,1,1), τ_1},     

  * ℛ(υ_0ρ^2)=
{ϖ^(2,2,1,1), ϖ^(2,1,2,1), ϖ^(1,1,0,0)τ _1},   

  * ℛ (υ_1ρ^2)=
{ϖ^(2,2,2,1), ϖ^(2,1,2,2), ϖ^(1,1,1,0)τ _1, ϖ^(1,1,0,1)τ_1,  ϖ^(2,1,1,1)τ_2},

  * ℛ (υ_2ρ^3)
={ϖ^(3,3,2,2), ϖ^(3,2,3,2), ϖ^(2,2,1,1)τ_1, ϖ^(2,1,2,1)τ_1, ϖ^(2,2,0,1)τ_1, ϖ^(2,1,1,2) τ_1, ϖ^(3,2,1,2)τ_2},   

  * ℛ (υ_3ρ^4)
={ϖ^(4,3,3,3), ϖ^(3,2,2,2)τ_1, ϖ^(4,2,2,3)τ_2},

  * ℛ
 (υ_4ρ^4)=
{ϖ^(4,4,2,2), ϖ^(4,2,4,2),  ϖ^(3,3,1,1) τ_1, ϖ ^ (3,2,0,1)τ_1, ϖ^(4,3,1,2)τ_2}. 
      

Moreover,  H'τ_i K ∈ H '   \ G / K are pairwise distinct for i = 0 , 1 , 2.   
      
  
A proof of this  is provided   in  <ref>.  
 
 A quick check on our lists of  representatives for each ℛ(w) above is through computing their classes in K \ G / K. These   should  return ϖ^λ on the diagonal where   λ corresponds to w in  Lemma    <ref>. The distinctness of our representatives  is also  easily checked using a  Cartan style decomposition proved in <ref>.  What is difficult however  is establishing that these  represent   all the orbits of U' on K w K / K and this is where bulk of the work lies.  
 
  
H ' \ H ' · (ℌ ) / K =   {τ_0, τ_1, τ_2}.  In particular if g ∈ G is such that  H'gK ≠ H'τ_iK for i = 0,1,2, then (H', g)-restriction of ℌ is zero.     
 
 
The is clear  from the expression (<ref>)  and Proposition  <ref>.    

For i = 0 , 1 , 2, we let 𝔞_i, 𝔟_i, 𝔠_i, _i∈𝒞_(U' \ H' /  H_τ_i') denote the (, τ_i)-restriction of 𝔄, 𝔅, ℭ, ℌ respectively.    
Here for g ∈ G, _g denotes   the compact open subgroup  H ' ∩ g K g^-1 of H '.   As before, we omit writing  for characteristic functions.  By Proposition <ref>, we have 
    (K ρ K ) = (U' ϖ^(1,1,1,1)K) + ( U' τ_1K ) .
  Since U' ϖ^λ K  ⊂ H' K for any λ∈Λ and U' τ_1 K ⊂ H' τ_1 K,  the (H', τ_i)-restrictions of (K ρ K) for i = 0, 1, 2  are given by    
    (U' ϖ^(1,1,1,1) U') ,       (U' H_τ_1') ,        0
 respectively. Proceeding in a similar fashion, we find that  
 

    𝔞_0    = (U' ϖ^(2,2,2,1) U' ) + (ϖ^(2,1,2,2)) +  2 (ϖ^(2,2,1,1)) + 2  (ϖ^(2,1,2,1)) + 4  (ϖ^(2,1,1,1)  ),
    𝔞_1    = (ϖ^(1,1,1,0)_τ_1 ) +  (U ' ϖ^(1,1,0,1)_τ_1 )  +  2 (ϖ^(1,1,0,0)_τ_1 ), 
    𝔞_2    = (ϖ^(2,1,1,1)_τ_2)  ,   
    𝔟_0    =  (ϖ^(3,3,2,2) ) + (ϖ^(3,2,3,2)) +  4  (ϖ^(3,2,2,2) ),  
    𝔟_1    =  (ϖ^(2,2,1,1)_τ_1 ) +  (ϖ^(2,1,2,1)_τ_1 )  +  (ϖ^(2,2,0,1)_τ_1 )  +   (ϖ^(2,1,1,2)_τ_1 )  + 4   (ϖ^(2,1,1,1)_τ_1 ) ,   
    𝔟_2    =   (ϖ^(3,2,1,2)  H_τ_2 ' )  , 
    𝔠_0    =   (ϖ^(4,3,3,3)  ) + (ϖ^(4,4,2,2)  ) +   (ϖ^(4,2,4,2)  )   ,  
    𝔠_1    =    (ϖ^(3,2,2,2)_τ_1 ) + (ϖ^(3,3,1,1)_τ_1) +   (ϖ^(3,2,0,1)_τ_1 )  ,  
    𝔠_2    =     (ϖ^(4,2,2,3)_τ_2 ) + (ϖ^(4,3,1,2)_τ_2).
 
Using expression (<ref>), we find that 

    _0   =  
    (1 + ρ^8) () - (1+ρ^6) (ϖ^(1,1,1,1) ) +(1+ 2 ρ^2 + ρ^4) 𝔞_0 -(1+ρ^2) 𝔟_0 +  𝔠_0 , 
    
     _1    =    - ( 1 + ρ^6 ) (U   ' H_τ_1') +  ( 1 +  2 ρ^2 + ρ^4 ) 𝔞_1 - ( 1 + ρ^2 ) 𝔟_1   +  𝔠_1  ,         
    _2     =   ( 1 + 2 ρ^2  +  ρ^4 )  𝔞_2  - ( 1 + ρ^2 ) 𝔟_2  + 𝔠_2

where the the central elements ρ^2k distribute over Hecke operators as before. 
 The particular choice of τ_1, τ_2 is motivated by the structure of the group H' ∩τ_i K τ_i^-1 which  is  convenient for decomposing double cosets involving these groups (see <ref>). Note that τ_i  very closely related to the “Schröder's representatives" for the double coset H'\ G/ K  given  in  <cit.>.  
 


§ RESTRICTION TO GL2 × GL2  × GL2 
          In this section, we  record  the twisted restrictions of _0, _1, _2 with respect to H.  For i = 0, 1, 2 and h ∈ H', we let   ℛ_i(h) denote the  double coset space U  \ U' h H_τ_i'/ H_τ_i'. The convention  used  in <ref> for listing elements of double  coset spaces will also be be applied to ℛ_i(h).    



 §.§ H-restrictions of _0
  To write the restrictions of 𝔥_0, we  introduce the following elements of H '= _2(F) ×_F^×GSp_4(F):

    ϱ_0 = 1_H',   ϱ_1  =  (  0.9( ϖ
         1    ) ,   ( [ ϖ     1;   ϖ 1;     1;       1 ] )  ) ,   ϱ _2 =   
     (  0.9( ϖ
       ϖ ) ,    
     ( [ ϖ^2           1;     ϖ^2   1;           1;               1 ] )   )
 
which we also view as elements of G via ι'.   
       With notations  and conventions  as above, we have  
   
   

  * ℛ_0(ϖ^(1,1,1,1)) =  {ϖ^(1,1,1,1), ϱ_1},   

  * ℛ_0(ϖ^(2,2,2,1))=  {ϖ^(2,2,2,1), ϖ^(2,2,1,2), ϖ^(1,1,1,0)ϱ_1}, 

  * ℛ_0(ϖ^(2,1,2,2)) = {ϖ^(2,1,2,2), ϖ^(1,0,1,1)ϱ _1, ϱ_2}, 

  * ℛ_0(ϖ^(3,2,3,2) ) = {ϖ^(3,2,3,2), ϖ^(3,2,2,3), ϖ^(2,1,2,1)ϱ_1,  ϖ^(2,1,1,2)ϱ_1,  ϖ^(2,1,2,0)ϱ_1,  ϖ^(1,1,0,1)ϱ_2},  

  * ℛ_0(ϖ^(4,2,4,2) ) = 
{ϖ^(4,2,4,2), ϖ^(4,2,2,4), ϖ^(3,1,3,1)ϱ_1, ϖ^(3,1,1,3)ϱ_1, ϖ^(2,1,2,0)ϱ_2}.

Moreover H ϱ_i U' ∈ H \ H' / U' are pairwise distinct for i = 0 , 1 , 2.  
 
   A  proof of this is given in <ref>. 

By Lemma <ref>, the representatives of ℛ_0(ϖ^λ) depend only on those for U_2\ U_2' ϖ^_2(λ) U_2'/U_2'. Then one easily obtains the following from Proposition   <ref>.  
   We  have 
 

  * ℛ_0(ϖ^(2,2,1,1) ) = {ϖ^(2,2,1,1)}   ,

  * ℛ_0(ϖ^(2,1,2,1)) =  {ϖ^(2,1,2,1), ϖ^(2,1,1,2), ϖ^(1,0,1,0)ϱ_1} ,

  * ℛ_0(ϖ^(3,3,2,2)) = {ϖ^(3,3,2,2),  ϖ^(2,2,1,1)ϱ_1}  ,

  * ℛ_0(ϖ^(4,3,3,3)) =   {ϖ^(4,3,3,3),  ϖ^(3,2,2,2)ϱ_1,  ϖ^(2,2,1,1)ϱ_2}  ,  

  * ℛ_0(ϖ^(4,4,2,2)) =   {ϖ^(4,4,2,2)}   .   

 
The last two results describe the the U-orbits of all the double coset spaces arising from  (<ref>) up to translation by the central element ρ^2. 
This implies the next  claim.  
 H \ H ·(𝔥_0) / U'  = {ϱ _0, ϱ _1, ϱ _2}.    
    
For i = 0 ,1 , 2, we let 𝔞_ϱ_i ,  𝔟_ϱ_i , 𝔠_ϱ_i ,  𝔥_ϱ_i∈𝒞_(U \ H /  H_ϱ_i) denote the (H, ϱ_i)-restriction of 𝔞_0, 𝔟_0, 𝔠_0, _0 respectively where as before,  we let H_ϱ_i denote H ∩ϱ_i   K   ϱ_i ^-1 as before. From    Proposition <ref> and Corollary <ref>, we find that 
    

    𝔞_ϱ_0    =
    (Uϖ^(2,2,2,1)U)+(Uϖ^(2,2,1,2)U)
    +(Uϖ^(2,1,2,2)U) +  2 (U ϖ^(2,2,1,1)U) + 2  (U ϖ^(2,1,2,1)  U )  
           +  2   (U ϖ^(2,1,1,2)   U )   +   
    4  (U ϖ^(2,1,1,1) U ) , 
    𝔞_ϱ_1    = (U   ϖ^(1,1,1,0) H_ϱ_1  ) +  (U  ϖ^(1,0,1,1)    H_ϱ_1)  +  2  ( U ϖ^(1,0,1,0)  H_ϱ_1  ) ,  
     
    𝔞_ϱ_2    = (U H_ϱ_2)  ,  
    𝔟_ϱ_0    =  (U  ϖ^(3,3,2,2) U ) +  (U ϖ^(3,2,3,2)   U )  +  (U ϖ^(3,2,2,3) U )  + 4   ( U  ϖ^(3,2,2,2) U )  ,     
    𝔟_ϱ_1    =  (U  ϖ^(2,2,1,1)   H _ϱ_1 ) + ( U  ϖ^(2,1,2,1)    H_ϱ_1 )  +  ( U ϖ^(2,1,1,2) H_ϱ_1 )   +   (U ϖ^(2,1,2,0) H_ϱ_1 )  + 4  (U ϖ^(2,1,1,1) H_ϱ_1)   ,    
    𝔟_ϱ_2    =   (U  ϖ^(1,1,0,1)  H_ϱ_2)    ,   
    𝔠_ϱ_0    =   ( U  ϖ^(4,3,3,3)   U ) + (U  ϖ^(4,4,2,2)   U  ) +   (U ϖ^(4,2,4,2) U  )      + ( U ϖ^(4,2,2,4) U )    ,   
    𝔠_ϱ_1    =    (U ϖ^(3,2,2,2) H_ϱ_1 ) + (U ϖ^(3,1,3,1)   H _ϱ_1) +   (U ϖ^(3,1,1,3) H _ϱ_1 )   ,   
    𝔠_ϱ_2    =     (U  ϖ^(2,2,1,1) H_ϱ_2 ) + (U ϖ^(2,1,2,0) H_ϱ_2)  .

From the expression (<ref>), we  get

    _ϱ_0   =  
    (1 + ρ^8) (U) - (1+ρ^6) ( U 
       ϖ^(1,1,1,1)  U ) +(1+ 2 ρ^2 + ρ^4) 𝔞_ϱ_0 -(1+ρ^2) 𝔟_ϱ_0 +  𝔠_ϱ_0  , 
    
     _ϱ_1    =    - ( 1 + ρ^6 ) (U H_ϱ_1) +  ( 1 +  2 ρ^2 + ρ^4 ) 𝔞_ϱ_1 - ( 1 + ρ^2 ) 𝔟_ϱ_1    +  𝔠_ϱ_1    ,       
    _ϱ_2     =   ( 1 + 2 ρ^2  +  ρ^4 )  𝔞_ϱ_2  - ( 1 + ρ^2 ) 𝔟_ϱ_2  + 𝔠_ϱ_2   .



 §.§ H-restrictions of _1
 
We consider  the following elements in H ': 

    σ_0 = 1_H' ,     σ_1 =  w_2 ,      σ_2  =  ϱ_1ϖ^-(1,1,1,1),     σ_3 =  ϱ_1 .
 
where ϱ_i are as in (<ref>).  For i = 0 , 1, 2 , 3, let ς_i∈ G denote σ_iτ_1. Also let   ψ  =    (  [ 1; 1 1 ] , 1  _  H_2 )   ∈ H.  
   With notations and conventions as above, we have  


  * ℛ_1(ϖ^(1,1,1,0))  =   {ϖ^(1,1,1,0), ϖ^(1,1,0,1)σ_1, ϖ^(1,1,1,0)σ_2}, 

  * ℛ_1(ϖ^(1,1,0,1))=  {ϖ^(1,1,0,1), ϖ^(1,1,1,0)σ_1, ϖ^(1,0,0,0)σ_2, ϖ^(1,1,0,1)σ_2, ϖ^(1,0,1,1)σ_2, ϖ^-(0,1,0,0)σ_3}, 

  * ℛ_1(ϖ^(1,1,0,0))= {ϖ^(1,1,0,0), ϖ^(1,1,0,0)σ_1, ϖ^(1,0,1,0)σ_2},

  * ℛ_1(ϖ^(2,2,1,1))= {ϖ^(2,2,1,1), ϖ^(2,2,1,1)σ_1, ϖ^(2,2,1,1)σ_2, ϖ^(2,0,1,1)σ_2},

  * ℛ_1(ϖ^(2,1,2,1))= {ϖ^(2,1,2,1), ϖ^(2,1,1,2)σ_1, ϖ^(2,1,2,1)σ_2, ϖ^(2,1,2,0)σ_2, ϖ^(2,1,1,0)σ_2, ϖ^(1,0,1,0)σ_3},

  * ℛ_1(ϖ^(2,2,0,1))=  {ϖ^(2,2,0,1), ϖ^(2,2,1,0)σ_1, ϖ^(2,0,2,1)σ_2, ϖ^(2,0,2,0)σ_2, ϖ^(2,0,1,0)σ_2, ϖ^(1,-1,1,0)σ_3},

  * ℛ_1(ϖ^(2,1,1,2))=   {ϖ^(2,1,1,2), ϖ^(2,1,2,1)σ _1, ϖ^(2,1,0,1)σ_2, ϖ^(2,1,1,2)σ_2, ϖ^(1,0,0,1)σ_3},

  * ℛ_1(ϖ^(2,1,1,1))=   {ϖ^(2,1,1,1), ϖ^(2,1,1,1)σ_1, ϖ^(2,1,1,1)σ_2},

  * ℛ_1(ϖ^(3,2,2,2)) = [t]1.0{ϖ^(3,2,2,2), ϖ^(3,2,2,2)σ_1, ϖ^(3,2,1,1)σ_2, ϖ^(3,2,2,2)σ_2, ϖ^(3,1,1,2)σ_2, ϖ^(3,2,1,2)ψσ_2,  ϖ^(2,1,1,1)σ_3}, 

 

  * ℛ_1(ϖ^(3,3,1,1))= {ϖ^(3,3,1,1), ϖ^(3,3,1,1)σ_1, ϖ^(3,0,2,1)σ_2}  ,

  * ℛ_1(ϖ^(3,2,0,1)) = {ϖ^(3,2,0,1),  ϖ^(3,2,1,0)σ_1, ϖ^(3,1,3,1)σ_2, ϖ^(3,1,2,0)σ_2, ϖ^(2,0,2,0)σ_3}. 
  

Moreover  Hσ_iH_τ_1'  ∈ H \  H'  / H_τ_i' are pairwise distinct  for i = 0,1,2,3.     
    
 A proof  of this  is provided in 
<ref>. 
 

 We also need ℛ_1(1) = {σ_0, σ_1, σ_2} but this is obtained from  ℛ_1(ϖ^(2,1,1,1)).  
 

 The appearance of ψ in one of the representatives listed  in ℛ_1(ϖ^(3,2,2,2)) seems unavoidable. Curiously,  U ϖ^(3,2,1,2)ψ H_ς_2 is  the only double coset arising form ℌ 
whose degree vanishes modulo q - 1. See Lemma <ref>. 
 
 
H \ H ·(𝔥_1 ) /  H_τ_1'  =  {σ_0,  σ _1, σ_2,  σ _3}.    

For i = 0 , 1, 2, 3,  let 𝔞_ς_i, 𝔟_ς_i, 𝔠_ς_i, 𝔥_ς_i∈𝒞_(U \  H / H_ς_i) denote the (H,σ_i)-restrictions of 
𝔞_1,  𝔟_1, 𝔠_1, 
 𝔥_1 respectively.
Proposition    <ref> implies that  

    𝔞_ς_0    = (U ϖ^(1,1,1,0) H_ς_0) + 
    (Uϖ^(1,1,0,1)H_ς_0)+ 2 (Uϖ^(1,1,0,0)H_ς_0),  
    𝔞_ς_2     =   ( U ϖ^(1,1,1,0) H_ς_2)   +  ( U ϖ^(1,0,0,0) H_ς_2) +  (U ϖ^(1,1,0,1) H_ς_2) + ( U ϖ^(1,0,1,1) H_ς_2) + 2( U ϖ^(1,0,1,0) H_ς_2)  ,  
    𝔞_ς_3     =   ( U ϖ^-(0,1,0,0) H_ς_3 )   , 
    𝔟_ς_0    =  (U   ϖ^(2,2,1,1)   H _ς_0 ) +  (U ϖ^(2,1,2,1)   H_ς_0) + (Uϖ^(2,2,0,1) H _ς_0) + (U ϖ^(2,1,1,2) H_ς_0) + 4 (Uϖ^(2,1,1,1)H_ς_0) , 
    𝔟_ς_2    = 
    (Uϖ^(2,2,1,1)H_ς_2)+    (Uϖ^(2,0,1,1)H_ς_2)+ (Uϖ^(2,1,2,1)H_ς_2)+ 
    (Uϖ^(2,1,2,0)H_ς_2)+ 
    (Uϖ^(2,1,1,0)H_ς_2)  +
             (Uϖ^(2,0,2,1)H_ς_2)+(Uϖ^(2,0,2,0)H_ς_2)+
    (Uϖ^(2,0,1,0)H_ς_2)+
    (Uϖ^(2,1,0,1)H_ς_2)+
    (Uϖ^(2,1,1,2)H_ς_2)  +
               4(Uϖ^(2,1,1,1)H_ς_2),
    𝔟_ς_3    =  (U  ϖ^(1,0,1,0) H_ς_3) +  (U ϖ^(1,-1,1,0)H_ς_3 )  +  (U ϖ^(1,0,0,1)H_ς_3)  ,     
    𝔠_ς_0    = 
    (Uϖ^(3,2,2,2)H_ς_0)+
    (Uϖ^(3,3,1,1)H_ς_0)+(Uϖ^(3,2,0,1)H_ς_0),
    𝔠_ς_2    =    
    (Uϖ^(3,2,1,1)H_ς_2)+
    (Uϖ^(3,2,2,2)H_ς_2)+   (Uϖ^(3,1,1,2)H_ς_2)  
    + ( U ϖ^(3,2,1,2)ψ  H_ς_2) 
    + ( U ϖ^(3,0,2,1)
    H_ς_2)   + 
                ( U ϖ^(3,1,3,1)    H_ς_2)  + ( U ϖ^(3,1,2,0)  H_ς_2) , 
    𝔠_ς_3    =     (U  ϖ^(2,1,1,1) H_ς_3 ) + (U 
     ϖ^(2,0,2,0) H_ς_3).

Using expression (<ref>), we  get

    _ς_0   =  -  (1+ρ^6) ( U   H_ς_0  ) +(1+ 2 ρ^2 + ρ^4) 𝔞_ς_0  -  (1+ρ^2) 𝔟_ς_0 + 𝔠_ς_0   , 
    
     _ ς_2    =    - ( 1 + ρ^6 ) (U H_ς_2)  +  ( 1 +  2 ρ^2 + ρ^4 ) 𝔞_ς_2 - ( 1 + ρ^2 ) 𝔟_ς_2     +  𝔠_ς_2     ,    
    _ς_3     =   ( 1 + 2 ρ^2  +  ρ^4 )  𝔞_ς_3  - ( 1 + ρ^2 ) 𝔟_ς_3  + 𝔠_ς_3Now observe that each ℛ(ϖ^λ) in Proposition <ref> contains a unique representative of the form ϖ^s_2(λ)σ_1. Moreover ς_1 = w_2ς_0 and w_2 normalizes  U (and H).  So  w_2 U ϖ^λ  H_ς_0w_2 = U ϖ^ s_2(λ)  H_ς_1 for all λ∈Λ. Therefore 𝔥_ς_1    = w_2𝔥_ς_0 w_2

where w_2 distributes over each double coset characteristic function. 






 §.§ H-restrictions of _2


For i = 0, 1, 2, denote  θ_i : = σ_i and θ_3 :  =  ϖ^-(1,1,1,1)σ_3 where σ_0, σ_1, σ_2, σ_3 are as in 
 (<ref>). For i = 0 ,1, 2, 3, set  
ϑ_i = σ_iτ_2∈ G.  Additionally for k ∈  []^∘ := [] ∖{ -1 }, we define θ̃_k = (1, η̃_k)  ∈ H' where   
    η̃_k =  0.9[     k     1;   k+1     1;                -1 k + 1;                 1   - k ]∈ H_2'
   
and set ϑ̃_k   = θ̃_kτ_2∈ G. Note that θ̃_0 = w_2 w_3θ_2 w_3 and w_3τ_2 =  τ_2w_3t_1 where t_1 = diag(1,1,-1,1,1,-1). So  
    ϑ̃_0 = θ̃_0τ_2 =  w_2 w_3θ_2 w_3τ_2 = 
    w_2 w_3ϑ_2  
     w_3 t_1.
    




        We have 
  
 

  * ℛ_2 ( ϖ^(0,0,0,0) )  = { 1,   θ_1 ,  θ_2 ,  θ̃_k  |  k ∈ []^∘},  

  * ℛ_2(ϖ^(3,2,1,2)) = 
[t]1.0
{ϖ^(3,2,1,2), ϖ^(3,2,2,1)θ_1, ϖ^(3,2,1,2)θ_2, ϖ^(3,1,2,1)θ_2, ϖ^(3,1,2,2)θ_2, ϖ^(3,1,2,2)θ_3}∪
{ϖ^(3,1, 2, 2)θ̃_0, ϖ^(3,1,1,2)θ̃_0, ϖ^(3,2,1,1)θ̃_k  |    k ∈   []^∘},
 

  * ℛ_2(ϖ^(4,3,1,2))   =   {ϖ^(4,3,1,2),  ϖ^(4,3,2,1)θ_1  ,    ϖ^(4,1,3,2)θ_2 ,      ϖ^(4,1,2,3)θ̃_0,    ϖ^(4,1,3,2)θ_3},       

  * ℛ_2(ϖ^(4,2,2,3)) =  {ϖ^(4,2,2,3), ϖ^(4,2,3,2)θ_1, ϖ^(4,2,2,3)θ_2, ϖ^(4,2,1,2)θ̃_0, ϖ^(4,2,2,3)θ_3}.

   
 The proof of this result is provided in <ref>   
  
 H \  H · (_2) / H_τ_2'   =   {θ_0, θ_1, θ_2 , θ_3 , θ̃_k  |   k ∈ []^∘}.  
  
 This follows by Lemma  <ref> and  Proposition  <ref>.  
 
For ϑ∈{ϑ_0, ϑ_1, ϑ_2, ϑ_3, ϑ̃_k  |   k ∈  []^∘}, we  let 𝔥_ϑ∈𝒞_(U \  H / H_ϑ   ) denote the (H,ϑτ_2^-1)-restriction of 𝔥_2. By the results above,

    𝔥_ϑ_0     =   
    (ρ^2 + 2ρ^4 +  ρ^6)  (U H_ϑ_0 ) -  ( 1 + ρ^2)     (U  ϖ^(3,2,1,2)  H_ϑ_0 )  +   ( U ϖ^(4,2,2,3) H_ϑ_0)  + ( U ϖ^(4,3,1,2)H_ϑ_0)  ,   
     𝔥_ϑ_2    =   
    ( ρ^2 + 2 ρ^4 +  ρ^6 )  (  U H_ϑ_2  )   - ( 1 + ρ^2 )  
     (     (  U    ϖ^(3,2,1,2)   H _ϑ_2  )  +   ( U   ϖ^(3,1,2,1)   H _ϑ_2  )  +    ( U    ϖ^(3,1,2,2)   H _ϑ_2  ) 
      )      +    
                 ( U ϖ^(4,2,2,3)  H_ϑ_2 )   + ( U ϖ^(4,1,3,2)  H _ ϑ_2  )  ,     
    𝔥_ϑ_3     =  
     ( U ϖ^(4,2,2,3)   H _ϑ_3 )   +  (U ϖ^(4,1,3,2)   H _ϑ_3 )    - ( 1 + ρ^2 ) (U ϖ^(3,1,2,2)  H_ϑ_3 ),  
    𝔥_ϑ̃_k       =   (ρ^2 + 2ρ^4 + ρ^6 ) ( U H_ϑ̃_k  ) - ( 1 + ρ^2 )   (   U ϖ^(3,2,1,1) H_ϑ̃_k) 
     where k ∈ [] ∖{ 0 , - 1 }. Observe that  H_ϑ_1 = w_2 H_ϑ_0 w_2 and that  in  each set appearing in  Proposition  <ref>, ϖ ^λ for some λ∈Λ is listed in that set if and only if ϖ^s_2(λ)θ_1 is.    
    So as in the case of 𝔥_ς_1, we have    
    𝔥_ϑ_1      = w _2𝔥_ϑ_0 w _2.
    Similarly we have H_ϑ_2 = w_2 w_3 H_ϑ̃_0 w_3 w_2 and ϖ^λϑ_2 appears in Proposition  <ref> if and only if ϖ^s_2s_3(λ)ϑ̃_0 does. Therefore  
    𝔥_ϑ̃_0     =  w_2 w_3𝔥_ϑ_2 w_3 w_2 .
 



§ HORIZONTAL NORM RELATIONS
 
 


Let X  = Mat_2× 1(F) be  the F-vector space of  size 2 column vectors over F. We view X as  a locally compact totally disconnected topological  vector 
 space. Define a right action  X × H   → H, (v⃗, h)     ↦_1(h)^-1·v⃗ where  dot denote matrix multiplication. Let  be an integral domain in which ℓ is invertible and let 𝒮_X =  𝒮_X, denote the -module  of all locally constant compactly supported functions X →𝒪. Then 𝒮_X  inherits a smooth left  H-action. We define 
    ϕ =  (  [ _F; _F ] )  ∈𝒮_X .
  For any compact open subgroup V of H, we let 𝒮_X(V) denote the submodule V-invariant functions. Let Υ_H denote the collection of all compact open subgroups of H and  𝒫(H, Υ_H) denote the category of compact opens (see <cit.>). Then 
    𝒮_X : 𝒫(H, Υ_H) →𝒪-Mod,        V ↦𝒮_X(V)

is a cohomological Mackey functor. Note that ϕ∈𝒮_X(U).  For g ∈ G, let H_g = H ∩ gK g^-1 as before and V_g⊂ H_g denote the subgroup of all elements h ∈ H_g such that sim(g) ∈ 1 +  ϖ_F.   For  g ∈ G, we denote  by 𝔥_g∈𝒞_(U \ H / H_g ) the  (H,g)-restriction ℌ.  
  For any g ∈ G, 𝔥_g,*(ϕ)  lies in the image of the trace map _* :  𝒮_X(V_g ) →𝒮_X(H_g ).  
  

 Since 𝔥_η g γ , *  = 𝔥_g, *∘ [η]_H_g, H_η g , *, 
it suffices to  prove the claim for g  ∈ H  \  H  ·(ℌ) /  K. By the results of the previous section, a complete system of representatives for this double quotient is the set {ϱ_0, ϱ_1, ϱ_2,  ς_0,  ς_1,  ς_2, 
 ς_3, ϑ_0, ϑ_1, ϑ_2, ϑ_3,  ϑ̃_k  |   k ∈ []^∘}.    By the results established in <ref>, 
    𝔥_g,*(ϕ) ≡  0 q  - 1
 for  all g ≠ϑ_3 in this  set and 𝔥_ϑ_3, * ( ϕ )  =    -   (    [ ϖ ^-1_F^×;  ϖ^-2_F^× ] ).  So it suffices to show that χ : =    (  [ ϖ_F^×;  _F^× ] )  ∈𝒮_X(H_ϑ_3)  is the trace of a function in 𝒮_X(V_ϑ_3). 
 By  <cit.>, it suffices to verify that for all v⃗∈(χ), the stabilizer Stab_H_ϑ_3 (v⃗) of v⃗ in H_ϑ_3 is contained in V_ϑ_3. So let v⃗ = [ x; y ]∈(χ) and h = (h_1, h_2, h_3 )  ∈Stab_H_ϑ_3(v⃗). If we write  h_1 = [ a b; c d ], then v⃗· h  = v⃗ is equivalent to v⃗·  h^-1 = v⃗ and so     
 
    ( a  - 1 ) x +  by   = 0 , 
    
      cx +   ( d - 1 ) y  = 0   .

By Lemma <ref>, h_1∈_2(_F)  and   b  ∈ϖ^2_F. Since x ∈ϖ_F^×, it follows that a ∈ 1 + ϖ_F. Similarly 
y ∈_F^×,  x ∈ϖ_F ^×  implies  d ∈ 1 + ϖ_F. Thus sim(h) =  ad - bc  ∈ 1 + ϖ_F and  so  h ∈  V_ϑ_3.   
 

Now  let : = ×_m, G̃ its group of F-points and K̃ its group of _F-points. Embed  into  via 1 ×sim and let ι̃ : → denote the embedding (1 ×sim) ∘ι. Fix a c ∈ and define  
    ℌ̃ = ℌ_spin, c(Frob) ∈𝒞_[q^-1](K̃\G̃ / K̃  )
 where Frob = ( ϖ_F^×  ).   Let L̃  =  K × ( 1 + ϖ_F ) ⊂K̃. Let Υ_G̃ denote the collection of all compact open subgroups of G̃ and 𝒫(G̃, Υ_G̃) the associated   category.  
      For any cohomological  Mackey 
functor M_G̃ : 𝒫(G̃, Υ_G̃) →𝒪-Mod 
and 
any Mackey  pushforward ι̃_* : 𝒮_X→ M_G̃, there exists a class y    ∈ M_G̃( L̃  )  such that 
    ℌ̃_*∘ι̃_U , K̃ ,   *(ϕ  )=  _L̃  , K̃ ,*(y )
  



 By the expression in Proposition <ref>, it is clear that (G, g)-restriction of ℌ̃ is non-zero only if g ∈ G K̃ and the (G, 1_G̃  )-restriction is ℌ_spin,c(1).   The  claim is then a  consequence of  Theorem <ref>, Proposition  <ref>  and 
 <cit.>. 
 



 §.§ Global relations
    We now  repurpose our notation for the global setup.  Let , = ×_m,  be as before.  Fix a set S of rational  primes. By _S, be mean the product ∏_ℓ∈ S _ℓ and by _f^S, we mean the group of finite rational adeles away from primes in S.  Let  G,  G̃, H denote the group of _S·_f^S points of ,  ,  
respectively.  Let Υ_G̃ denote the collection of all neat compact open subgroups of G̃ and Υ_H denote the collection of compact open subgroups of the form H ∩L̃ where L̃∈Υ_G̃. Let 𝒫(H, Υ_H), 𝒫(G̃, Υ_G̃) denote the corresponding categories  of compact opens. These satisfy axioms (T1)-(T3) of <cit.>. 
 
Next fix a neat compact open subgroup K ⊂ G such that if ℓ∉ S is a rational prime, K = K^ℓ K_ℓ where K_ℓ =  G(_ℓ) as before and K^ℓ = K / K_ℓ⊂(_f^ℓ ) is the group at primes away from ℓ.    Let 𝒩 denote the set of all square free  products of primes away from S  where the empty product  means  1.   For each n ∈𝒩, let 
    K[n]  = K ×∏_ℓ∤ n_ℓ^×∏_ℓ| n  ( 1+ ℓ_ℓ)  ∈Υ_G̃ .
 
We also denote K[1] as K̃.   
Let X = Mat_2 × 1 (_f) ∖{0⃗} and let H act on X in a manner analogous to  the local situation. Let 𝒪 be a characteristic zero integral domain such that ℓ∈𝒪^× for all ℓ∉ S. Let  𝒮_X =  𝒮_X, 𝒪 denote the set 
of all 
functions   χ : X  →𝒪 such that χ   = 
f_S⊗χ^S where f_S is a fixed locally constant  compactly supported function on  Mat_2 × 1 (_S) that is invariant under (_S ) and χ  ^S is any locally constant compactly supported  function on Mat_2× 1(_f^S).   Then 
    𝒮_X : 𝒫(H, Υ_H) →𝒪-Mod ,      V ↦𝒮_X(V)
 is a CoMack functor with Galois descent.  Let U = H ∩K̃ and ϕ∈𝒮_X(U) be the  function  
f_S⊗(^S) where ^S = ∏_ℓ∉ S _ℓ denotes integral adeles away from S. Note that ϕ^S is the restricted tensor product of ⊗_ℓ∉ S ϕ_ℓ where ϕ_ℓ  =     ( [ _ℓ; _ℓ ] ). Fix an integer c and for each ℓ∈ S, let 
    ℌ̃_ℓ   =  ℌ_spin,c,ℓ(Frob_ℓ)  ⊗( K̃^ℓ) ∈𝒞_[ℓ^-1]  ( K̃\G̃ / K̃)

where Frob_ℓ  =   ( ℓ_ℓ ^  × ) is as before.  

 

      For any cohomological Mackey functor  M_G̃ : 𝒫(G̃, Υ_G̃) →𝒪-Mod 
and any Mackey pushforward   ι̃_* :   𝒮_X→   M_G̃,  there exists a collection of classes y_n∈ M_G̃(K[n]) indexed by integers n ∈𝒩 such that y_1 = ι̃ _U,K̃,*(ϕ) and 
    ℌ̃_*     
    ( y_n  )   = pr_K[nℓ],  K [n],*(y_nℓ)

for all n , ℓ∈𝒩 such that ℓ is a prime and ℓ∤  n.   
    
 Combine  Theorem <ref>, <cit.> and the  results  referred to in Corollary <ref>.  
  


PART: 
Proofs 


§ DOUBLE COSETS OF   GSP6
    




Throughout, we maintain the notations introduced in Part 1. 



 §.§ Desiderata
 The embedding ι' : → identifies the set Φ_H' of roots of 𝐇'   with   
    {±α_0 , ±α_2 , ±α_3 , ± ( α_2 + α_3) , ± ( 2 α_2 + α_3 )  }⊂Φ.
  
The Weyl group W' of H ' is then the subgroup of W generated by s_0 , s_    2      ,  s_3 and  W'  ≅   S_2× ( ( / 2  ) ^2⋊ S_2  ). We let Φ_H'^+ = Φ^+∩Φ_H be the set of positive roots. The base is then Δ_H' = {α_0, α_2, α_3}  
and the corresponding Iwahori subgroup I' of H ' equals the intersection I ∩ G. 
Since the normalizer N_H' (A) of A in H' equals the intersection N_G(A) ∩ H', the Iwahori Weyl  group 
W_I' = N_H'(A)/A^∘ is also identified  with a subgroup of W_I. We let W_aff' denote the affine Weyl group of H '. 

For notational convenience in referring to the roots  corresponding to the projection _2' = GSp_4 of ', we will denote  

    β_0  =  2 e_2 - e_0  ,      β_1 : = e_2 - e_3 ,     β_2 = 2 e_3  - e_0,
 and  let    r_0, r_1, r_2 denote the reflections associated with β_0, β_1  ,  β_2 respectively. In this notation, the generators of W_aff' of    are given by  S_aff'  =   {   s_0 , t (  f_1 ) s_0,  r_1, r_2 , t( f_2 ) r_0}  . 
The group W_I' is equals the semidirect product of W_aff' with the  cyclic subgroup Ω_H'⊂ W_I generated  by ω_H'   :=  t(-f_0) s_0 r_2 r _    1 r_2∈ W_I. The action of ω_H' on S_aff' is given by  s_0↔ t(f_1) s_0, r_2↔ t(f_2) r_0  and fixing r_1. It can be  visualized as the  order 2 automorphism of the extended Coxeter-Dynkin diagram

    [extended, Coxeter, edge length = 1cm, labels =  t(f_1) s_0  , s_0]A1    [extended,Coxeter,
    edge length=1cm,
    labels=t(f_2)  r_0    , r_1  ,  r_2]
    C2
 
A representative element in N_H'(A) for ω_H' is given by (ρ_1, ρ_2) ∈_2(F) ×_F^×GSp_4(F) where  
    ρ_1 =  
    [   1; ϖ ]  ,     ρ_2 = 
    [       1;     1  ;   ϖ  ; ϖ       ].

Note that ρ normalizes I'.  




 §.§ Intersections with H'
          In this subsection, we record some results on the structure of the twisted intersections H' ∩τ _i K τ_i^-1.  
      
If h ∈ H ', we will often write  h  =    (  [   a           b    ;     a_1 a_2     b_1 b_2;     a_3 a_4     b_3 b_4;   c           d;     c_1 c_2     d_1 d_2;     c_3 c_4     d_3 d_4 ]  ) or h =  (   (  a      b 
 c     d   )  ,     (   [ a_1 a_2 b_1 b_2; a_3 a_4 b_3 b_4; c_1 c_2 d_1 d_2; c_3 c_4 d_3 d_4 ] )   )  .    
 
 
                           H   ' K, H '  τ_1 K and H '  τ_2 K 
are pairwise disjoint.   
  
 
If τ _i K =  τ _j K for distinct i and j,   then τ_i^-1 h  τ_j∈ K for some h ∈ H.
Requiring the entries of k  : = τ_i ^-1 h  τ _j to be in _F, one easily deduces that ( k  ) ∈ϖ_F^×, a contradiction.    For instance,    
    τ_1^-1 h τ_2   =  [          a          *          *          * a - d_1ϖ^2          *;       -  c          *          *          *          *          *;                     *          *          *          *          *;        c ϖ                                *         cϖ           ;                     *          *          *       d_1ϖ          *;                     *          *          *          *          * ]

where a * denotes an expression in the matrix entries of  h and the empty spaces are zeros.  From the entries displayed above, we see that a , c  ∈ϖ_F and so the  first column is an integral  multiple of ϖ.      
  

     
  This also follows by an analogue of Schröder's decomposition  proved in <cit.>. 
 


    We let W^∘⊂ W' be the Coxeter subgroup generated by T : = 
 S_aff'  \{ s_0 , r_1} and U^∘  =  I'
W^∘     I'  the  corresponding maximal parahoric subgroup  of H'.    We let λ_∘  = (1, 1, 1, 1) and  τ_∘  = ϖ^-λ_∘τ_1.  

As usual,  we denote _τ_∘ : = ∩τ_∘  K τ_∘ ^-1. Then H_τ_1 ' is the conjugate of H_τ_∘ ' by ϖ^λ_∘.  Note that 
U^∘ is exactly the  subgroup of  whose elements lie in 
    [     _F ϖ^-1_F;    ϖ_F     _F ]×[     _F     _F ϖ^-1_F     _F;    ϖ_F     _F     _F     _F;    ϖ_F    ϖ_F     _F    ϖ_F;    ϖ_F     _F     _F     _F ]
 
 and whose similitude is in _F^×.    
      _τ_∘ is a subgroup of  U^∘ and '_2( _τ_∘ )  = '_2( U^∘ ). 
    
 Let h ∈  H _ τ_∘  ' and write h as in Notation <ref>. Then   
 
    τ_∘^-1 h τ_∘     =   [            a     -   c_1ϖ        -c_2ϖ b -  c_1/ϖ^2       a-d_1ϖ       - d_2ϖ;        -  cϖ          a_1          a_2    a_1 - d ϖ b_1  -  cϖ^2          b_2;                       a_3          a_4        a_3/ϖ          b_3          b_4;            c                                      d           cϖ             ;                       c_1          c_2         c_1ϖ          d_1          d_2;                       c_3          c_4        c_3 ϖ          d_3          d_4 ]∈  K
   
From  the  matrix  above, one sees that h satisfies all the conditions that are satisfied by elements of    U^∘, e.g., c ∈ϖ_F and  b ∈ϖ^-1_F and (h) = (τ_∘ h τ_∘^-1  ) ∈(K) ⊂_F^×. Therefore   _τ_∘⊂  U^∘. In  particular,  _2'( _τ_∘ ) ⊆_2   '  ( U^∘ ). To see the reverse inclusion, say h  = ( h_1 , h_2 )  ∈  U^∘   
and again  write h as in Notation <ref>.  Clearly, 
a_1 d_1 - b_1 c_1∈_F.
Since 
    sim(h_2)      =  a_1 d_1 - b_1 c_1 + a_3d_3 - b_3 c_3
       ∈  a_1 d_1 - b_1 c_1 + ϖ_F ,
 
we may find a', d' ∈_F, b ' ∈ϖ^-1_F and c   '     ∈ϖ_F such that a' - d_1/ϖ, a_1 - d' /ϖ, b' - c_1/ϖ^2, b_1 -     c'/ϖ^2 are all integral and a' d' - b ' c ' =  sim(h_2).    Then h '   =   (     (  [ a' b'; c' d' ] ) , h_2 ) ∈_τ_∘ and _2'(h') =  h_2.       
    


  We let  U^⊂ U'   denote the compact open subgroup of all elements     whose  reduction modulo ϖ  equals  (𝐇()).        
 

        _τ_2 is a subgroup of U'  and _2(_τ_2) = _2 ( U^ ). 
  
  If we write h ∈ H_τ_2' as in  <ref>, then 

    τ_2^-1 h τ_2   =   [            a         -c_1        -c_2ϖ   b  -c_1ϖ^2    a-d_1/ϖ^2       - d_2ϖ;          - c          a_1         a_2ϖ a_1 - d ϖ ^2  b_1 - c ϖ^2         b_2ϖ;                         *          a_4         a_3ϖ         b_3ϖ          b_4;            *                                      d            c             ;                         *            *            *          d_1            *;                         *          c_4        c_3 ϖ         d_3ϖ          d_4 ]∈  K
   
From  the  matrix  above, one sees that all the entries of h are integral. Since H_τ_2 is compact, sim(h) ∈_F^× and so   h ∈ U'.  Similarly, it is easy to see from the matrix above that _2' ( H_τ_2 ) ⊂_2'( U^ ).  For the reverse inclusion, say y ∈_2( U^) is given. Choose any h ∈ H ' such that '_2(h) = y and write h as in Notation <ref>.   Then 
    sim (y)        =  a_1 d_1 -  b_1c_1 + a_3 d_3 - b_3 c_3
       ∈  a_1 d_1 - b_1 c_1  +   ϖ^2_F
   
We may therefore find a', b', c' , d' ∈_F which are congruent to d_1, c_1, b_1, a_1 modulo ϖ^2 such that a'd' - b'c' =   sim (     y).  Then h ' =   (   ( [  a'  b'; c ' d ' ] ) ,   y   ) ∈_τ_2 and _2 '  ( h')  = y. 
 
   Let _τ :  _2→ be the embedding  given by   the  embedding  
    [ a b; c d ]↦ (  (  a     b 
     c     d   )  , 1.1(   [            ;           d                       c;                       1;           b                       a;                                     ad    -  bc ] ) ) .
 
We let  𝒳_τ :=  _τ(_2(_F)  ) and _τ∈𝒳_τ denote    _τ (  [   1; 1   ]  ). 
    
     For i = 0, 1, 2, 𝒳_τ is a subgroup of _τ_i.   In  particular,  '_1( _τ_i)  =  _2(_F ).    
    
 The first claim is easily verified by checking that τ_i^-1𝒳τ_i⊆ K for each i.  For the second, note that _1'(H_τ_i') are  compact open subgroups of H_1 = _2(F) that contains _2(_F ) and  U_1 = _2(_F) is a maximal compact open subgroup of H_1.        
  
         If h ∈ H_τ_i, a_1 - d , a - d_1 , b_1 - c, b - c_1∈ϖ^i_F. 
 
 Follows by   matrix  computations  above.   



 §.§ Cartan decompositions
      Throughout this article, we  let    ϖ^Λ denote the subset {ϖ^λ  |  λ∈Λ} of A.  
For i = 0, 1, 2, define  
    p_i  : Λ   → U ' ϖ^Λτ_i  K,      λ↦   U '   ϖ^λτ_i K  .
 
By <cit.>, we have an identification U ' ϖ^Λ H_τ_i '   U' ϖ^Λτ_i K given by U '  ϖ^λ_τ_i↦  U  ' ϖ^λτ_i  K.  
So we may equivalently view p_i as a map to U ' ϖ^Λ_τ_i. For i = 0, Cartan decomposition for H' implies the following. 
   p_0 induces a bijection W' \Λ  U' ϖ^λ K.    
 

Observe that _τ∈ N_H'(A^∘ ) is a lift of the element s_0 r_0∈ W '. 
Moreover 
    [   1            ;       1        ;           0           ϖ;               1        ;                   1;         -1ϖ           0 ]∈ H_τ_1' ,        [  1         ;     1      ;        0        1;           1      ;              1;       -1        0 ]∈ H_τ_2' .
 
Thus   p_1 factors through ⟨ s_0r_0, t(-f_3) r_2⟩\Λ and p_2 factor through   ⟨ s_0r_0, 
  r_2⟩\Λ.  

 For i = 1, 2, p_i ( λ) is distinct from p_i ( s_0λ ) if λ∉{ s_0λ ,   r_0λ}.            
   
  Write λ  = ( a_0, a_1, a_2, a_3 ).  Since p_i factors through ⟨ s_0 r_0⟩\Λ, we may assume by replacing λ with s_0(λ) etc.,  that 2 a_1≥ a_0 and 2 a_2≥ a_0.   Then we need to show that U' 
 ϖ^λ H_τ_i≠ U' ϖ^s_0(λ)_τ_i  whenever  2 p_1  > p_0 and 2p_2 >  p_0.  Assume on the contrary that  there  exists an h  ∈  U' such that γ  : =  ϖ^-λ  h  ϖ^s_0(λ)∈  
 H_τ_i. Write h = (h_1, h_2 ) as in Notation <ref>. Then γ  = (γ_1 ,  γ_2 ) satisfies  
 
    γ_1 =  [  a  ϖ^p_0 - 2p_1                b;                c d  ϖ^2p_1 -  p_0 ]  ,     γ _2  =  [               *               *               *               *;                               *               *               *; c_1ϖ^2p_2 - p_0               *               *               *;                               *               *               * ]  .

Lemma <ref> implies that   a ϖ ^  p_0  - 2p_1∈_F  and Corollary <ref> implies that  b - c_1ϖ^2p_2 -  p_0∈ϖ^i_F. Thus a ,   b   ∈ϖ_F. Since c, d ∈_F as h ∈ U', we see that sim(h) =  (h_1) = ad - bc ∈ϖ_F,  a  contradiction.   
    
Recall that λ_∘∈Λ denotes the cocharacter (1,1,1,1).  
         If  the W  ^ ∘-orbits of λ + λ_∘ and μ  + λ_∘ are distinct,   p_1(λ ) is distinct from p_1( μ  ).  
    
 Since W^∘ is a  Coxeter subgroup of the Iwahori Weyl group, there is a bijection 
    W^∘\  W_I' / W'         U  ^  ∘ϖ^Λ U'      
              W  ^ ∘ w  W ' ↦ U ^∘ w U ' .
     Recall that we have an isomorphism W_I'≃Λ⋊ W  ' which sends ϖ^λ∈ W_I' to (t(-λ), 1). Via this isomorphism, we obtain  bijection W^∘\Λ→ U ^∘ϖ^λ U' given by W ^∘λ↦  U ^∘ϖ^-λ U ' and hence a bijection 
     W^∘\Λ     U ' ϖ^λ U ^∘ ,        W^∘λ↦  U 'ϖ^λ U ^∘.
 
Now H_τ_∘⊂  U^∘ by Lemma  <ref>. So (the inverse of) the bijection above induces  a well-defined   surjection   U '  ϖ^Λ   H_τ_∘ '  →   U'   ϖ^Λ  U^∘   W^∘\Λ. Thus if λ_1, μ_1∈Λ are in different W^∘-orbits, U ' ϖ^λ_1 H_τ_∘' is distinct from U ' ϖ^μ_1 H_τ_∘ '.  Now  apply this to λ_1 : = λ  +  λ_0 and μ_1 : = μ + λ_∘ and use that H _τ_1 '  = ϖ^λ_∘ H_τ_∘  'ϖ^-λ_∘.        
 


 If the W'-orbits of λ, μ are distinct, p_2(λ) is distinct from p_2(μ). 
 

 This follows similarly since H_τ_2 '    ⊂   U'. 
    


 We denote  W_τ_1 ' = ⟨ s_0 r_0 , t(-f_3) r_2⟩⊂ W_I'  and  W_τ_2' = ⟨  s_0 r_0 , r_2⟩. We also denote W' by W_τ_1' for consistency.  

                  For i = 0, 1, 2,  the maps p_i induce bijections W _ τ_i '  \Λ U   ' ϖ^λτ_i  K. 
 


  Follows  from  the results above. 
  









 §.§ Schubert cells
  The decompositions of various double cosets is  accomplished by a recipe proved in <cit.>. Below, we    provide its formulation in the special case of  G = GSp_6(F).   

Recall that I denotes the Iwahori subgroup of G contained in U whose reduction modulo ϖ lies in the Borel of () determined by Δ.  For i = 0 ,1 ,2 ,3, let  
x_i : _a→𝐆 denote the root group maps 

    x_0   :      u ↦ (   [   1                    ;       1                ;           1            ; ϖ u           1        ;                   1    ;                       1 ] ),       
        x_1 : u  ↦[   1   u            ;       1                ;           1            ;               1        ;             - u   1    ;                       1 ],     x_2   : u   ↦[  1               ;     1  u         ;        1         ;           1      ;              1   ;             -u  1 ],       
    x_3 :  u  ↦[ 1          ;   1        ;     1     u;       1    ;         1  ;           1 ]

and let g_i : [  ] → G be the maps κ↦ x_i(κ ) w_i. Then Iw_i I/ I =  _κ∈ []  g_i(κ) I for i = 0,  1, 2, 3.  
For w ∈  W_I, 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 and  define  
    𝒳_w :  [   ] ^ ℓ(w)    →  G 
     
     (κ_1 ,   …,   κ_ℓ(w)  )   ↦    g_s_w,1 ( κ_1 )  ⋯  g_s_w, ℓ(w)  ( κ_ℓ (w)    )   ρ_w
    
Here, we have suppressed the dependence on the choice of the reduced word decomposition in light of the following result, 
which is a consequence of the braid relations in Iwahori Hecke algebras.     
     I w I =   _ κ⃗∈ []^ℓ(w)𝒳_w() I. If w has minimal possible length in w W, then I w K = _κ⃗∈ []^ℓ(w)𝒳_w(κ⃗)K.     
 
Thus the image of 𝒳_w modulo I is independent of the choice of decomposition and we have     |        im ( 𝒳_w)I/I      |     =  q ^ ℓ(w). Moreover,  the  same facts  holds with right K-cosets if w has the aforementioned   minimal length property. For such w,  ℓ(w) =    ℓ_min( t(-λ_w )  ) where λ_w∈Λ is the   unique     cocharacter   such that w K  =  ϖ^λ_w K. We refer to the image of 𝒳_w as a Schubert cell since these images are reminiscent of the Schubert cells that appear in the stratification of the classical  Grassmannians. 

Now given a λ∈Λ^+, a set of representatives for U' \ K ϖ^λ K / K can be obtained by studying U  '-orbits on a decomposition for K ϖ^λ K / K. Let W^λ denote the stabilizer of λ in W.  The next result shows that the study of such   orbits  amounts to studying U'-orbits on certain Schubert cells. 

 There exists a unique w =  w_λ∈ W_I of minimal possible length such that K ϖ^λ K = K w  K.  If [W / W^λ] denotes the set of minimal length representatives in W for W / W^λ, then   

    K ϖ^λ K  = _τ_κ⃗∈ []^ℓ(τ w  )𝒳_τ w  ( κ⃗  )    K  .

Moreover, ℓ ( τ w ) = ℓ (τ) + ℓ(w) for all τ∈   [ W / W^λ ].  
 
In what follows, we will write these Schubert cells for various words in W_I. Note  W / W^λ is identified with the orbit  W λ of λ.  The  set of possible reduced  words decompositions for τ∈ [W / W_λ] can be   visualized by a Weyl orbit diagram. This is the Hasse diagram on the subset [ W / W^λ ] ⊂ W under the weak left Bruhat order. Via the bijection [W/ W^λ ] ≃ Wλ,  the nodes of this diagram can be viewed as elements of W λ and its edges are labelled by  one of the simple reflections in Δ =  {  s_1 , s_2 , s_3}.  The unique minimal element of this diagram is λ^opp (the unique anti-dominant element in Wλ)  and  the unique maximal element in this diagram is λ. 
        Let λ = (2,2,1,1). Then λ ^ opp = (2,0,1,1) and the Weyl orbit diagram  is  
 
 


(2,0,1,1) [r, "s_1"]     (2,1,0,1) [r, "s_2"]     (2,1,1,0) [r, "s_3"]     (2,1,1,2)  [r, "s_2"]     (2,1,2,1)  [r, "s_1"]     (2,2,1,1)


 
By Lemma <ref>, we have w_λ = w_0ρ^2. So the decomposition of K ϖ^λ K / K can be given by six Schubert cells, corresponding to the reduced words 
    w_0ρ^2,     w_1 w_0ρ^2 ,     w_2 w_1 w_0ρ^2,      w_3 w_2 w_1 w_0ρ^2,         w_2  w_3 w_2 w_1 w_0ρ^2 ,       w_1   w_2  w_3 w_2 w_1 w_0ρ^2

which are obtained by  “going down" the Weyl orbit diagram. Each cell down this diagram can be obtained from one preceding it by applying two elementary row operations, one for the reflection and one for the root group  map.    We also apply an optional column operation to “match" the diagonal with  the value of the cocharacter at ϖ at each node (for aesthetic reasons).   For instance, let ε_0 = w_0ρ^2 and ε_1 = w_1ε_1. We have  
    im( 𝒳_ε_0) K / K          =   0.8*([   1                    ;       ϖ                ;           ϖ        ; x ϖ         ϖ^2        ;                   ϖ    ;                       ϖ ])   K    
    x   ∈  []
    im( 𝒳_ε_1)K/ K      =   
      0.8*([     ϖ     a                        ;           1                        ;                 ϖ            ;                       ϖ            ;         x ϖ       - a ϖ   ϖ^2      ;                                   ϖ ])   K    
    a  ,    x  ∈  []    
    .
 
Note that for ε =  w_2 w_3 w_2 w_1 w_0ρ^2, our recipe gives 

    im(𝒳_ε) K / K      
     =   
      0.8*([               ϖ                                                               a                ;                             ϖ^2            c_1ϖ           a   ϖ z  + cc_1 + ϖ x             c ϖ;                                               ϖ                                ;                                                               ϖ                                ;                                                                               1                ;                                                                           - c_1               ϖ ])   K    
    a  , c, c_1 ,     x  ,  z  ∈  []
 
However, we can replace z + cc_1 + ϖ x with a variable y running over [_2], since for a fixed value of c, c_1 and a,   the expression z + c c_1 + ϖ x runs over such a set of representatives of _F /ϖ^2_F  and a column operation between fifth and second columns allows us to choose any such set of representatives. In what follows, such replacements will be made without further  comment. 
 

Convention. To save space, we will often write the descriptors of parameters below the Schubert cells rather than within the set.  We will also write 𝒳_ε for the Schubert cell  where we really mean im(𝒳_ε)K/K and omit writing K next to the matrices.  When drawing Weyl orbit diagrams, we remove all the labels of the nodes as they can be read off by following the labels on the edges.  
   That the listed representatives are distinct follows by Lemma   <ref> and Lemma <ref>.     The goal therefore  is  to show that  the Schubert  cells reduce to the claimed  representatives in each case.   For each of the words w, we will draw the Weyl orbit diagram beginning in the anti-dominant cocharacter λ_w associated with w.  In   these   diagrams, we pick the first vertex and the vertices that only have one incoming arrow labelled s_1 (all of which we mark on the diagrams)  and study the U'-orbits on Schubert cells corresponding to these vertices. This suffices  since the orbits of U  ' on the remaining cells are contained in these by the recursive nature of the cell maps. 
We list all of the  relevant  cells and record all of our conclusions. However since the reduction steps involved are just  elementary row and column operations[row operations coming from _2(_F) ×__F^×GSp_4(_F) and column operations coming from GSp_6(_F)],  we only provide detailed justifications for one cell in each case, and leave the remaining for the reader to verify (all of which are  completely  straightforward).    
    



∙ w  =  ρ. Here  λ_w = (1,0,0,0) and the Weyl orbit diagram is as follows.  




                                                                                                     [rd, "s_1"  ]                                                             

 [r, "s_3"]      [r, "s_2"]      [ru, "s_3"] [rd, "s_1"' , "∘" marking   ]                                  [r, "s_2"]      [r, "s_3"]                   

                                                                                                     [ru, "s_3"']                                                           


 Thus there are two cells of interests, corresponding to the words ε_0 = ρ and ε_1 =  w_1 w_2    w_3ρ. The cell 𝒳_ρ   obviously reduces to ϖ^(1,1,1,1). As for ε_1, we have      
    𝒳_ε_1   =   0.8*([   ϖ   a   c   z        ;       1                ;           1            ;               1        ;              -a   ϖ;             - c       ϖ ])        a, c, z ∈   [     ]

We can eliminate z via a row operation. Then we conjugate by reflections w_3 and v_2 =  w_2 w_3 w_2 to make the diagonal ϖ^(1,1,1,1) which puts the entries a, c in the top right 3 × 3 block. Conjugation by w_1 switches a, c and one execute Euclidean division (using row/column operations) to make one of a or c equal to zero. Conjugating by an  element of A^∘ if necessary, we get ϖ ^(1,1,1,1) or τ_1 as  possible  representatives from this cell.      
     


∙ w = w_0ρ^2. The Weyl orbit diagram of λ_w =  (2,0,1,1) is 

    


 [r, "s_1",  "∘" marking]     [r, "s_2"]      [r, "s_3"]      [r, "s_2"]      [r, "s_1",   "∘" marking  ]          


 There are three cells of interests corresponding to ε_0 =  w_0ρ^2, ε_1 = w_1ε_0 and ε_2 =   w_1 w_2 w_3 w_2ε_1.  The cells 𝒳_ε_0, 𝒳_ε_1 were recorded in Example <ref> and 

    𝒳_ε_2 = 0.8*([    ϖ ^2    a_1ϖ    c_1ϖ z + ϖ x    a  ϖ   c   ϖ;               ϖ               a                ;                       ϖ       c                ;                               1                ;                            -a_1       ϖ        ;                            -c_1               ϖ ])        a ,  a_1 ,  c , c_1,  
           x,  z   ∈  []   .

We claim that the U  '-orbits  on 
      

  * 𝒳_ε_0 are represented by  ϖ^(2,2,1,1), 

  * 𝒳_ε_1  
are represented by ϖ ^(2,1,2,1), ϖ^(1,1,0,0)τ_1,    

  * 𝒳_ε_2 are represented by ϖ^(2,2,1,1)  ,  ϖ^(1,1,0,0) τ_1.  

We record our steps for reducing 𝒳_ε_2.    Eliminate the entry z + ϖ x using a row operation. Conjugation by w_3∈ U (resp., w_2w_3w_2∈ U   ') switches a_1 , a (resp., c_1 , c) and keeps the diagonal ϖ^(2,2,1,1). Using row/column operations, we may make one a, a_1  (resp., c ,  c_1) zero while still keeping the diagonal ϖ^(2,2,1,1). Without loss of generality,  assume a_1 , c_1 are zero.  Conjugation by w_2∈ U   '  switches  a, c and we may again apply row-column operations to make one of a, c zero, say c. Normalizing by an appropriate  diagonal matrix in A^∘, we get the representatives ϖ^(2,2,1,1) or ϖ^(1,1,0,0)τ_1 depending on whether a = 0 or not. 
   






∙ w = υ_1ρ^2.  We have λ_w =    (2,0,0,1) and the Weyl orbit diagram is 




                                                                                                                         [r, "s_1",   "∘" marking  ]      [rd, "s_2"]                                                                                  

  [r, "s_2"]      [r, "s_3"] [rd, "s_1"',  "∘" marking  ]      [rd, "s_1"    ] [ru, "s_2"]                                                                                [r, "s_3"]        [r, "s_2"]      

                                                                            [r, "s_3"']                          [r, "s_2"']      [ru, "s_1"] [r, "s_3"']      [ru, "s_1"']                                 
 

 
So we need to study the U-orbits on the analyze Schubert cells corresponding to the words 
ε_0 =  w, ε_1 = w_1 w_2 w and ε_2 = w_1 w_2 w_3 w_2 w .. The cells corresponding to these words are  
    𝒳_ε_0  =   0.8*([     1                              ;           1                        ;                 ϖ                  ;  x_1ϖ   a ϖ         ϖ^2            ;   a ϖ - x ϖ               ϖ^2      ;                                   ϖ ]),   𝒳_ε_1  =   0.8*([       ϖ     a_1       c                        ;               1                                ;                       1                        ;                               ϖ                ;            x_1ϖ     a ϖ   -a_1ϖ ϖ  ^  2        ;             a ϖ   - x ϖ   - c ϖ            ϖ ^2 ])
 

    𝒳_ε_2  =   0.8*([           ϖ^2      a_1 +a ϖ        c_1  ϖ       z + ϖ x                         c ϖ;                           1                                                        ;                                         ϖ             c                            ;                                                       1                            ;                      x_1  ϖ               -(a_1 +  a ϖ)           ϖ^2              ;                                                    -c_1                           ϖ ])
  
where a, a_1, c , c_1 , x , x_1,   z ∈ []. 
We claim that the U  '-orbits  on 
    
     

  * 𝒳_ε_0 are given by ϖ^(2,2,2,1), ϖ^(1,1,1,0)τ_1, 

  * 𝒳_ε_1 are given by ϖ^(2,1,2,2), ϖ^(1,1,0,1)τ_1, 

  * 𝒳_ε_2 are given by ϖ^(2,2,2,1)   ,  ϖ^(1,1,1,0)τ_1   ,  ϖ^(2,1,1,1)τ_2.   
    
   We record our analysis for 𝒳_ε_2. 
Begin by eliminating  the entries  z + ϖ x and x_1ϖ using row operations.   Conjugation by w_3∈  U ' switches c_1, c while keeping the diagonal ϖ^(2,2,0,1) and we can apply row-column operations to make either c or c_1 zero, say c_1. Conjugating by r_0 =   w_2 w_3 w_2∈  U    ', we arrive at   

    0.8([      ϖ^2                            a_1 +a ϖ      c ϖ;               ϖ^2          a_1 +a ϖ                  ;                          ϖ        c                  ;                                   1                  ;                                            1         ;                                                     ϖ ])

for some a, a_1, c ∈ []. We now divide in two case.   Suppose first that c is zero. Then (a_1 + a ϖ ) is in _F^×, ϖ_F^× or is equal to zero, and we  can normalize by conjugating with an element of A^∘ to get the representatives ϖ^(2,2,2,1), ϖ ^(1,1,1,0)τ_1, ϖ^(2,1,1,1)τ_2.  Now suppose  that  c ≠ 0.  Then we  may assume a = 0 by applying  row-column operations. If now  a_1≠ 0, we may make c = 0 and normalizing by A^∘  
leads us to the representative  ϖ^(1,1,1,0)τ_1. 
If a_1 = 0 however, then conjugating by w_2 and normalizing by A^∘ gives us the representative   ϖ^(1,1,0,1)τ_1. 
      



 


∙  w =  υ_2ρ^3.  Here λ_w = (3,0,1,1) and the Weyl orbit diagram is    






                                                                                                                                                 [r, "s_1"]                             [r, "s_2"]                            [rd, "s_3"]                                                                                     

                                                                                                  [ru, "s_3"] [r, "s_1", "∘" marking   ]      [rd, "s_2"'] [ru, "s_3"']                                                                                  [rd, "s_2"]                                                         

                                                 [rd, "s_1"] [ru, "s_2"]                                                                                                        [rd, "s_3"]                                                                                                       [rd, "s_1"]         

   [rd, "s_1"',  "∘" marking]  [ru, "s_3"]                                                       [r, "s_2"]                           [rd, "s_3"'] [ru, "s_1"]                                                       [r, "s_2"]        [rd, "s_1"'] [ru, "s_3"]                                  

                                                   [ru, "s_3"'] [rd, "s_2"']                                                                                                      [ru, "s_1"'] [rd, "s_2"]                                                                                  [ru, "s_3"']        

                                                                                                  [rd, "s_3"']                                                                                                                          [r, "s_1"]        [ru, "s_2"']                                                        

                                                                                                                                                 [r, "s_2"']                            [r, "s_1",swap,   "∘" marking   ] [ru, "s_3"]          [ru, "s_3"']                                                                                   


  
There are four cells of interest corresponding to words  ε_0 =  w_0 w_1 w_2 w_3ρ^3, ε_1 =  w_1ε_0, ε_2 =  w_1 w_2  w_3ε_0 and ε_3 = w_1  w_2 w_3 w_2  w_1ε_0.    Their   Schubert cells are  

    𝒳_ε_0    = 0.8*([     1                              ;           ϖ                        ;                 ϖ                  ;    yϖ a ϖ^2 c ϖ^2   ϖ^3            ;   a ϖ                     ϖ^2      ;   c ϖ                           ϖ^2 ]),      𝒳_ε_2 : = 0.8*([           ϖ^2      a_1 +c ϖ        c_1  ϖ           ϖ z                            ;                           1                                                        ;                                         ϖ                                          ;                                                       ϖ                            ;                        - yϖ        -a ϖ^2 -ϖ (a_1 +c ϖ)           ϖ^3;                        -a ϖ                     -c_1  ϖ                         ϖ^2 ])
    𝒳_ε_1     =    0.8*([         ϖ       a_1                                        ;                   1                                        ;                             ϖ                              ;                 a ϖ                 ϖ^2                    ;     a ϖ^2      y  ϖ     c ϖ^2 -a_1  ϖ^2       ϖ^3;                 c ϖ                                     ϖ^2 ]),      𝒳_ε_3 = 0.8*([               ϖ^3     a ϖ^2 +a_2  ϖ     c ϖ^2 +c_2  ϖ                 z          a_1  ϖ^2          c_1  ϖ^2;                                   ϖ                                 a_1                                    ;                                                     ϖ               c_1                                    ;                                                                       1                                    ;                                                        - (a_2 + a ϖ   )               ϖ^2                  ;                                                       -( c_2  +  c ϖ  )                                 ϖ^2 ])

where a, a_1,  a_2  ,  c, c_1, c_2∈ [], y ∈ [ _2] and z ∈ [_3].  Then we claim that the U'-orbits on   
         

  * 𝒳_ε_0 are represented by ϖ^(3,3,2,2), ϖ^(2,2,1,1)τ_1, 

  * 𝒳_ε_1 are represented by ϖ^(3,2,3,2) ,  ϖ^(2,1,2,1) τ _1,   

  * 𝒳_ε_2 are represented by ϖ^(3,2,3,2),  ϖ^(2,1,2,1)τ _1, 
ϖ^(2,1,1,2)τ _1,       ϖ^(3,2,1,2)τ_2, 

  * 𝒳_ε_3 are 
represented by  ϖ^ (3,3,2,2) , ϖ^(2,2,1,1) τ _1,  ϖ^(2,2,0,1)τ  _1,  ϖ^ (3,2,1,2)  τ  _2.  

    
 We record our reduction steps for 𝒳_ε_3. 
Begin by eliminating the entry y by a  row operation.  Observe  that if a_1 (resp., c_1) is not zero, then we can assume a (resp.,  c) is zero by row column operations. Moreover, conjugation by w_2 switches the places of a,  a_1,  a_2 by c, c_1, c_2 respectively  and   keeps  the  diagonal  ϖ ^  ( 3,3,1,1).    We have three cases to discuss.   
    

Case 1. Suppose a_1 = c_1 = 0. Apply row column operations to replace a ϖ^2 + a_2ϖ, c ϖ^2 + c_2ϖ by their greatest common divisor (with the other entry being zero).  Since we can  swap entries by w_2, let's assume that  a ϖ^2 + a_2ϖ = 0. We may normalize the gcd by an element of A^∘ so that the greatest common divisor is  0 or ϖ or ϖ^2.  Now conjugate by  s_2 r_0 r_2 = s_2 (s_1 s_0s_1) s_3∈  U to   makes the diagonal ϖ^(3,3,2,2) and put the non-diagonal entries in right place.  Thus this case leads us to    representatives  ϖ^(3,3,2,2)τ_1, ϖ^(3,3,2,2)τ_2.   
    



Case 2.  Suppose exactly one of a_1, c_1 is non-zero. Since we can swap these, we may assume  wlog  a_1≠ 0, c_1 = 0.  Then we are free to make  a = 0. Now if  a_2≠ 0, it  can be used to replace the entries  a_1 , c , c_2 by zero.   Conjugating by r_0 r_2 =  w_2 w_3 w_2 w_3 and normalizing by A^∘ gives us ϖ^(3,3,2,2)τ_2. If however a_2 = 0, then we can conjugate by w_3 to make the diagonal ϖ^(3,3,1,2) while moving the c ϖ^2 + c_2ϖ entry corresponding to the root group of e_1 + e_3 - e_0. As a_1≠ 0, we are free to eliminate c_1.  There are now two further sub-cases. If c_2 = 0, we obtain the representative ϖ^(3,3,1,2)τ_1 after normalizing by an element of A^∘. If however c_2≠ 0, we can replace a_1 = 0 and  conjugating by w_2 w_3∈ U and normalizing by A^∘ gives us ϖ^(3,3,2,2)τ_2.  
 


Case 3.  Suppose both a_1, c_1 are non-zero.   Then we may assume a, c are zero. If a_2 (resp., c_2) is not zero, we can  eliminate entries containing  a_1 (resp.,  c_1).  Then an argument similar to Case 2 yields ϖ^(3,3,2,2)τ_2, ϖ^(3,3,1,2)τ_1 as representatives.     
 

∙      w =  υ_3ρ^4.  The Weyl orbit diagram  for λ_w   =    (4,1,1,1) is the same as for (1,0,0,0) and so we have to analyze cells of length ε_0 =  w_0 w_1 w_0 w_2 w_1 w_0ρ^4 and ε_1 =  w_1 w_2 w_3ε_0. The two cells are as follows: 

    𝒳_ε_0     =  0.8*([        1                                             ;                 1                                    ;                          1                           ;    x_2 ϖ   a_1  ϖ      c ϖ      ϖ^2                  ;   a_1  ϖ -  x_1 ϖ    - a ϖ               ϖ^2         ;      c ϖ   -  a ϖ   x 
      ϖ                        ϖ^2 ]) ρ^2     a ,  a_1 ,  c, x    
          x_1, x_2∈   [ ]
    𝒳_ε_1    =    0.8*([                ϖ^2         a_2 + c  ϖ         c_1  +a  ϖ           z +  x ϖ                                      ;                                     1                                                                            ;                                                        1                                                         ;                                                                           1                                      ;                             - x_2   ϖ             - a_1ϖ    - ( a_2 + c ϖ )                ϖ^2                   ;                               -  a_1ϖ            x_1   ϖ -  ( c_1 +  a ϖ  )                                   ϖ^2 ]) ρ^2      a ,  a_1  , a_2 ,  c, c_1 , 
          x ,  x_1, x_2,  z       ∈   [ ]
 We claim that   the U'-orbits on 
             

  * 𝒳_ε_0 are given by  ϖ^(4,3,3,3), ϖ^(3,2,2,2)σ _1,   

  * 𝒳_ε_1 are given ϖ^(4,3,3,3), ϖ^(3,2,2, 2)τ_1, ϖ^(4,2,2,3)τ _2,  
     
 
We record our analysis for orbits on 𝒳_ε_1. 
We can eliminate the entries involving a_1,  x , x_1 , x_2 , z using row  operations. Conjugating by w_3 and w_2 w_3 w_2 gives us  
    0.8([         ϖ^2                                        a_2 +c ϖ c_1  +  a ϖ;                     ϖ^2                a_2 +c ϖ                        ;                                 ϖ^2  c_1 +  a ϖ                        ;                                               1                        ;                                                           1            ;                                                                       1 ])   ρ^2
 
and one can apply Euclidean algorithm to the entries c_1 + a ϖ, a_2 + c ϖ to replace one of them with 0 and the other by the greatest common divisor which is either 0, 1 or ϖ.   Conjugating by w_2 and normalizing by A^∘ if necessary, we obtain  the  three representatives.  
 

∙ w =  
υ_4ρ^4. We have λ_w = (4,0,2,2) and the Weyl orbit diagram is the same as for (2,0, 1,1).  We need to analyze  the 
Schubert cells  corresponding to ε_0 = w_0 w_1 w_2 w_3 w_2 w_1 w_0ρ^4, ε_1 =  w_1  w and ε_2 =   w_1 w_2 w_3 w_2 w_1  w. These cells are 

    𝒳_ε_0 =   0.8*([        1                                             ;      a ϖ      ϖ^2                                    ;      c ϖ               ϖ^2                           ;       yϖ a_1  ϖ^3 c_1  ϖ^3      ϖ^4   -a ϖ^3   -c ϖ^3;   a_1  ϖ                                 ϖ^2         ;   c_1  ϖ                                          ϖ^2 ]),   𝒳_ε_1 = 0.8*([                ϖ^2           a_2 +a ϖ                                                                            ;                                     1                                                                            ;                                   c ϖ                ϖ^2                                                         ;                                a_1  ϖ                                   ϖ^2                                      ;           a_1  ϖ^3               y  ϖ           c_1  ϖ^3 - (a_2 + a  ϖ) ϖ^2                ϖ^4             -c ϖ^3;                                c_1  ϖ                                                                         ϖ^2 ])


    𝒳_ε_2 = 0.8*([                ϖ^4 a_1  ϖ^3 +a_3  ϖ^2 c_1  ϖ^3 +c_3  ϖ^2                       a ϖ^3 +a_2  ϖ^2    c ϖ^3 +c_2  ϖ^2;                                   ϖ^2                              a_2 +a ϖ                                      ;                                                      ϖ^2           c_2 +c ϖ                                      ;                                                                           1                                      ;                                                                -a_3 -a_1  ϖ                ϖ^2                   ;                                                                -c_3 -c_1  ϖ                                   ϖ^2 ])
 
where a , a_1, a_2, a_3,  c , c_1 , c_2 , c_3∈  [] and y ∈ [_3].  We claim that  the U '-orbits on  
       

  * 𝒳_ε_0   are given by  ϖ^(4,4,2,2), ϖ^(3,3,1,1)τ _1,  

  * 𝒳_ε_1 are given by 
ϖ^(4, 2, 4, 2 ), ϖ^(3,2,0,1)τ_1,  ϖ^(4,3,1,2)τ _2     

  * 𝒳_ε_2 are given by   
ϖ^(4,4,2,2), ϖ^(3,3,1,1)τ  _1,  
   
   
Let us record our steps for the reduction of 𝒳_ε_1. 
We begin by eliminating the entries involving y, c, c_1  using row operations. If a_1 = 0, then conjugating r_0 = w_2 w_3 w_2 and normalizing by an appropriate element of A^∘, we obtain     ϖ^(4,2,4,2), ϖ^(3,1,3,1)τ_1, ϖ^(4,1,3,2)τ_2 depending on the valuation of a_2  + a ϖ. Now 
    U  ' ϖ^(3,1,3,1)τ_1 K =  U '  ϖ^(3,2,0,1)τ_2 K,      U ' ϖ^(4,1,3,2)    K   =  U ' ϖ^(4,3,1,2)τ_2 K
 
by Proposition <ref>. If however a_1≠ 0, then a can be made zero via row-column operations. We then  have two  further 
subcases. If a_2 = 0, then we can conjugate by s_0 = 
s_α_0 and normalize by A^∘ to obtain ϖ^(3,1,3,1)τ_1 which is the same as ϖ^(3,2,0,1). On the other hand, if a_2≠ 0,  then a_1 can be made zero and   normalizing by A^∘ gives ϖ^(4,1,3,2)τ_2 which is the same as ϖ^(4,3,1,2)τ_2.   
 


     If one instead tries  to directly study the U-orbits on the double cosets in the proof above, one needs to study far more Schubert cells and distinguish an enormous number of representatives from each other. For instance for w = υ_2ρ^3, one would need to study 12  cells instead of 4.  
  


§ DOUBLE COSETS OF  GL2 X GSP4
 

In this section, we record the proofs of various  claims involving the action of U on double cosets spaces of H '. Since both U and U' have a common _2(_F ) component, the computation of orbits is facilitated   by  studying the orbits of U_2 on double cosets of H _2  '. This in turn is  achieved by techniques analogous to the one used in <ref> for decomposing double cosets of parahoric subgroups of an unramified group.  



      
If h ∈ H_2⊂ H_2  ', we will often write  
    h  =    (  [   a       b;     a_1     b_1;   c       d;     c_1     d_1 ]  )      or        h =  (   (  a      b 
     c     d   )  ,     (    a_1    b_1
     c_1    d_1 )   )    .

We let Λ_2 denote f_0⊕ f_2⊕ f_3. Given λ = a_0 f_0 + a_2 f_2 + a_3 f_3∈Λ_2 as (a_0, a_2, a_3) and let ϖ^λ denote the element  diag(ϖ^a_2, ϖ^a_3, ϖ^a_0 -a_2,  ϖ^a_0 - a_3)  ∈ H_2'.    




 §.§ Projections
      
Let s : '_2→' denote the section of _2' given  by γ↦ (   (   [ sim(γ);             1 ] )  ,    γ  ). Fix a compact open subgroup V ⊂ H ' such that _1'(V) = _2(_F ) and an arbitrary  element h = (h_1, h_2 ) ∈ H '. Denote V_2 = _2'(V).  We  refer to   
    _h, V   :  U \ U' h  V / V 
       → U_2\ U_2' h_2  V_2 / V_2 , 
      
    U  γ  V      ↦ U_2_2(γ )   V_2
  
as the projection map. We are interested in the fibers of pr_h,V.  




   



    Suppose h_1∈_2(F) is diagonal and either s(V_2) ⊂ V or h_1 is central.  If η∈ H_2' has the same similitude as h and U_2η  V_2∈ U_2\ U_2 ' h_2 V_2/ V_2, then {  U (h_1, η) V    } = _h,V^-1(U_2η V_2 ). In particular, pr_h,V is a bijection.  

 
 
Note that  any element of U  \ U ' h V / V  can be written as  U (1, γ) h V for some γ∈Sp_4(_F) and similarly for elements of U_2\  U_2' h_2 V_2/ V_2.  This immediately implies that _h,V is surjective.  

Suppose now that γ∈Sp_4(_F )  is such that   U (1, γ ) h V maps to  U_2η V_2 under pr_h,V.  
Then  there  exist u_2∈ U_2, v_2∈ V_2  such that η  =  u_2γ  h_2 v_2. Taking similitudes, we see that sim(u_2)   = sim(v_2) ^-1.   Let u_1 = diag(1, sim(u_2) )  ∈_2(_F) and set u = (u_1 , u_2) ∈  U.
Take v = s(v_2)  ∈ V if s(V_2) ⊂ V or an arbitrary element in (_2')^-1(V) if h_1 is central.  Write v = (v_1, v_2). Then  

    U (1,γ) h V     = U u (1,γ) h v V 
        = U (u_1h_1 v_1, u_2γ_2 h_2 v_2 ) V
        =  U ( u_1v_1 h_1, η ) V  
       
    = U (h_1, η) h V

where we used that h_1 commutes with v_1 in both cases and  that  (u_1v_1, 1) ∈ U_2.  

In case h_1 is non-central or s(V_2) ⊄V,  one needs to perform an additional check to determine the fibers of _h, V.  Define 

S^-      = {  (   [ 1  ; x 1 ]  )   
|     x ∈_F  } ,                S^+     = 
{
([     1; -1 ])  
( [  1 ϖx;     1 ] )   |   x ∈_F      }.
 
For a positive integer a, define S_a^- to be the subset S^- where we require the variable x to lie  in [_a ] (see <ref> for notation)  and S_a^+ the subset of S^+ where we require  x to lie in [_a-1   ]. 
We also  denote S^± = S^-∪ S^+ and S^±_a = S^-_a∪ S^+_a. 

   Suppose   h_1   =   diag  (ϖ^u, ϖ^v) with  u > v and  η∈ H_2' is such that  U_2η V_2∈   U_2\  U_2' h V_2 / V_2 with   sim(η) =    ϖ^ u+v. Then 
    ^-1_h ,  V ( U_2η V_2 ) =   {  U (h_1χ ,   η )   V    |  χ∈ S^± _ u -  v and     U'  (h_1χ , η ) V = U' h V    }
  


 In the proof of Lemma <ref>, one obtains the equality U ( 1, γ ) h V = U (h u_1 v_1 , η V ) with u_1 v_1∈_2(_F ). Now u_1 v_1 can be replaced with a   representative in the quotient 
    ( _2(_F )  ∩ h_1  ^-1_2(_F  )  h_1  )    \_2(_F)
 
and S^±_u-v forms such a set of representatives.        

 We will need to use the last result for V ∈{ H_τ_1' , H_τ_2' } when lifting coset representatives η  for U_2\ U_2' h_2 V_2/ V_2 to U \ U' h V/ V.  In almost all cases,  it will turn out that there is essentially one choice of γ∈ S^± that satisfies U ' (h_1γ , η)V = U' h V. If  there are more than one element in the fiber, we will invoke  a suitable Bruhat-Tits  decomposition for parahoric double cosets to distinguish them.         









 §.§ The GSp_4-players
        
Recall that the roots of H_2' = GSp_4  are  identified with 
    {±β_0, ±β_1, ±β_2 , ± (β_1 + β_2)    } .
     To compute these decompositions,  we let 

    v_0 =   0.9(   
          1ϖ
        1       
    ϖ         
              -1 
     ) ,      
    v_1 =   0.9( 
         1
     
    1         
              1 
           1     ) ,       
    v_2 =  
    0.9(  1    
              1  
           - 1 
        1        )
 
which respectively represent the reflections t(f_2) r_0, r_1, r_2 which generate the affine Weyl group  W_2, aff' of H_2'.  We also denote v_β_0 = diag(ϖ, 1, ϖ^-1, 1) v_0 which represents the reflection r_0   in the root β_0.   For i = 0, 1, 2, let y_i : _a→_2' be the maps 
    y_0 : u ↦0.9(   1    
         1        
       u  ϖ       1     
              1   ),     
    y_1 : u ↦0.9(    1      u 
         1 
           1 
           - u     1   ) ,    
    y_2  : u ↦0.9(  
     1     
         1         u 
           1 
              1     )   .

If I_2' ⊂ H'_2 denote the Iwahori subgroup given by _2(I'), then y_i([]) forms a set distinct q  representatives for the quotients I_2'/ I_2' ∩ v_i I_2' v_i.  For each i = 0, 1, 2, let  
    h_r_i : [] → H_2 ' ,   κ↦ y_i(κ) v_i .

Let W_I_2' denote the Iwahori Weyl group of H_2' and  l : W_I_2'→ denote the length function induced by _2(S_aff') = { r_1, r_2 , t(f_2) r_0}.   For v ∈ W_I_2' and v = r_v,1 r_v,2⋯  r_v, lω_v (where r_v,i∈_2(S_aff'), ω_v∈_2( Ω_H') is a power of _2(ω_H'))  is a reduced word decomposition, we set 

    𝒴_v : []^l(v)   ⟼  H_2 ' 
    
    (κ_1, …,  κ_l(v))      ↦ h_r_v,1(κ_1)    ⋯ h_r_v,l(v)(κ_l(v)) ρ_2,v
  
where ρ_2,v∈ H is the element representing ω_v.   For a compact open subgroup V ⊂ H_2', we let 𝒴_v/V to denote the coset space  im(𝒴_v)V/V, which we will also refer to as a Schubert cell.  



 §.§ Orbits on U' h U'/U'


Let W_2 denote the Weyl group of H_2 =  _2×__m_2. We can identify W_2 as the subgroup of W_2' generated by r_0 and r_2. For η∈ H_2', denote H_2∩η U_2' η^-1 by H_2,η.  Then the map 

    U_2ϖ^Λ_2η U_2' → U_2ϖ^Λ_2 H_2,η   U_2ϖ^λη U_2 ' ↦ U_2ϖ^λ H_2,η
 
is a bijection.  Let η_1, η_2 denote the projection of ϱ_1, ϱ_2 given in  (<ref>) to H_2 '.  Explicitly,  

    η_1
    =   0.9(  ϖ          1 
       ϖ    1 
            1  
              1  ) ,   η_2 =  0.9( ϖ^2          1  
       ϖ^2    1  
           1 
              1  )
 
 
    The cosets H_2 U_2', H_2η_1 U_2' and H_2η_2U_2' are pairwise disjoint. 
 
 This is similar to Lemma   <ref>. See also Remark <ref>.  
 
    The map W_2\Λ_2→ U_2ϖ^Λ_2 U _2' given by W_2λ↦  U_2ϖ^λ U_2' is a bijection.   If λ, μ∈Λ_2 are not in the same W_2-orbit, then  U_2ϖ^λη_1  U_2' is distinct from  U_2ϖ^μη_1  U_2'. 
 
   The first claim follows by the bijection (<ref>)  and Cartan decomposition for H_2. It is easily verified that H_2,η_1⊆ U_2, so  the second claim also follows by  Cartan decomposition for H_2.    
 
 

    For i = 1 ,2 and any λ∈Λ_2, U_2ϖ^λη_i U_2' = U_2ϖ^r_0r_2(λ)   η_i U_2'.  
 

 This follows by noting that η_i^-1 v_β_0 v_2η_i∈ U_2' for i = 1 , 2 and v_β_0 v_2∈ U_2.   
   
 
 For h ∈ H',   Let ℛ(h) denote the double coset space U_2\ U_2' h U_2'/ U_2'. 
By Lemma <ref>, it suffices to establish that 
   

  * ℛ(ϖ^(1,1,1)) = 
{ϖ^(1,1,1),  η_1}, 

  * ℛ(ϖ^(2,2,1)) = {ϖ^(2,2,1),  ϖ^(2,1,2), ϖ^(1,1,0)η_1},

  * ℛ(ϖ^(2,2,2)) = 
{ϖ^(2,2,2), ϖ^(1,1,1)η_1, η_2} 

  * ℛ(ϖ^(3,3,2)) = 
{ϖ^(3,3,2), ϖ^(3,2,3), ϖ^(2,2,1)η_1, ϖ^(2,2,0)η_1, ϖ^(1,1,0)η_2}, 

  * ℛ(ϖ^(4,4,2))  = {ϖ^(4,4,2),  ϖ^(4,2,4), ϖ^(3,3,1)η_1, ϖ^(2,2,0)η_2}. 
 
It is easy to check using Lemma <ref> and  Lemma <ref>  that the listed elements in each case represent distinct double cosets. It remains to show that they form a complete set of representatives. Here we again use the recipe given by <cit.>. As before, we will write the parameters of the below them and omit writing U_2 ' next to the matrices.  

 

(a) & (b) These were calculated in <cit.>.   


(c) We have U_2' ϖ^(2,2,2) U_2' = U_2' v_0v_1v_0ρ^2 _2  U_2' and v_0 v_1 v_0ρ^2_2 is of minimal possible length. The Weyl orbit diagram of (2,2,2) is 

 


 [r, "r_2"]     [r, "r_1"]      [r, "r_2"]        



So we need to analyze the cells corresponding to the first and the third node, which are of length 3 and 5 respectively. Let  ε_0 = v_0v_1v_0ρ^2_2 and ε  _1 = v_1 v_2v_0v_1v_0ρ^2_2 be the words corresponding to these nodes. We have 
    𝒴_ε_0  / U_2'      = 0.9*([    1               ;         1          ; x_1ϖ  a ϖ  ϖ^2     ;  a ϖ  x ϖ       ϖ^2 ]),     𝒴_ε_1   / U_2'    = 0.9*([          ϖ^2       a_1+aϖ        y+ϖ x;                         1                          ;                                      1;                      x_1ϖ -(a_1 + a ϖ)          ϖ^2 ])
 
where a, a_1 , x , y run over []. 
For the first cell, eliminate ϖ x_1, ϖ x via row operations and conjugate by v_α_0v_0.  For the second, eliminate y+ ϖ x, ϖ x_1 similarly and conjugate by v_2. The resulting matrices are 
    0.9( 
    ϖ^2          aϖ
       ϖ^2    aϖ
           1 
              1
     ),     0.9( 
    ϖ^2          a_1 + aϖ
       ϖ^2    a_1 + aϖ
           1 
              1
     )

respectively. By conjugating with appropraite diagonal matrices,  the left matrix can be simplified to ϖ^(2,2,2) or ϖ^(1,1,1)η_1 depending on whether a is zero or not. Similarly the second one simplifies to one of ϖ^(2,2,2), ϖ^(1,1,1)η_1, η_2.  


(d)  We have  U_2 ' ϖ^(3,3,2) U_2'  = U_2' v_0v_1v_2ρ^3_2   U_2' with v_0 v_1 v_2ρ^3_2 of minimal  possible length. The Weyl orbit diagram of (3,3,2) is


 

     [r, "r_2"]        [r,"r_1"]        [rd,"r_2"]          

[ru,"r_1"][rd,"r_2"]                    

     [r, "r_1"]      [r, "r_2"]       [ru, "r_1"]         


 
There are four cells to analyze which have lengths 3, 4, 5 and 6.  These correspond to ε _1 = v_0 v_1 v_2ρ^3_2, ε  _  2 = v_1ε  _1, ε _3  =  v_1 v_2ε  _1 and  ε _4 = v_1 v_2 v_1ε _ 1  .      
The matrices in the corresponding cells are as follows: 
  𝒴_ε_0/ U_2'       =  0.9*([     1                  ;           ϖ            ;   z ϖ a ϖ^2   ϖ^3      ;   a ϖ               ϖ^2 ]),    
 𝒴_ε_2  / U_2'      = 0.9*([          ϖ^2     a_1 + aϖ         y_1ϖ             ;                         1                          ;                                      ϖ             ;                       z ϖ (a_1 + a ϖ)ϖ          ϖ^3 ]),  

𝒴_ε_1 / U_2'      = 0.9
*([       ϖ     a_1                ;               1                ;             a ϖ     ϖ^2        ;   a ϖ^2     z ϖ -a_1ϖ^2     ϖ^3 ]),  
   
    𝒴_ε_3 / U_2 '        =  0.9
*([           ϖ^3  (a + a_2ϖ) ϖ    y_1 +  z ϖ        a_1ϖ^2;                           ϖ           a_1              ;                                         1              ;                             -(a_2+  a ϖ )           ϖ^2 ])   
 
where a, a_1 , a_2 , y_1∈ [] and z ∈  [_2 ].    From these matrices and using elementary row/column operations[A slightly non-obvious operation is ϖ^(2,0,2)η_2→ϖ^(2,2,0)η_2 obtained from  Lemma <ref>.]  arising from U_2, U_2',  one  can deduce that the orbits of U on  
 

  * 𝒴_ε_0/ U_2 ' are given by  ϖ^(3,3,2), ϖ^(2,2,1)η_1, 

  * 𝒴_ε_1/ U_2' are given by ϖ^(3,2,3), ϖ^(2,2,0)η_1,  ϖ^(2,1,2)η_1,   

  * 𝒴_ε_2/ U_2' are given by ϖ^(3,2,3), ϖ^(2,1,2)η_1, ϖ^(1,1,0)η_2,  

  * 𝒴_ε_3/ U_2 ' are given by   ϖ^(3,3,2), ϖ^(2,2,0)η_1, ϖ^(2,2,1)η_1, ϖ^(1,1,0)η_2.  
 


(e)  We have   U_2'  ϖ^(4,2,2)  U_2'  =  U_2' v_0v_1v_2v_1v_0ρ^4_2 U_2' and v_0 v_1v_2 v_1v_0ρ^4_2 is of minimal possible length. The Weyl orbit diagram for (4,2,2) is   

 

 [r, "r_1"]     [r, "r_2"]      [r, "r_1"]        



So we have three cells to check, corresponding to π_1 = v_0 v_1 v_2 v_1 v_0ρ^4_2, σ_2 = v_1σ_1 and σ_3 = v_1 v_2 v_1σ_1. The matrices in the corresponding cells are as follows:  


    𝒴_ε_0 /U_2 ' =  0.9*([      1                     ;    a ϖ   ϖ ^2              ;    z ϖ a_1ϖ^3    ϖ^4 - aϖ^3;   a_1ϖ                  ϖ^2 ]),  
      𝒴_ε_1/ U_2'   = 0.9*([               ϖ^2          a_2 + aϖ                                    ;                                   1                                    ;                                a_1ϖ               ϖ^2                  ;            a_1ϖ^3               z ϖ - ( a_2 + a ϖ)ϖ^2               ϖ^4 ])


    𝒴_ε_2/U_2'  =     
       0.9*([             ϖ^4 (a_1 + a_3ϖ)ϖ^2       y_1 + z ϖ  (a_2 + aϖ) ϖ^2;                             ϖ^2       a_2 + a ϖ                ;                                               1                ;                                    (a_3 + a_1ϖ)             ϖ^2 ])
 
where a, a_1, a_2, a_3, y_1∈ [] and z ∈  [_3 ]. From these, one deduces that  the orbits of U on  
 

  * 𝒴_ε_0/U_2' are given by ϖ^(4,4,2), ϖ^(3,3,1)η _1, 

  * 𝒴_ε_1/U_2' are given by   ϖ^(4,2,4), ϖ^(3,1,3)  η_1, ϖ^(2,2,0)η_2

  * 𝒴_ε_2/U_2  ' are given by  ϖ^(4,4,2), ϖ^(3,3,1)η_1, ϖ^(2,2,0)η_2. 

Note that we make use of U_2ϖ^(2,2,0)η_2 U_2' = U_2ϖ^(2,0,2)η_2 U_2' which holds by Lemma <ref>.      
 


 §.§ Orbits on U'h H_τ_1'/H_τ_1'
             The proof of  Proposition <ref> is based on Lemma  <ref>. To compute  the decompositions of the projections of U'ϖ^λ H_τ_1' to H_2', it will be convenient to work the with the conjugate H_τ_∘' of H_τ_1' introduced in Notation <ref>. This is done since the projection U_2^∘  :   =  _2'(H_τ_∘') is a (standard) maximal  parahoric subgroup of GSp_4(F). It is possible to perform these computations with _2'(H_τ_1') instead, but this requires us to introduce a different Iwahori subgroup of GSp_4. 


Recall that W_2' denotes the Weyl group of H_2' and  W_I_2' the Iwahori Weyl group. Let W^∘_2 denote Coxeter  subgroup of W_I_2' generated by T_2 := { t(f_2)r_0 ,  r_2 }. Each coset W_2' w W_2^∘∈ W_2' \ W_I_2' / W_2^∘ contains a unique element of minimal possible length which we refer to as (W_2', W_2^∘)-reduced element. We let [W_2' \ W  _ I_2' / W_2^∘] denote the subset of W_I_2' of all (W_2', W_2^∘)-reduced elements. If w ∈ W_I_2' is such a  reduced element, the intersection 
    W_2, w' := W_2' ∩ w W_2^∘ w^-1
 is a Coxeter subgroup of W_2' generated by T_2,w  :=  w T_2w^-1∩  W_2'. Then each coset in W_2'/ W_2,w'  contains a unique element of minimal possible length. The set of all  representatives elements for W_2'/W_2,w' of minimal length  denoted by [ W_2' / W_2,w'  ].   
Then the   decomposition  recipe of <cit.> says the following. 
 For any w ∈ [ W_2' \ W_I_2' / W_2^∘ ], 
    U_2' w  U^∘_2  = _τ_∈ []^l(τ w)𝒴_τ w()  U_2^∘
 
where τ runs over [W_2'/W_2,w']. 
 
 Note that l(τ w) = l(τ) + l(w) for τ∈ [W_2' / W_2,w'] and w ∈   [W_2'\ W_I_2'/ W_2^∘].    
 
              For each λ∈Λ_2^+, the element w = w_λ∈ W_I_2' specified is the unique element in W_I_2' of minimal possible length such that U_2' ϖ^λ  U_2^∘  = U_2' w  U_2  ^  ∘ 
 

  * λ = (1,1,1), w =  ρ_2

  * λ = (2,2,2), w  =   v_0 v_1ρ^2_2 

  * λ = (3,3,2), w  =  v_0  v_1ρ^3_2  

  * λ = (3,2,3), w = v_0 v_1 v_2 v_1ρ^3_2

  * λ = (4,4,2), w = v_0 v_1 v_2 v_1ρ^4_2 
  
 
 It is easy to verify the equality of cosets for each λ and w. To check that the length is indeed minimal, one can proceed as follows.     Under the isomorphism, U_2' \ H_2'/U_2^∘≃ W_2'\  W_I_2'/ W_2^∘, the coset U_2' ϖ^λ  U_2 ^  ∘ corresponds to W_2' t(-λ) U_2 ^∘. The minimal possible length of elements in W_2' t(-λ ) U_2 ^  ∘ is the same as that for U_2 ^∘ t(λ) W_2' (taking inverse establishes a bijection). One can then use analogue of (<ref>)   for GSp_4 to find the 
minimal possible length in each of γ t(λ) W_2' for every  γ∈ W^∘_2   = { 1, r_2 , t(f_2) r_0, t(f_2)r_0r_2}. For instance, 
    W^∘_2  t(3,2,3) = { t(3,2,3), t(3,2,0) }
  and the minimal lengths of elements in  t(3,2,3)W_2' is 4 while that of  t(3,2,0)W_2'  is 5.       
 
    1, v_1, η_1 and η_2 
represent distinct classes in H_2\ H_2' /  U_2^∘.    
    
 We need to show that for distinct  γ  , γ'  ∈{ 1, v_1, η_1, η_2}, γ ^-1 h γ '  ∉ H for any h ∈ H.  Writing h as in Notation <ref>,   we  have 

    hv_1   =   0.9[       a       b; a_1     b_1    ;       c       d; c_1     d_1     ] ,     
     h η_i    =  0.9[   a ϖ^i               b       a;          a_1ϖ^i     a_1     b_1;   c ϖ^i               d       c;         c _1ϖ^i     c_1     d_1 ],     
    v_1  h η_i  = 
    0.9[         a_1ϖ ^i     a_1     b_1;  a ϖ ^i               b       a;         c_1ϖ ^i     c_1     d_1;     c ϖ               d       c ]
 
where i = 1, 2 and  

    η^-1_1 h η_2 =   
    0.9[        a ϖ     - c_1ϖ    b-c_1/ϖ a - d _1/ϖ;       -c ϖ       a_1ϖ a_1 -d_1/ϖ    b_1-c/ϖ;      c ϖ^2                     d          c;               c _1ϖ^2        c_1        d_1 ].
  
If any of hv_1∈  U_2^∘,  then a_1, c_1∈ϖ_F. Since all entries of hv _1 are integral, this would mean (h v_1 ) ∈ϖ_F, a contradiction. If hη_i∈ U_2^∘, then all entries of h  excluding b are integral and b ∈ϖ^-1_F. Since the  first two columns of h η_i are integral multiples of ϖ, this would still make ( h η_i ) ∈ϖ_F, a contradiction.  Similarly for v_1 h η_i.  Finally, η_1^-1 h η_2∈  U_2^∘ implies  that c, d, c_1, d_1∈_F and the top right  2 × 2 block implies a , a_1∈_F. So again, the first two columns are integral multiples of ϖ making ( η_1 ^-1 h η_2 ) ∈ϖ_F, a contradiction.        
 
 For this subsection only, we  let  ℛ_V(h), denote the double coset space U_2\   U_2' h V / V   where   h ∈ H_2' and V ⊂ H_2' a compact open subgroup. 
 
     We have 
 

  * ℛ_U_2^∘(ϖ^(2,1,1)) = {ϖ^(2,1,1),  ϖ^(2,1,1)v_1 ,  ϖ^(1,1,0)η_1} 

  * ℛ_U_2^∘(ϖ^(2,2,2)  )   = {ϖ^(2,2,2),   ϖ^(2,2,2)v_1,  ϖ^(1,0,0)η_1,  ϖ^(1,0,1)η_1,  ϖ^(1,1,1)η_1, η_2}

  * ℛ_U_2^∘(ϖ^(3,2,3)) = {ϖ^(3,2,3),  ϖ^(3,3,2)v_1,  ϖ^(2,0,1)η_1,  ϖ^(2,1,2)η_1,  ϖ^(1,0,1)η_2}  
  
and ℛ_U_2^∘(ϖ^(2,2,1)) =  ℛ_U_2^∘(ϖ^(2,1,1)). 
   
 

 That the representatives are distinct follows by Lemma <ref> and by checking that H_2∩η_1 U_2^∘η_1^-1 is contained in an Iwahori subgroup of H_2 (see e.g., the argument in Lemma <ref>).    As usual, we show  that all the orbits are represented by  studying the U_2-orbits on  Schubert cells. Note that 
    W_2' = { 1, r_1, r_2, r_2r_1, r_1 r_2, r_1r_2r_1, r_2r_1r_2,  r_2r_1r_2r_1}

and (r_2r_1)^2 = (r_1 r_2)^2.  
 

(a) w = ρ_2^2.  We have W_2,w ' = W_2' ∩ W_2 ^∘  = ⟨ r_2⟩, so W_2'/W_2,w =  
{ W_2,w' , r_1 W_2,w', r_2r_1 W_2,w' , r_1r_2 r_1 W_2,w' } .
 So [W_2'/W_2,w]  = { 1 , r_1, r_2 r_1 , r_1  r_2  r_1}. Thus to study  ℛ_U_2^∘(w),  it suffices to study the U_2-orbits on cells corresponding to ε_0 =  ρ_2^2,  ε_1 =  r_1ρ_2^2 and ε_2 = r_1 r_2 r_1ρ_2^2. Now 𝒴_ε_0/U_2^∘ = ϖ^(2,1,1)U_2^∘ and 

    1/U_2^∘ =   *0.9[  a ϖ    ϖ;    ϖ;                   ϖ;              ϖ -a ϖ ]      2/U_2^∘   =  0.9*[     y  a_1ϖ     ϖ - a ϖ;   a ϖ     ϖ;     ϖ; -a_1ϖ                -ϖ ]

where a, a_1, y ∈ []. For 1/U_2^∘, the case a = 0 clearly leads to ϖ^(2,1,1)v_1. If a ≠ 0, then we can multiply by diag(a^-1,1,1,a^-1  ) on the left and diag(1,a,a,1) on the right to assume a = 1. We then hit with v_β_0∈ U_2 on left and v_0∈ U_2^∘ on right to arrive at diag(-1,1,1,-1) (which we can ignore) times 

    [               ϖ;           1;       ϖ   1; ϖ^2           ϖ ].

Now a simple column operation and a left multiplication by a diagonal matrix in the compact torus transforms this into ϖ^(1,1,0)η_1. As for 2/U_2^∘, begin by eliminating y with a row operation. Then note that conjugation by v_2 swaps a with a_1 and reverses all signs. So  after applying operations involving second and  fourth row and columns, we may assume that wlog that a_1 = 0. Right multiplication by v_2 yields the matrix 
    [ ϖ^2         a ϖ;       ϖ   a;           1;               ϖ ] .
 
which results in either ϖ^(2,2,1) (which represents the same class as  ϖ^(2,1,1)) or ϖ^(1,1,0)η_0. So all in all, we have three representatives: ϖ^(2,1,1), ϖ^(2,1,1)v_1, ϖ^(1,1,0)η_1. 


(b) w = v_0 v_1ρ^2_2. Here w W^∘_2w^-1 = ⟨ t(f_3)r_2, t(f_2)r_0⟩, 
so W_2,w' is trivial. So we  need  to analyze cells corresponding to ε_0 = v_0v_1ρ_2^2, ε_1 = v_1ε_0, ε_2 = v_1 v_2ε_0 and ε_3 =  v_1 v_2v_1ε_0.  
The corresponding cells are   
  𝒴_ε_0/ U_2 ^  ∘         =  
0.9*([                           1;      ϖ                     ; a  ϖ^2    ϖ^2           ϖ x;                  - ϖ    a ϖ ]),    
 𝒴_ε_2  / U_2^∘       =
 0.9*([                    y ϖ                                           - ϖ              a_1 + a ϖ;                                                                                           1;                     -ϖ                                                                     ; (a_1 +   a   ϖ  )    ϖ                    ϖ^2                                          x  ϖ ]),


𝒴_ε_1 / U_2 ^∘      = 0.9
*([     ϖ               a_1;                       1;          -ϖ         a ϖ; a ϖ^2   ϖ^2  a_1ϖ   x ϖ ]),  
   
    𝒴_ε_3 / U_2^∘        =  0.85
*([  (a_2 + aϖ) ϖ           ϖ^2          a_1ϖ       y + ϖ x;             ϖ                                       a_1;                                                       1;                                         ϖ -(a_2+  a ϖ ) ])   
 
where a, a_1, a_2, x, y ∈ [].  Using similar arguments on these, one deduces that the orbits of  U_2 on 
 

  * 0/U_2^∘ are represented by ϖ^(2,2,2)v_1, ϖ^(1,0,0)η_0,

  * 1/U_2^∘ are represented by ϖ^(2,2,2), ϖ^(1,0,1)η_0, ϖ^(1,1,1)η_1, 

  * 2/U_2^∘  are  represented by ϖ^(2,2,2), ϖ^(1,1,1)η_1, η_2 

  * 3/U_2^∘ are represented by ϖ^(1,0,0) η_1, ϖ^(1,0,1)η_1, η_2. 
     
  

(c)  w = v_0 v_1 v_2 v_1 ρ_2^3. Here w W^∘_2w^-1 =  ⟨ r_2 , t(3f_2)r_0⟩ which means that  W _2, w' = ⟨  r_2⟩. So as in part (a), we have [W_2'/W_2,w] = { 1, r_1 ,  r_2 r_1, r_1 r_2 r_1}.  Again,  we   have three cells to analyze, which correspond to ε_0 = v_0v_1v_2v_1ρ_2^3, ε_1 =  v_1ε_0 and ε_2 = v_1 v_2 v_1ε_0.  The  corresponding  cells  are      

    𝒴_ε_0 /U_2  ^ ∘  =   0.9*([                                               1;                                   ϖ         a ϖ;      - aϖ^3         ϖ^3      a_1ϖ^2 (x + y ϖ) ϖ;         ϖ^2                                a_1ϖ ]),  
      𝒴_ε_1 /U_2^∘    = 0.9*([                                               ϖ        a_2 + aϖ;                                                               1;             ϖ^2                                            a_1ϖ; -(a_2 + aϖ) ϖ^2             ϖ^3          a_1ϖ^2   ϖ ( x + ϖ y ) ])
   

    𝒴_ε_2/U_2^∘  = 
       0.9*([ -(a_2 + aϖ) ϖ^2             ϖ^3  (a_3 + a_1ϖ) ϖ               z;                                               ϖ       a_2 + a ϖ;                                                               1;           - ϖ^2                                  - (a_3 + a_1ϖ) ])
 
where a,a_1,a_2, a_3 , x,y ∈ []. From these, we deduce that
  

  * 0/U_2^∘  are represented by   ϖ^(3,3,2) v_1, ϖ^(2,0,1)η_1, 

  * 1/U_2^∘  are  represented  by ϖ^(3,2,3),  ϖ^(2,1,2) η_1, ϖ^(1,0,1)η_2, 

  * 2/U_2^∘  are  represented by ϖ^(3,3,2) v_1, ϖ^(2,0,1)η_1, ϖ^(1,0,1)η_2.  
 
We can use Proposition <ref> to obtain representatives for the remaining words computed in  Lemma  <ref> without computing Schubert cells.  
   We have 


  * ℛ_U^∘_2(ϖ^(1,1,1)) = {ϖ^(1,1,1) v_1,   ϖ^(1,1,1) ,  η_1} 

  * ℛ_U^∘_2(ϖ^(3,3,2))  = {ϖ^(3,2,3)v_1 ,  ϖ^(3,3,2),  ϖ^(2,2,1)η_1, ϖ^(2,2,0)η_1, ϖ^(2,1,0)η_1, ϖ^(1,1,0)η_2}  

  * ℛ_U^∘_2 ( ϖ^(4,4,2)  )   = {ϖ^(4,2,4)v_1  ,  ϖ^(4,4,2), 
ϖ^(3,3,1)η_1,  ϖ^(3,2,0)η_1,  ϖ^(2,2,0)η_2}  
 

 Since the class of ρ_2 normalizes W_2^∘ (see diagram  (<ref>)) and ρ_2 normalizes the Iwahori subgroup I_2', it normalizes U_2^∘. Thus  for any integer k, the representatives for ℛ_U_2^∘(hρ^k_2) can be obtained from ℛ_U_2^∘(h)  by multiplying representatives on the right by ρ^k_2.  Now we have the following relations:

    ρ_2 U_2^∘   =  ϖ^(1,0,1) v _1 U_2^∘,        v_1ρ_2 U_2^∘  =  ϖ^(1,1,0) U_2^∘,      η_iρ_2 U_2^∘  = v_α_0v_2ϖ^(1,1,0)η_i   U_2^∘

for i = 1 , 2.    
By Lemma <ref>, parts  (a), (b) and  (c) are obtained by the corresponding parts of  Proposition <ref>. For instance, ϖ^(1,1,0)η_1∈ℛ_U_2^∘(ϖ^(2,1,1)) corresponds to ϖ^-(1,0,1)η_1ρ_2 and   
    U_2ϖ^-(1,0,1)η_1ρ_2 U_2     =  U_2ϖ^-(1,0,1)  v_α_0 v_2ϖ^(1,1,0)η_1  U_2^∘
        = U_2v_α_0 v_2η_1 U_2^∘  = U_2η_1 U_2^∘

which gives the representative η_1 in ℛ_U_2^∘(ϖ^(1,1,1)).          

Let us denote  U_2^† : = _2'(H_τ_1').   Since H_τ_1 is the conjugate of H_τ_∘ by ϖ^(1,1,1,1),  U_2^† is the conjugate of U^∘_2 by ϖ^(1,1,1). Set 

    η_0:  = η_1ϖ^-(1,1,1) = 0.9(  1            1 
        1      1 
            1  
              1  )  .
 
Then η_iϖ^-(1,1,1) = η_i-1 for i = 1 , 2. Moreover v_1  commutes with ϖ^(1,1,1). So  by Proposition <ref> and Corollary <ref>, we obtain the following.     
       We  have    
 

  * ℛ_U_2^†(ϖ^(1,0,0)) = {ϖ^(1,0,0),  ϖ^(1,0,0) v _1,  ϖ^(1,1,0)η_0}, 

  * ℛ_U_2^†(ϖ^(1,1,1)) = {ϖ^(1,1,1),  ϖ^(1,1,1) v _1,  ϖ^(1,0,0)η_0 ,  ϖ^(1,0,1)η_0,  ϖ^(1,1,1)η_0 ,  η_1}

  * ℛ_U_2^†(ϖ^(2,1,2)) = {ϖ^(2,1,2),   ϖ^(2,2,1) v _1,   ϖ^(2,0,1)η_0,  ϖ^(2,1,2)η_0,  ϖ^(1,0,1)η_1}

  * ℛ_U_2^†(ϖ^(0,0,0))  =  { 1,  v _1,   η_0}

  * ℛ_U_2^†(ϖ^(2,2,1)) =  {ϖ^(2,1,2) v _1 ,  ϖ^(2,2,1),  ϖ^(2,2,1)η_0,
ϖ^(2,2,0)η_0, ϖ^(2,1,0)η_0,    ϖ^(1,1,0)η_1}          

  * ℛ_U_2^†(ϖ^(3,3,1)) = {ϖ^(3,1,3)v _1  ,  ϖ^(3,3,1), 
ϖ^(3,3,1)η_0,  ϖ^(3,2,0)η_0,  ϖ^(2,2,0)η_1} 
  
 

 
     Note that Proposition  <ref> implies that for 
ℛ_U^†_2(ϖ^λ) =  ℛ_U_2^† (ϖ^r_0(λ))  = ℛ_U_2^†(ϖ^r_2(λ) - (0,0,1) ) 
Thus Corollary <ref> records the decompositions of the all the projections in  Proposition  <ref>. 
 


Next,   we study the fibers of the projection _λ :  ℛ_1(ϖ^λ)  →ℛ_U_2^†(ϖ^_2(λ)) and use Corollary <ref> to calculate coset representatives given in Proposition <ref>.  Let us denote by Λ^α_0 > 0 the set of all λ∈Λ such that α_0(λ)  > 0. We first  specialize Corollary 
<ref> to the case of H_τ_1'.    
    For any λ  = (a, b, c ,  d    ) ∈Λ^α_0 > 0 and η∈ H_2' such that U_2η U_2^†∈  U_2\ 
 U_2 ' ϖ^_2(λ) U_2^†  / U_2^† with sim(η) = a, the fiber of _λ above U_2η U_2^† is 
    { U  
     ( ϖ^(a,b)χ ,   η  ) H_τ_1'    |  χ∈ S^±_1 and  U ' (ϖ^(a,b)χ, η ) H_τ_1' =  U' ϖ^λ H_τ_1'  }
  
 

 This follows since ( [ 1  ; x 1 ] , 1  ),  (  [ 1 x;   1 ] , 1  ) lie in H_τ_1' if x ∈ϖ_F. 
     
For x ∈_F,  define 
    ϰ^+(x) =     (  0.9(  1     x 
        1   )  , 1.1(   [  ; 1    ;   1; x   1;       1 ] ) )  ,         ϰ^-(x) =     ( 0.9(  1    
     x     1   ) , 1.1(   [  ; 1   x;   1;     1;       1 ] ) )   .
 
Note that ϰ^±(x) ∈ H_τ_1 ' as these elements are in the subgroup 𝒳_τ⊂_τ_1 introduced in Notation <ref>.  We let κ^±_1(x) = _1'( ϰ_± (x)) and κ^±_2(x) = _2' (ϰ^∓(x)) denote their projections.  


   If   λ∈Λ_α_0^>, then   
U ϖ^λχ   H_τ_1' ∈{ U ϖ^λ H_τ_1',      U ϖ^s_0(λ) H_τ_1'} for any χ∈ S_1^±×{ 1 }. 



    Write χ  = (χ_1, 1 ). If χ_1∈ S^+ = {[       1; - 1 ]}, the claim is clear. If χ∈ S_1^-, let x ∈_F be such that χ_1 =   κ_1^-(x). The case x = 0 is also obvious, so we assume that x ∈_F^×.  Observe  that  
    U ϖ^λχ 
    H_τ_1'      = U ϖ^λχϰ^-(-x) H_τ_1'   
        =  
    U ϖ^λ ( 1 ,  κ_2^+(-x)  )  H_τ_1' . 
    If 2c - a ≥ 0, the conjugate of ( 1 ,  κ^+_2(-x)  ) by ϖ^λ lies in U and so our double coset equals U ϖ^λ H_τ_1'. 
    If  2c - a < 0 however, then the conjugate of  ( κ^+_1(-x^-1) ,  κ^-_2(x^-1)   ) by ϖ^λ lies in U.  
    So U ϖ^λγ H_τ_1' 
          = U ϖ^λ  (κ_1^+(-x^-1),  κ_2^-(x^-1)κ_2^+(-x)   ) H_τ_1'

Now note that 
    (κ_1^+(-x^-1),  κ_2^-(x^-1)κ_2^+(-x)   ) ·ϰ ^+(x^-1) 
    =  (  1 ,  
    [          -x;       1; 1/x;               1 ] ) .
 
From this, it follows that U ϖ^λχ    H_τ_1 ' = U ϖ^r_0(λ)H_τ_1' which equals U ϖ^s_0(λ) H_τ_1  '.       
 
           For λ∈Λ^α_0  >, 
U ϖ^λσ_1χ  H_τ_1' ∈{ U ϖ^λσ_1 H_τ_1',    U ϖ^s_0(λ)σ_1   H_τ_1'   } 
for any χ∈ S^±   _    1 ×{ 1 }. 
 
 
Since  v   _1 normalizes U_2 and σ_1 = (1,  v_1  ) commutes with χ 
we see that U ϖ^λσ_1χ H_τ_1' =  σ _1  U ϖ^r_1(λ)χ 
 H_τ_1  '. The claim  now 
 follows from part by  noting that r_1 commutes with s_0.
 
Next we record results on double cosets involving σ_2. 

   H ∩σ_2 H_τ_1' σ_2^-1  =  H ∩ς_2 K ς_2 ^-1 is contained in the Iwahori subgroup  J_ς_2 of  triples (h_1, h_2, h_3) ∈ U where h_2 reduces to an upper triangular matrix modulo ϖ and h_1, h_3 reduce to lower triangular matrices.   
 
 This follows by a stronger result established in  Lemma  <ref>.  
 

    Suppose λ = (a, b, c, d ) ∈Λ_α_0^> and let χ = (χ_1, 1) ∈ S^±_1×{ 1 }.  
   

  * If χ _1 =   [ 1; x 1 ]∈ S^-_1,  then  
    U '  ϖ^λσ_2χ  H_τ_1'  =    
    U  'ϖ^λ H_τ_1'     if       x ∈_F^×,    c + d ≥ a ,    2c  ≥ a    or if      x = 0,     c+ d ≥ a  ,      
    
    U' ϖ^r_0(λ) H_τ_1'     if     x ∈_F^× ,    c + d ≥ a > 2c  , 
    
    U   '    ϖ^r_1 r_0(λ)  H_τ_1'      if     x ∈_F^×  ,           2d > a > c+ d  , 
    
    U  '  ϖ^r_0r_1 r_0(λ) H_τ_1'      if     x ∈_F^×,     a > c + d,      a ≥ 2d   or if    x  = 0,         a > c + d .
 



  * If χ_1 =    [         1; -  1      ]∈ S^+_1, then   
    U '  ϖ^λσ_2χ  H_τ_1'  =     U  'ϖ^r_0(λ) H_τ_1'     if    c + d ≥ a    
    
    
    
    U  '  ϖ^r_1 r_0(λ) H_τ_1'      if  
    
    a > c + d
      
 
 
 For x ∈_F,  define  
    ν_2^+  = [   1          -1;       1 - 1;           1;               1 ] ,        ν_2^-  = [  1   ;     1;    -1  1; -1     1 ]
  and set ν^±  = (1, ν_2^±). Note that  ν ^   -  ∈ H_τ_1'. Now if c + d ≥ a, we have ϖ^λν^+ϖ^-λ∈ U'. Thus 
    U' ϖ^λσ_2χ H_τ_1'  = U ' ϖ^λν^+σ_2χ H_τ_1' = U ' ϖ^λχ 
     H_τ_1'
  
since ν^+σ_1 = ( 1, ν_2^+η_0 ) = (1, 1). If on the other hand a > c + d, then ϖ^λν^-ϖ^-λ∈ U', and so 

    U ' ϖ ^ λσ_2χ H_τ_1'         = U ϖ^λν^-σ_2χν^-   H_τ_1'    
        = U' ϖ^λ ( χ_1 ,  ν_2^-[           1;     1;    -1  1; -1        1 ] )  H_τ_1
        =   U' ϖ^λ ( χ_1 ,    [           1;     1;    -1   ; -1          ] )  H_τ_1
        = U ' ϖ^r_0 r_1 r_0 (λ)χ  H_τ_1'  .
 
The rest of the proof now proceeds along exactly the same lines as Lemma  <ref> which determines the classes of U ϖ^μχ H_τ_1' for any χ∈ S^±  _  1 ×{ 1 }, μ∈Λ and  hence those of U' ϖ^μχ H_τ_1'.  
 

   Suppose  λ∈Λ^α_0 >0 satisfies  either β_0(λ) ≥ 0 or β_2(λ) ≤ 0. Then  Uϖ^λσ_2χ H_τ_1 =  U ϖ^λσ_2  H_τ_1'  for any χ∈ S^-_1×{ 1 }.  
 
 Write χ =  (χ_1, 1) and χ_1 =  [ 1  ; x 1 ]. If  2c ≥ a.  replace ϖ^λσ_2χ with ϖ^λσ_2ϰ^-(-x)  = ϖ^λσ_2κ_2^+(-x). If 2d ≥ a, replace ϖ^λσ_2χ with ϖ^λσ_2χν^-ϰ^-(-x)  =   ϖ^λσ_2ν^-κ_2^+(-x) where ν^-∈ H_τ_1' is as in  proof     Lemma <ref>.  Explicitly if λ = (a,b,c,d), then  
    ϖ^λσ_2κ_2^+(-x) =   (  ϖ^(a,b)  ,  ( [      ϖ^c          - x  ϖ^c      ϖ^c;               ϖ^d      ϖ^d;                      ϖ^a-c;                               ϖ^a-d ] )   ) ,    ϖ^λσ_2ν^-κ_2^+(-x)   =   ( ϖ^(a,b) ,    ( [                                     ϖ^c;                           ϖ^d;               ϖ^a-c     ϖ^a-c;     ϖ^a-d           - x ϖ^a-d     ϖ^a-d ])   )
  
Now an obvious row operation transforms these into ϖ^λσ_2.   

 





          Suppose λ = (a,b,c,d) ∈Λ^α_0 > 0 satisfies  2c  + 1 ≥ a and c + d ≥ a. Then   



    U ϖ^λσ_3χ  H_τ_1'  =    U ϖ^λσ_3 H_τ_1'     if χ∈ S_1^-×{ 1 }
    
     U ϖ^s_0(λ) - f_1σ_3      H_τ_1'       if χ∈ S_1^+×{ 1 }
 
Moreover U ' ϖ^λσ_3  H_τ_1' =  U' ϖ^λ+  λ_∘ H_τ_1'  and U' ϖ^s_0(λ)-f_1σ_3 H_τ_1' = U' ϖ^s_0(λ) - f_1 + λ_∘ H_τ_1' = U ' ϖ^s_0(λ+ λ_∘) H_τ_1'.    




 
   
  This is  entirely similar Lemma <ref> and <ref>.  
    
     The proof in each case 
goes by 
applying either Lemma <ref> or  Corollary <ref> to  the coset representatives computed in  Corollary <ref>. In the latter case, we   will need to  determine the fibers  of the projection   
    _μ :   ℛ_1( ϖ^μ ) →ℛ_U^†_2(ϖ^_2(μ))
 for a given μ = (u_0, u_1, u_2,  u_3) ∈Λ above each ϖ^(a,c,d)γ∈ℛ_U_2^†(ϖ^_2(μ)) where γ∈{ 1, v_1 , η_0,  η_1}. Let λ = (a, b, c, d ) where b  = u_1  if γ∈{ 1, v_1  , η_0}  and u_1     - 1 if γ =  η_1. Let  i ∈{0,1,2,3} be the unique integer 
such that _2(σ_i) = γ. Then the fiber consists of cosets of the form U ϖ^λσ_iχ  H_τ_1' where χ∈ S^±_1.   

Let us first address the case where γ∈{ 1,  v_1}.  
Note that in each of the projections computed in Corollary <ref>,  there is a unique element of the form ϖ^(a,c,d)γ. So the projection of ℛ_1(μ) in each case has a unique element of this form (see Remark <ref>). Lemma <ref> and Corollary <ref>  tell us that the fiber  of  _ 2, μ  above each such element is   contained in    {ϖ^λγ , ϖ^s_0(λ)γ}. If λ≠ s_0(λ), then Corollary <ref> implies 
that only one of ϖ^λγ or ϖ^s_0(λ)γ can belong to the fiber. 
Thus the fiber is necessarily a  singleton. 
Now U ϖ^μ H_τ_1', U  ϖ^r_1(μ)σ_1  H_τ_1 ' are clearly subsets of U' ϖ^μ  H_τ_1' and their projections  ϖ^(u_0, u_2, u_3), ϖ^(u_0,u_3,u_2) v_1 respectively  have the desired form[that is,  Uϖ^(u_0,u_2,u_3) H_τ_1' = U ϖ^(a,c,d) H_τ_1' and U ϖ^(u_0,u_3,u_2)v_1H_τ_1'  = U ϖ^(a,c,d)v_1H_τ_1']. So  we  are free to choose ϖ^μ as the representative element in the fiber if γ = 1 and ϖ^s_2(μ)σ_1  if γ = v_1. 



The case where γ = η_0 requires a closer  case-by-case analysis.   Here we need study the possible values χ∈ S_1^±  such that U' ϖ^λσ_iχ H_τ_1 ' =  U ' ϖ^μ H_τ_1'.  
We let C(λ) = {λ,  r_0(λ) , r_1 r_0 (λ), r_0 r_1  r_0( λ)  }. In each case, we compute the intersection C(λ) ∩ W_τ_1'μ,  using which we read off the possible values of χ from Lemma <ref>, i.e., we only consider χ for which U 'ϖ^λσ_2χ  H_τ_1' = U' ϖ^μ' H_τ_1' for μ' ∈ C(λ) ∩ W_τ_1' which is a necessary condition by Proposition  <ref>.  We then use Lemma <ref> to simplify these cosets if possible. In most cases, this results in a single element in the fiber. For γ = η_1, the analysis is similar but much easier and we will  only need Lemma <ref> to decide the elements of the fiber.  

    


∙ μ =  (1,1,1,0). 
    The projection is ℛ_U_2^†(ϖ^(1,1,0) )  =  ℛ_U_2^† ( ϖ^(1,0,0) ) = {ϖ^(1,0,0)  , ϖ^(1,0,0)   v_1  ,    ϖ^(1,1,0) η_0}. To determine the lift of  ϖ^(1,1,0)η_0,  let λ  : = (1,1,1,0). Then C(λ) = { (1,1,1,0), (1,1,0,0) } and ϖ^(1,1,0,0)∉ U' ϖ^μ H_τ_1'  by Lemma    <ref>.    Lemma  <ref> tells us for χ∈ S^±_1, U' ϖ^λσ_2χ H_τ_1 = U' ϖ^μ H_τ_1' only when 
χ∈ S^-.    But then U ϖ^λσ_2χ  H_τ_1 ' = U ϖ^λσ_2 H_τ_1'  by  Lemma   <ref>.   Thus ϖ^(1,1,1,0)σ_2 is the unique element of the fiber  above  ϖ^(1,1,0)η_1.       


∙  μ  =  (1,1,0,1 ). 
  
We have  ℛ_U^†_2(ϖ^(1,0,1)) =  ℛ_U^†_2(ϖ^(1,1,1))      = {ϖ^(1,1,1) ,  ϖ^(1,1,1)   v_1 ,  ϖ^(1,0,0)η_0 ,  ϖ^(1,0,1)η_0 ,  ϖ^(1,1,1)  η_0 ,  η_1} . 
Let λ_1 = (1,1,0,0), λ_2 = (1,1,0,1), λ_3 = (1,1,1,1). Then  
C(λ_i)  ∩ W_τ_1μ
 = 
{ (1,1,0,1)  }.    
For λ_1 and λ_3, the only choice is χ = [     1; -1 ] and so the unique elements in the fibers above ϖ^(1,0,0)η_0 and  ϖ^(1,1,1)η_0  are respectively ϖ^s_0(λ_1)σ_2   = ϖ^(1,0,0,0)σ_2 and  ϖ^s_0(λ_3)σ_2  =  ϖ^(1,0,1,1)σ_2.  For λ_2,  
the only  choice is χ = [ 1;   1 ]∈ S^-, ϖ^(λ_2)σ_2 is the unique element in the fiber above  ϖ^(1,0,1)η_0.


For η_1, the unique element above is ϖ^-f_1σ_3 by  Lemma  <ref>,  since λ_∘ = (1,1,1,1) does not belong to W_τ_1' μ but (1,0,1,1) =  s_0( λ_∘) does.  
   



∙   μ =  (1,1,0,0).  
   This is similar to the first case  except now we work with ϖ^(1,1,0,0)∈ C(λ) where λ = (1,1,1,0). In this case, the only possible choice for χ  =    [     1; -1    ].  The fiber is therefore U ϖ^λσ_2χ H_τ_1' = U ϖ^s_0(λ) σ_2 H_τ_1 and we take the representative ϖ^(1,0,1,0)σ_2.  



∙  μ  =  ( 2,2,1,1) . 

The projection is ℛ_U_2^†(ϖ^(2,1,1)) = {ϖ^(2,1,1), ϖ^(2,1,1) v_1,  ϖ^(2,1,1)η_0}. Lemma <ref> implies that U '  ϖ^(2,2,1,1)σ_2χ  H_τ_1' coincides with U ' ϖ^(2,2,1,1) Hτ_1' for any χ∈ S^±_1. Now if χ∈ S^-, then U ϖ^(2,2,1,1)σ_2χ H_τ_1 ' = U ϖ^(2,2,1,1)σ_2 H_τ_1' by Lemma <ref>.    If however χ = [     1; -1 ], then U ' ϖ^(2,2,1,1)σ_2 H_τ_1' = U' ϖ^(2,0,1,1) H_τ_1'. Thus the fiber  above ϖ^(2,1,1)η_0 consists of 
    U ϖ^(2,2,1,1)σ_2 H_τ_1'     and    U ϖ^(2,0,1,1)σ_2 H_τ_1'.
  These are distinct elements of the fiber, since U ϖ^(2,2,1,1) H_ς_2⊂ U ϖ^(2,2,1,1)  J_ς_2, Uϖ^(2,0,1,1) H_ς_2⊂ U ϖ^(2,0,1,1) J_ς_2 by  Lemma <ref> and U \ H / J_ς_2≃Λ.  



∙ μ = ( 2,1,2,1), (2,1,1,2),  (2,1,1,1). 
  These are handled by Lemma <ref>. 



∙ μ = (2,2,0,1)
  
The projection is ℛ_U_2^†(ϖ^(2,2,1)) = {ϖ^(2,1,2)v_1 ,  ϖ^(2,2,1),  ϖ^(2,2,1)η_0,
ϖ^(2,2,0)η_0, ϖ^(2,1,0)η_0,    ϖ^(1,1,0)η_1}. Let λ_1 = (2,2,2,1), λ_2 = (2,2,2,0), λ_3 = (2,2,1,0). Then for  χ∈ S^±_1 and any i = 1, 2,  3, the double coset   U ' ϖ^λ_iσ_2χ H_τ_1' coincides with 
U ' ϖ^(2,2,0,1) H_τ_1' only when χ  =  [       1; - 1 ]. This gives the three desired representatives. 

As for ϖ^(1,1,0)η_1, the unique element in the fiber  is ϖ^s_0(1,1,1,0) -  f_1σ_3  =   ϖ^(1,-1,1,0)σ_3  by Lemma <ref>  since (1,1,1,0) + λ_∘ =  (2,2,2,1)   ∉ W_τ_1' μ  but  (2,0,2,1) = s_0(1,1,1,0) + s_0 ( λ_∘   ) = ( 2,0,2,1) ∈ W_τ_1'.  
  


∙   
μ = (3,2,2,2)   
 
We have ℛ(ϖ^(3,2,2)) = ϖ^(2,1,1)ℛ_U_2^†(ϖ^(1,1,1)), so

    ℛ_U_2^† (ϖ^(3,2,2)) =  {ϖ^(3,2,2),  ϖ^(3,2,2)v_1,   ϖ^(3,1,1)η_0,  ϖ^(3,1,2)η_0,  ϖ^(3,2,2)η_0,  ϖ^(2,1,1)η_1}.
 
Let λ_1 = (3,2,1,1), λ_2 = (3,2,1,2), λ_3 =(3,2,2,2). Then C(λ_i) ∩ W_τ_1' μ =  {  (3,2,2,2) } for all i. For  λ_1 and λ_3, Lemma <ref> forces χ to be in  S^-_1, and  Lemma <ref>   allow us to conclude that  U ϖ^λ_1σ_2 H_τ_1', Uϖ^λ_2σ_2  H_τ_1' are the only elements of the fibers above ϖ^(3,1,1)η_0, ϖ^(3,2,2)η_0 respectively.  For λ_2, the possible choices are χ = [     1; -1 ] or χ = [ 1  ; x 1 ] for x ∈_F  ^×. In the latter case, we have U ϖ^λ_2σ_2χ  H_τ_1' = U ϖ^λ_2σ_2ψ H_τ_1'
since the conjugate of ϖ^λσ_2χ by diag(x, 1, x , 1 , x, 1) ∈ U ∩  H_τ_1' equals ϖ^λ_2σ_2ψ. So the fiber above ϖ^(3,1,2)η_0 contains 
    U ϖ^s_0(λ_2)σ_2  H_τ_1'   and     U  ϖ^λ_2σ_2ψ H_τ_1' .

Since  U ϖ^λ_2ψ J_ς_2  = U ϖ^λ_2 J_ς_2, so the same argument used in the case μ = (2,2,1,1) shows that the two displayed elements are distinct. 



For ϖ^(2,1,1)η_1, the only element in the fiber is ϖ^(2,1,1,1)σ_3 by Lemma <ref>, since (3,2,2,2) = (2,1,1,1) + λ_∘ belongs to W_τ_1' but (3,1,2,2) =  s_0 (2,1,1,1) + s_0 ( λ_∘  ) does not.     
      

∙ μ =  (3,3,1,1) 
 
The projection is ϖ^(2,1,1)·ℛ_U_2^†(ϖ^(1,0,0)) = {ϖ^(3,1,1), ϖ^(3,1,1) v_1, ϖ^(3,2,1)η_0}. For  λ = (3,3,2,1), the only possibility is χ = [     1; -1 ] which gives the representative ϖ^(3,0,2,1)σ_2 in the fiber above ϖ^(3,2,1)η_0.   
          

∙  μ   =  ( 3 , 2 , 0 ,   1 ). 
 
The projection is ℛ(ϖ^(3,3,1)) = {ϖ^(3,1,3)v_1, ϖ^(3,3,1), ϖ^(3,3,1)η_0,  ϖ^(3,2,0)η_0,  ϖ^(2,2,0)η_1}. Set  λ_1 : = (3,2,3,1) and λ_2  : = (3, 2,2,0).  Then C(λ_i)  ∩ W_τ_1'μ = {μ} for i = 1, 2. In both cases, the only possibility is χ  =  [     1; -1    ] which gives the rerpresentatives ϖ^(3,1,3,1)σ_2, ϖ^(3,1,2,0)σ_2 above ϖ^(3,3,1)η_0, ϖ^(3,2,0)η_0 respectively. 

For ϖ^(2,2,0)η_1, the unique element in the fiber is ϖ^(2,0,2,0)σ_3 since (2,1,2,0) + λ_∘∉ W_τ_1' but (3,1,3,1) =  s_0 (2,1,2,0) + s_0(λ_∘) ∈ W_τ_1'μ.       
  


 §.§ Orbits on U' h H_τ_2'/ H_τ_2'
    
Let E   =  _2(U^) denote the projection of the group U^. Thus     E ⊂ U_2  ' is the endohoric[a portmanteu of Iwahori and endoscopic] subgroup of all  elements whose reduction modulo ϖ lies in 𝐇_2 ()  =  _2() × _^×_2().  
For a ,  b ∈_F, let   
    γ(u, v)    =    
        [  1  u     v;     1  v;        1;       -u  1 ]  .


    

  I_2 ' 
E  / E   = _a,b ∈  [] γ (a, b ) E.  
 
  Let 𝐍'_2 (resp., 𝐍_2)  denote the unipotent radical of the Borel subgroup of _2' (resp., _2) determined by {β_0 , β_2}.   Let Z ⊂ E the subgroup of all elements  that  reduce modulo ϖ to the Borel subgroup of H_2. 
 Then  Z  = I_2' ∩ E and so 
    I_2' E /  E   ≃  I_2'/  Z    ≃𝐍_2' ()/ 𝐍_2() .
 
 Now | 𝐍_2'() | = q^4  and | 𝐍_2() | = q^2 and so | I_2 ' / Z |  = q^2 and it is easily seen that the reduction of γ(u,v) for u, v ∈ [] form a complete set of representatives for 𝐍_2'() / 𝐍_2().  
    
Let v_1 be as in <ref>  and η_0, η_1 be as in (<ref>),  (<ref>). Recall (<ref>) that for k ∈ [], we  denote 

    η̃_k =  0.9[     k     1;   k+1     1;                -1 k + 1;                 1   - k ]
 
and  []^∘  = [] ∖{ -1 }.  
     1, v_1, η_0, η_1, and η̃_k for k ∈ []^∘ represent pairwise  distinct classes in H_2\ H_2'/ E.          
  
 
This is 
handled as in Lemma <ref>.  The matrix formulas shown  therein for η_i, i =1 , 2  also apply for i = 0 and it is easy to deduce the pairwise distinction for 1, v_1, η_0, η_1 from these formulas. Let us distinguish the class of η̃_k for k ∈ []^∘ from γ∈{ 1, v_1 ,η_0, η_1}. Write h ∈ H as in Notation <ref>. Then  

    h η̃_k  
    = [     ak      a     -b b(k+1);             *      *      *;             *      *      *;             *      *      * ],    η_i^-1 h η̃_k  
    = [       *       *       *       *;               *       *       *;      ck       c      -d d (k+1);       *       *       * ]
 
for i = 0, 1.  
If h η̃_k∈ E, we see from the entries shown above that the first row is a multiple of ϖ which makes (hη̃_k) ∈ϖ_F, a contradiction. Since v_1 just swaps the rows of hη̃_k, the same argument applies to v_1 h η̃_k. Similarly for η_i^-1 h η̃_k.  Finally for k ,k' ∈  []^∘ and k ≠ k', we see from the matrix    
    (η̃_k)^-1 h η̃_k'     =       [             *      a_1 -  a             *  - b - b_1 k'; a k'  - a_1 k             *   - b - b_1 k             *;                           *             *             *;                           *             *             * ]
 
 that (η̃_k)^-1 h η̃_k' lies in E only if a, b ∈ϖ_F. But since η̃_k , η̃_k'∈ U_2' and E ⊂ U_2', we also have h ∈ U_2'. But then a, b ∈ϖ_F implies that sim(h) = ad - bc ∈ϖ_F, a contradiction. 
  

 Note that U v_2η̃_-1v_2 E = U η_0 E.    
 
          For γ∈{η_0,  η̃_0}, the map Λ_2→ U_2ϖ^Λγ E, λ↦  U_2ϖ^λγ   E is a bijection.   
 
 It is easy to see that  H_2∩γ E γ^-1 are  contained in certain  Iwahori subgroups of H_2.  So the Bruhat-Tits decomposition along with the identification U_2ϖ^Λγ E   U ϖ^Λ (H_2∩γ E γ^-1) implies the result.     
 
      If λ∈Λ_2  is such that  β_1(λ), β_2(λ)  ∈{ 0 ,1 }, then U_2' ϖ^λ E  =  U_2' ϖ^λ  I_2'  E.     
  
 The conditions ensure that ϖ^λ I_2' ϖ^-λ⊂ U_2'.      
 
 For this subsection only, we  let  ℛ_E(h), denote the double coset space U_2\   U_2' h E/ E   for   h ∈ H_2'.   
 
     We have  
    

  * ℛ_E (ϖ^(0,0,0)) =  
{   1  ,  v_1 , η_0 ,    η̃_k  |    k ∈ []^∘}.   

  * ℛ_E(ϖ^(1,1,1)) = 
[t]1.0 
{ϖ^(1,1,1), ϖ^(1,1,1)v_1, ϖ^(1,0,1)η_0, ϖ^(1,1,0)η_0, ϖ^(1,1,1)η_0, η_1}∪
*ϖ^(1, 1, 1)η̃_0, ϖ^(1,0,1)η̃_0, ϖ^(1,0,0)η̃_k k ∈ []^∘ 


  * ℛ_E(ϖ^(2,2,1)) 
=   * ϖ^(2, 2, 1),  ϖ^(2,1,2)   v_1 ,   ϖ^(2,2,1) η_0 ,   ϖ^(2,1,2)η̃_0  ,   ϖ^(1,1,0) η_1.    
      
 
 By Lemma <ref> and <ref>, the elements listed in part (a), (b), (c) represent distinct classes.  We show that these also form a full set of representatives. 
Say for λ∈Λ_2   is such that 0 ≤β_1 ( λ) , β_2(λ)  ≤  1 and say U_2' ϖ^λ I_2'  =   _γ∈ΓγĨ_2  for some finite set Γ.   
Then by Lemma <ref> and <ref>,  


    U _2   '  h  E    =    U_2' h  I_2' E        =       ⋃_γ∈Γ  U_2γ   I_2'  E    =      ⋃_γ∈Γ
      u ,v  ∈ []   U_2γγ_u, v    E
 
Since (0,0,0), (1,1,1), (2,2,1) satisfy the condition of Lemma <ref>, the decomposition (<ref>) applies. Now  we can compute the set Γ for each λ by replacing ϖ^λ with w ∈ W_I_2' of  minimal  possible  length   such that U_2' ϖ^λ E = U_2'w E and invoking the analogue of Proposition <ref>    for GSp_4. 
Since we are only interested in computing the double cosets U_2γγ_u, v  E appearing in U_2'   w  E,  we only need to study the  cells corresponding to 
    ε_0 : =  w ,   ε_1 :  = r_1w,   ε_2 : = r_1 r_2w ,   ε_3 :  =  r_1r_2r_1w  .
 
   Thus we need to study the classes in  U_2\ H_2' / E of  {𝒴_ε_i(κ⃗) γ_u,v E   |   κ⃗∈ []^l(ε_i) ,     u, v ∈ []     } for each i = 0    ,    1, 2 , 3. We will refer to these sets Schubert cells  as  well and as usual,  abuse notation to  denote them by  iE/E. 
    

(a)  Here w = 1 and the four cells are  
  𝒴_ε_0E/ E      =  
0.9*([   1   u       v;       1   v    ;           1    ;         - u     ]),    
 𝒴_ε_2  E /  E      =
 0.9*([        a   y + au   vy - u   av + 1;        1        u                 v;                 1        v         ;               - a - (av+1)          ]),

𝒴_ε_1 E / E      = 0.9
*([      a au + 1      v     av;      1      u             v;                  - u      1;               au + 1     -a ]),  
   
  𝒴_ε_3 E  / E      =  0.9
*([               z      u z +  a_1 a u + a_1 v + 1          vz - a;               a          au + 1               v              av;               1               u                               v;            -a_1          -a_1 u               u       -a_1v - 1 ])   
 
where a, a_1, u, v , y ∈   [] and z  :  = y + aa_1.  Note that the  ε_1-cell is obtained from ε_0-cell by multiplying on the left by y_1(a)v_1. If a = 0, the orbits of U_2 are v_1 times  those of ε_0-cell since v_1 normalizes U_2. Similarly we can assume that a ≠ 0 in ε_2-cell and a_1≠ 0 in ε_3-cell.   

Consider the ε_0-cell. Conjugation by v_2 swaps the entries u, v and row column operations arising from U_2, E   allow us to make at least one of  u, v zero. So say u = 0. Then we obtain either identity or η_0 as representative from this cell.  Next consider the ε_1-cell. As observed above,  the case  a = 0 leads to orbits of v_1 and v_1η_0 and we have U_2 v_1η_0 E = U_2η̃_0 E. If a ≠ 0,  we apply the following sequence of row-column operations:  
    0.9([      a au + 1      v     av;      1      u             v;                  - u      1;               au + 1     -a ])⟶0.9([      a au + 1            av;      1      u   -v/a      v;                  - u      1;               au + 1     -a ])⟶0.9([      a au + 1            av;      1      u     uv       ;                  - u      1;               au + 1     -a ])
 
    ⟶0.9([      a au + 1   a uv     av;      1      u              ;                  - u      1;               au + 1     -a ])⟶0.9([      a au + 1              ;      1      u              ;                  - u      1;               au + 1     -a ])⟶0.9([      1 au + 1              ;      1     au              ;                 - au      1;               au + 1     -1 ]).
 
Let us denote k = au. The structure of Y allows us to restrict k ∈ []. Conjugating this matrix by v_β_0v_β_2 and scaling by -1 gives us the matrix η̃_k if k ∈ []^∘, i.e., au ≠ - 1. If au = -1 however,  then conjugating by v_2 further  gives us η_0. So the ε_1-cells decomposes into U_2-orbits of v_1,  η_0 and η̃_k for k ∈ []^∘. 

For the case of ε_2-cell and a ≠ 0, use   

    0.9([        a   y + au   vy - u   av + 1;        1        u                 v;                 1        v         ;               - a - (av+1)          ])⟶0.9([        a            -auv-u   av + 1;        1        u                 v;                 1        v         ;               - a - (av+1)          ])⟶0.9([        a                     av + 1;        1        u uv + u/a        v;                 1        v         ;               - a - (av+1)          ])
  
    ⟶0.9([        a                     av + 1;        1                          v;                 1        v         ;               - a - (av+1)          ])⟶0.9([                       1           v            ;                       a    (av + 1)            ;          -a                         - (av +  1);           1                                   v ])⟶0.9([      v      1              ; (av+1)      a              ;                  - a av + 1;                    1    - v ])

and multiply on the left by diag(a, 1,  1, a ) and diag(1, a^-1  ,  a^-1  ,1) on the right to arrive at the same situation as the ε_1-cell.  Finally the case for ε_3-cell with a_1≠ 0, use 

    0.9([               z      u z +  a_1 a u + a_1 v + 1          vz - a;               a          au + 1               v              av;               1               u                               v;            -a_1          -a_1 u               u       -a_1v - 1 ])⟶0.9([                             a_1 a u + a_1 v + 1             - a;               a          au + 1               v              av;               1               u                               v;            -a_1          -a_1 u               u       -a_1v - 1 ])
 

    0.9([                             a_1 a u + a_1 v + 1             - a;                               1 (au + a_1v)/a_1          -a/a_1;                               u                               v;            -a_1          -a_1 u               u       -a_1v - 1 ])⟶0.9([                                     a_1     a u + a_1 v + 1                    ;                                       1      (au+ a_1v)/a_1                    ;                   1                   u                     (au + a_1 v ) / a_1;                -a_1              -a_1 u                   u  - (au + a_1v + 1 ) ])    .

Next  substitute k_1  =  au + a_1 v and use 

    0.8([                                   a_1           k_1  + 1                   ;                                     1            k_1/a_1                   ;                  1                  u                    k  _   1     / a_1;               -a_1             -a_1 u                  u         - k_1  - 1 ])⟶0.8([                                             a_1                 k_1 + 1                        ;                                               1               k_1  /a_1                        ;                       1                         - u ( k_1 + 1 )  /  a_1               k_1 / a_1;                    -a_1                  -a_1 u                       u              - k_1  - 1 ])⟶0.8([                 a_1   k_1 + 1          ;                   1 k_1 / a_1          ;         1                     k_1 / a_1;      -a_1    -a_1 u     -uk_1 - k_1 - 1 ])
 
    ⟶0.8([                 a_1   k_1 + 1          ;                   1 k_1 / a_1          ;         1                     k_1 / a_1;      -a_1                     - k_1 - 1 ])⟶0.85([                     a_1    k_1  + 1            ;                       1 k_1   / a_1            ;           1                         k_1   / a_1;        -a_1                         - k _1  - 1 ])⟶0.8([       k_1 + 1           a_1                            ;       k_1/a_1             1                            ;                                         1 - k_1   / a_1;                                     - a_1     k _1  + 1 ])  .

Now multiply by diag(-1,-a_1,a_1,1) on the left, diag(1,-a_1^-1,-a_1^-1,1) on the right and use the substitution k = -k_1 - 1. If a_1 = 0 in the ε_3-cell, then one gets v_1, 1 , v_1η_0, v_1η̃_k and the latter two can be replaced with η̃_0, η̃_k' where k' = -(k+1).   


(b) We have w = ρ _2 and the four cells are  
  𝒴_ε_0E/ E      =  
0.85*([         - u   1;           1    ;       ϖ v ϖ    ;   ϖ u ϖ     v ϖ ]),    
 𝒴_ε_2  E /  E      =
 0.85*([         ϖ       u ϖ    y - au   a + v ϖ;                           - u         1;                             1          ;                 - ϖ - a - v ϖ           ]),


𝒴_ε_1 / U_2 ^∘      = 0.85
*([                     1 - au        a;                        - u        1;        ϖ     u  ϖ               v ϖ;     -a ϖ (1-au) ϖ     v  ϖ -a   v ϖ ]),  
   
  𝒴_ε_3 E  / E      =  0.85
*([              - a  ϖ         ( 1 - au) ϖ a_1  + v  ϖ  -  u z            z - av ϖ;                                                      1 - au                   a;                                                          -u                   1;                 - ϖ             -  u  ϖ               a_1 u          -a_1 - v ϖ ])   
 
where a, a_1 , y , u , v  ∈ [] and z  = y + a a_1 in the ε_3-cell.    Using  analogous arguments on these cells, one deduces that the U_2-orbits on 

    
  * 0E/ E  are  represented by ϖ^(1,1,1)v_1, ϖ^(1,0,1)η̃_0,  ϖ^(1,1,1)η̃_0, 
    
  * 1E/E are  represented by   ϖ^(1,1,1), ϖ^(1,1,1)η_0, ϖ^(1,1,0)η_0 when a equals zero[these are obtained by applying v_1 to the representatives of the ε_1-cell] and ϖ^(1,0,0)η̃_k for k ∈ [] when a is non-zero,  
    
  * 2E/E are  represented by   ϖ^(1,1,1), ϖ^(1,1,1)η_0, ϖ^(1,1,0)η_0 when a equals zero and η_1 when a ≠ 0 
    
  * 3E/E are  represented by   ϖ^(1,1,1)  v_1, ϖ^(1,0,1)η̃_0, ϖ^(1,1,1)η̃_0, η̃_k for k ∈ [] when a_1 equals zero and  η_1 when a_1≠ 0. 
   
   

(c) In this case, w = v_0ρ^2_2 and the four  cells are    
  𝒴_ε_0E/ E      =  
0.85*([                 1      ;           ϖ  v  ϖ      ; ϖ ^ 2 u ϖ^2   x ϖ v ϖ^2;              u  ϖ   - ϖ ]),    
 𝒴_ε_2  E /  E      =
 0.85*([                          y  ϖ  a + (u + vy)ϖ            - ϖ;                                            1               ;                             ϖ            v ϖ               ;          - ϖ^2  - ( a + u ϖ ) - (x + av )  ϖ       - v ϖ ^2 ]),


𝒴_ε_1 / U_2 ^∘      = 0.85
*([                                 ϖ          a + v ϖ                 ;                                                  1                 ;                                                u ϖ              - ϖ;              ϖ^2          u  ϖ ^2 ( x -  a u )   ϖ ( a +  v   ϖ ) ϖ ]),  
   
  𝒴_ε_3 E  / E      =  0.85
*([                  ϖ^2 (  a _1 + u ϖ   )  ϖ                    z        ( a + v ϖ ) ϖ;                                         ϖ           a  +  v  ϖ                     ;                                                              1                     ;                                                   -  a_1 - u ϖ                    ϖ ])   
  
where a, a_1, x , y, u, v ∈ [] and z denotes y + aa_1 + ( x - au + a_1 v ) ϖ  in the ε_3-cell.   From  these,  one   deduces that the orbits of U_2 on 
 

  * 0E/E are represented by ϖ^(2,2,1), ϖ^(2,2,1)η_0,  

  * 1E/E are represented by ϖ^(2,1,2)    v_1, ϖ^(2,1,2)η̃_0 when a  = 0 and   ϖ^(1,1,0) when a ≠ 0,

  * 2E/E are represented by ϖ^(2,1,2)    v_1, ϖ^(2,1,2)η̃_0 when a  = 0 and   ϖ^(1,1,0) when a ≠ 0, 

  * 3E/E are represented by ϖ^(2,2,1), ϖ^(2,2,1)η_0 when both a,  a_1 are 0 and  η_1 when at least  one of a, a_1 is non-zero.    


 The result above implies that (the reductions of) 1, v_1 and η̃_k for k ∈ [] form a complete system 
 of representatives for  𝐇_2() \𝐇_2'() / 𝐇_2().     
 

    ℛ_E(ϖ^(2,1,2)) = {ϖ^(2,2,1) v_1  ,   ϖ^(2,1,2),   ϖ^(2,0,1)η̃_0  ,  ϖ^(2,1,2)η_0 ,  ϖ^(1,0,1)η_1} 
 
 
First note that  U ' ϖ^(2,1,2) = U' v_0 v_1ρ_2^2.  Since v_1 normalizes E and ρ_2^2∈ H' is central, U_2 '  v_0 v_1ρ_2^2 E = U_2 ' v_0ρ_2^2 E v_1.  So the result follows by  Proposition <ref> (c).  
 
Now we address the lifts of these cosets to H'. Let S_1^± be as in    <ref> 
   Suppose λ is in Λ^+.  Then for any χ∈ S_1^±, 
U ϖ^λχ H_τ_2' ∈{ U ϖ^λ H_τ_2' , U ϖ^s_0(λ)H_τ_2'  } and U ϖ^r_1(λ)θ_1χ H_τ_2' ∈{ Uϖ^r_1(λ)θ_1 H_τ_2' , U ϖ^s_0r_1(λ)θ_1 H_τ_2' }. 
 
  This first part is  proved in the same manner as Lemma <ref>. Since θ_1 = σ_1 = w_2 normalizes U,  commutes with χ, w_α_0   and w_2ϖ^λ =   ϖ^r_1(λ) w_2,  the second claim also follows easily.     
 
    Let λ∈Λ^α_0 > 0 and χ =  ( χ_1, 1 ) where χ_1∈ S^±_1.    
  


  * Suppose (β_1 + β_2)(λ) ≥ 0. Then  
    U '  ϖ^λθ_2χ  H_τ_2' =    U' ϖ^λ H_τ_2'     if  χ_1 = 1   or if   χ_1∈ S^-_1∖{ 1 },  β_0(λ) ≥ 0  
     
    U  ' ϖ^s_0(λ)        
    H_τ_2'        if χ_1∈ S^-_1∖{ 1 },  β_0(λ) < 0   or if   χ_1∈ S_1^+
 

  * Suppose  β_1 ( λ) ≤ 
0. Then 
    U ' ϖ^λθ̃_0χ   H_τ_2' =  U '  ϖ^r_1(λ)  H_τ_2  '       if  χ_1 = 1   or if   χ_1∈ S_1^-∖{1 },  β_2(λ) ≥ 0, 
    
      U ' ϖ^s_0r_1(λ)  H_τ_2'     if  χ_1∈ S_1^-∖{ 1 },   β_2(λ) < 0    or  if χ_1∈ S^+_1
 

  * Suppose β_1( λ) = 0.  Then for any k ∈ [], 
    U'  ϖ^λθ̃_kχ H_τ_2'  =          U' ϖ^λ  H_τ_2'      if  χ_1 = 1  or if   χ_1∈ S_1^-∖{ 1 },  β_0(λ) ≥ 0    
      U'  ϖ^s_0(λ) H_τ_2'      if χ_1∈ S^-_1∖{ 1 },  β_0(λ) < 0  or if   χ_1∈ S^+ .
  
 
  * Suppose (β_1 + β_2)(λ) ≥ 1. Then  
    U '  ϖ^λθ_3χ  H_τ_2' =    U' ϖ^λ H_τ_2'     if  χ_1 = 1   or if   χ_1∈ S^-_1∖{ 1 },  β_0(λ) ≥  0 
     
      U  ' ϖ^s_0(λ 
      )        
     H_τ_2'        if χ_1∈ S^-_1∖{ 1 },  β_0(λ) <  0   or if   χ_1∈ S_1^+
 
  

    
 In each of the parts (a), (c) and (d), the assumption made implies  the equality U' ϖ^λγ = U' ϖ^λ where γ denotes σ_2, σ^k , σ_3. In part (b), the assumption implies that U' ϖ^λσ^0 = U'ϖ^r_1(λ). Using this and the fact that the matrix ϰ^-(-x) in (<ref>) lies in H_τ_2' for x ∈_F, one easily deduces each of the claims. 
  


  For ℛ_2(1) (resp., ℛ_2(ϖ^(4,2,2,3))), the result is obtained by applying Lemma <ref> to  Proposition <ref> (resp., Corollary <ref>). The other two cases are handled by studying the fibers of the projection 
    _μ : ℛ_2(ϖ^μ) →ℛ_E(ϖ^_2(μ))
 
using Corollary <ref>. That is, if μ∈{ (3,2,1,2), (4,3,1,2) } and  ϖ^(a,c,d)γ lies in ℛ_E(ϖ^_2(μ)) for some γ∈{ 1, v_1, η_0,   ϖ^-(1,1,1)η_1,  η̃_k  |      k ∈  []^∘}, the fiber _μ   above ϖ^(a,c,d)γ consists of all elements of the form ϖ^λγ̂χ  where  γ̂∈{1, θ_1, θ_2,  θ_3,  θ̃_k  |   k ∈ []^∘} satisfies _2(γ̂) = γ, the cocharacter  λ = (a,b,c,d) ∈Λ^α_0 > 0 is such that b  = _1'(ϖ^μ) and χ∈ S_α_0(μ)^± is arbitrary.  Note that α_0(μ)   = 1 for both μ.    


∙ μ = (3,2,1,2) 
   The projection is  ℛ_E(ϖ^(3,1,2) ) = ℛ_E(ϖ^(3,2,2))  = ϖ^(2,1,1)ℛ_E(ϖ^(1,1,1)) which by Proposition <ref>(b),  equals    
    {ϖ^(3,2,2), ϖ^(3,2,2)v_1, ϖ^(3,1,2)η_0, ϖ^(3,2,1)η_0, ϖ^(3,2,2)η_0, ϖ^(2,1,1) η_1,  ϖ^(3, 2, 2)η̃_0, ϖ^(3,1,2)η̃_0, ϖ^(3,1,1)η̃_k  |       k ∈ []^∘}
 
By Lemma  <ref> and Proposition <ref>, the fibers above ϖ^(3,2,2) and ϖ^(3,2,2)v_1 are singletons. Since ϖ^(3,2,1,2), ϖ^(3,2,2,1)σ_1 clearly belong to ℛ(ϖ^μ), we choose these as the representative elements above the corresponding fibers.    For the remaining elements of ℛ_E(ϖ^(3,2,2)), one deduces from Lemma <ref> that χ must be either identity or in S_1^+ in each case (but not both), and the corresponding unique representative in the fiber is  easily obtained.    

 
∙ μ = (4,3,1,2) 
  
The projection ϖ^(2,1,1)·ℛ_E(ϖ^(2,2,1)) = {ϖ^(4,3,2), ϖ^(4,2,3)v_1, ϖ^(4,3,2)η_0, ϖ^(4,2,3)η̃_0, ϖ^(3,2,1)η_1}. Again, we decide the lifts for ϖ^(4,3,2), ϖ^(4,2,3)v_1 using Lemma <ref> and use Lemma <ref> to show that χ∈ S^-_1 is the only possible for choice  for each of the remaining representatives in ℛ_E(ϖ^(4,3,2) ).  
 
 

§ CONVOLUTIONS
   



    


Recall that X denotes the topological  vector   space  Mat_2 × 1(F) and 𝒮 =  𝒮_𝒪, X denotes the set of all locally constant compactly supported 𝒪-valued  functions on X.  The space X  admits a continuous  right action of H_1 =  _2(F) via left matrix multiplication by inverse and we extend this action to H via  _1 : H → H_1. These induce left actions of H_1 and H on 𝒮.  If 𝔭 is an ideal of 𝒪 and ξ_1, ξ_2∈𝒮, we write ξ_1≡ξ_2𝔭 if ξ_1(x) - ξ_2(x) ∈𝔭 for all x ∈ X.    If V is a compact open subgroup of H_1 or H, we let 𝒮(V) denote the space of V-invariants of 𝒮. 
If m, n are integers, we let 

    X_m,n      =  {[ x; y ]  |   x ∈ϖ^m_F ,     y ∈ϖ^n_F}
 
which are  compact open subset of X. We denote  
    ϕ_(m,n) : = (X_m,n ) ,     ϕ̅_(m,n) = ϕ_(-m,-n).
   We let z_0 denote the inverse of the central element ρ_1^2 =  diag(ϖ, ϖ) ∈   H_1. 
For n a positive integer, we let U_ϖ^n denote the subgroup of all elements in U  whose reduction modulo ϖ^n is identity in 𝐇(/ϖ^n ). For λ∈Λ, we define the depth of λ to be dep(λ) : = max{±α_0(λ) , ±β_0(λ) ,  ±β_2(λ) }. Then for λ of depth at most n,   ϖ^-λ  U_ϖ^nϖ^λ⊂ U.     

      
We will often write h  = (h_1, h_2, h_3) ∈_2(F) ×_F^×_2(F) ×_F^×_2(F)  ⊂GSp_6(F) as  
    h  =    (  [   a           b;     a_1         b_1;         a_2         b_2;   c           d;     c_1         d_1;         c_2     d_2 ]  )      or        h =  (   (  a      b 
     c     d   )  ,     (    a_1    b_1
     c_1    d_1 )  ,       (    a_2    b_2
     c_2    d_2 )    )  .
  
If we wish to refer to another element in H, we will write h ' and all its   entries will be 
 adorned with a prime.  Given a, b ∈, we write ϖ^(a,b) to denote diag(ϖ^b , ϖ^a-b   ) ∈_2(F).  




    





 §.§ Action of _2
  It will be useful to record a few general results on convolution of Hecke operators of _2(F) with ϕ. Let 𝒯 _u,v denote the double coset Hecke operator [U_1 diag(ϖ^u, ϖ^v )     
U_1].   
It acts on 𝒮(U_1) and in particular, on ϕ∈𝒮(U_1). It is clear that 𝒯_u,v(ϕ) = 𝒯_v,u(ϕ) and 𝒯_u,u(ϕ) = ϕ_(u,u).   
     𝒯_u,v(ϕ) =  ϕ_(v,v) + q^u-vϕ_(u,u) + ∑_ i=1^u-v-1 (q^i - q^i-1) ϕ_(i+v, i+v) when u > v.  Here the sum in the expression  is zero if u - v  = 1. 
 
 Let ξ = 𝒯_u,v(ϕ) =  ∑_γγ·ϕ where γ runs over representatives of of U_1diag(ϖ^u , ϖ^v )  U_1 / U _1.  Translating everything by (z_0)^v, it suffices to establish our formula  when  v = 0. Then u ≥ 1 and 
    U_1[ ϖ^u;       1 ]U_1 / U_1 =  _κ∈ [_u] [ ϖ^u   κ;       1 ]  U_1⊔ _κ∈ [_u-1] [   1;  ϖκ ϖ^u ] U_1  .
 
From the  decomposition above, we see that  ξ(v⃗) = q^i whenever  v  ∈ ( X_i,i∖  X_i,i+1 )  ∪ ( X_i,i+1∖ X_i+1,i+1  )  = X_i,i∖ X_i+1,i+1 for all i ∈{ 0, 1 , …, u  - 1 }  
and that  ξ(x⃗ ) = q^u + q^u-1 when x⃗∈ X_u,u.  
 

Let 𝒯_u,v,* : = 𝒯_-u,-v =  [U_1diag(ϖ^u, ϖ^v)U_1]_*  denote the dual (or transpose)  of 𝒯_u,v.   
       If u ≠ v, then 𝒯_u,v,* (ϕ) ≡ (z_0^u + z_v^v) ·ϕq - 1 and 𝒯_u,u,*(ϕ) = z_0^u·ϕ.     
 
  This is clear by Lemma <ref>. 
 
Let I_1^+ denote the Iwahori subgroup of U_1 = _2(_F) of upper triangular matrices and I_1^- the Iwahori subgroup of lower triangular matrices. For u,  v integers, let ℐ_u,v^± denote the double coset Hecke operator [I_1^±diag(ϖ^u,ϖ^v) U_1 ].



        Let u, v be integers. Then 

    ℐ_u,v^+ (ϕ)  = q^u-vϕ_(u,u)  +  ∑_i=0^u-v-1 
      q^iρ_1^2(i+v)· ( ϕ - ϕ_(0,1) )     if     u ≥  v  
       
    
    q^v-u-1ϕ_(v-1, v)  +  ∑_i=0^v-u-2 
      q^iρ_1^2(i+u)· ( ϕ_(0,1) - ϕ_(1,1))     if       u <  v

and 

    ℐ_u,v^- (ϕ)  =  q^v-uϕ_(v,v)  +  ∑_i=0^v-u-1 
      q^iρ_1^2(i+u)· ( ϕ - ϕ_(1,0))    if   u ≤  v  
     q^u-v-1ϕ_(u,u-1)  +  ∑_i=0^u-v-2 q^iρ_1^2(i+v) ·  (ϕ_(1,0) - ϕ_(1,1))    if   u >   v

  where ρ _1^2  =   z_0^-1   =   [ ϖ  ;   ϖ ].  
 
 




  The first equality is established in the same manner as Lemma <ref> using the  decompositions 
    I^+_1[ ϖ^u;       1 ]   U_1 / U_1  =    _κ∈ []_u[ ϖ^u   κ;       1 ],                I_1^+[   1;     ϖ^v ]   U_1 / U_1  =    _κ∈ []_v-1[   1;  κϖ ϖ^v ]
 
which hold for integers u ≥ 0, v  ≥  1. The second is obtained from the first by notation that I_1^-, I_1^+ are conjugates of each  other by the reflection matrix [   1; 1   ].  
 



 §.§ Convolutions with restrictions of 𝔥_0
  This subsection is devoted to computing 𝔥_ϱ_i,*(ϕ) for i = 0 ,1 ,2.   Recall that 
    ϱ_0 =  
    1.1([ 1        ;   1    ;     1  ;       1;         1;           1 ])
    ,     ϱ_1 =  1.1([ ϖ        ;   ϖ       1;     ϖ   1;       1;         1;           1 ]) ,        ϱ_1 =  1.1([   ϖ                ;     ϖ^2               1;         ϖ^2       1;               ϖ;                   1;                       1 ]).



    Modulo q - 1,  
 


  * 𝔞_ϱ_0,*(ϕ) ≡    (  6  +  16  z_0  +  6z_0^2 ) ϕ 

  * 𝔟_ϱ_0,*(ϕ) ≡  4 ( 1 + z_0 ^3  +  6 z_0 + 6 z_0^2 )  ϕ

  * 𝔠_ϱ_0,*(ϕ)  ≡   (   ( z_0 + 1 )^4 - 2 z_0^2  ) ϕ 
  
and 𝔥_ϱ_0,*( ϕ ) ≡   0 

 For λ = (a,b,c,d) ∈Λ, the map 
    U ϖ^λ U / U     ⟶  ( U_1ϖ^(a,b) U_1/ U_1 )  × (U_1ϖ^(a,c)U_1/U_1) × (U_1ϖ^(a,d) U_1 / U_1)  
     
    (h_1, h_2, h_3 ) U    ⟼ (h_1U_1 , h_2 U_1 , h_3 U_1)
 
is a bijection.  Corollary <ref> implies that  
    [U ϖ^λ U / U  ] ( ϕ ) =    | U_1ϖ^(a,c) U_1/U_1 | · | U_1ϖ^(a,d)U_1/ U_1 | ·  (z_0^b + z_0^a-b)  ϕ  .

Now |  U_1ϖ^(u,v) U_1/U_1 | ≡  1  or  2  q-1 depending on whether 2v - u = 0 or not. So parts (a)-(c) are all easily obtained.   
Now recall from (<ref>) that  
    𝔥_ϱ_0,*(ϕ) = 
    (1 + ρ^8) (U) - (1+ρ^6) ( U 
    ϖ^(1,1,1,1)  U ) +(1+ 2 ρ^2 + ρ^4) 𝔞_ϱ_0 -(1+ρ^2) 𝔟_ϱ_0 +  𝔠_ϱ_0
  
Using our formulas, we find that 

    𝔥_ϱ_0,*(ϕ)    ≡ ( (  1+  z_0^4 )   - 4 ( 1 + z_0 ^3   ) ( 1 + z_0 ) +  ( 1 + z_0)^2 ( 6 + 16 z_0 + 6z_0^2) - 4( 1 + z_0 ) (  1 + z_0^3 + 6z_0 + 6z_0^2 )   
             + (z_0+1)^4 - 2z_0^2 ) ϕ
   
and one  verifies that  the  polynomial  expression in z_0  above    is  identically  zero.  



  
    

 Let 𝐏 : = _2×__m_2  and   define embeddings   
 _ϱ_1   : 𝐏       ↪𝐇 ,          _ϱ_1 :  𝐏      ↪𝐇  
 
(γ_1, γ_2)     ↦( ∂γ_1 ∂^-1 ,   γ_2 ,   γ_2 )               (γ_1, γ_2 )    
    ↦( ∂γ_1 ∂^-1,    γ_2 ,  γ_2   ) 
 
where  =    [   1; 1 ] and ∂  :  = ρ_1  =  [   ϖ; 1 ]. We let 𝒳_ϱ_1 denote the common image _ϱ_1(P^∘ ), _ϱ_1(P^∘  ).  We denote by M_ϱ_1 (resp., M_ϱ_1') denote the subgroup of U_ϖ in which the first and second (resp., first and third) components are identity.  We also let 
    _2,3 : →𝐏      (h_1, h_2, h_3) ↦ (h_2, h_3 )  .
 
Finally, we let 
 𝒴_ϱ_1, L_ϱ_1, L'_ϱ_1, P_ϖ^∘ denote respectively  the projections of  𝒳_ϱ_1, M_ϱ_1, M'_ϱ_1,  U_ϖ  under _2,3.     
  






     H_ϱ_1 =   𝒳_ϱ_1  M_ϱ_1    =  𝒳_ϱ_1  M _ ϱ_1  '.   
 


  Writing h ∈ H as in <ref>, we see that  
    ϱ_1 ^-1 h  ϱ_1  = 
    [             a                                        bϖ                            ;                         a_1          -c_2                b_1 -  c_2 ϖ  a_1 -  d_2 ϖ;                        -c_1           a_2               a_2  -  d_1 ϖ  b_2  - c_1 ϖ;           c ϖ                                         d                            ;                      c_1  ϖ                                       d_1           c_1;                                    c_2  ϖ                         c_2           d_2 ]
  
From this, one immediately sees that h = (h_1, h_2, h_3 ) ∈ H_ϱ_1 if and only if ∂^-1  h_1∂, h_2, h_3∈ U_1 and the modulo ϖ reductions of h_2, h_3 coincide. So H_ϱ_1⊃𝒳_ϱ_1, M_ϱ_1,  M_ϱ_1'. To see that H_ϱ_1 equals the stated products, we note that for any h = (h_1, h_2, h_3 ) ∈ H_ϱ_1, ι_ϱ_1 ( ∂^-1 h_1^-1∂ ,  h_2 ^-1 ) · h ∈  M_ϱ_1 and _ϱ_1(∂^-1 h_1 ^-1∂ , h_3 ^-1  ) ·  h ∈  M_ϱ_1'.  
 
    Modulo q - 1, 
 

  * 𝔞_ϱ_1,*(ϕ) ≡ 2 ( 1 + 3z_0 + z_0^2)  
ϕ,   

  * 𝔟_ϱ_1,*(ϕ)  ≡  ( 1 + 10 z_0   + 10 z_0^2 + 
 z_0^3  ) 
ϕ   

  * 𝔠_ϱ_1,*  (ϕ) ≡ 2z_0  ( 1   +  z_0  )  
ϕ 
 
and 𝔥_ϱ_1,*(ϕ)  ≡ 0.  
 
  
For λ = (a,b,c,d)  ∈Λ, let ξ_λ =   [ U ϖ^λ H_ϱ_1]_*( ϕ).  Then  Lemma <ref> implies that 
    ξ_λ =   |  
    P^∘\ P^∘ϖ^(a,c,d)_2,3(H_ϱ_1)   | ·   [ U_1ϖ^(a,b)∂ U_1∂^-1]_* (ϕ )
 
Now Corollary  <ref> implies that 
    [ U_1ϖ^(a,b)∂ U_1∂^-1]_* (ϕ )         =   ∂·𝒯_b+1, a -  b, *(ϕ) 
    ≡   (z_0^b+1 + z_0^a-b )   ϕ _(0,1)    if  a ≠ 2b + 1 
     
     z_0^b+1·ϕ_(0,1)       if  a = 2b + 1

If  moreover  |β_0(λ)|,| β_2(λ) | ∈{ 0,1 },  then P^∘ϖ^(a,c,d)_2,3( H_ϱ_1 ) simplifies to P^∘ϖ^(a,c,d)𝒴_ϱ_1. So in this case, 
    |  P^∘\  P^∘ϖ^(a,c,d)_2,3(H_ϱ_1) | = [ 𝒴 _ϱ_1 :  𝒴 
     _ϱ_1∩ P^∘_(a,c,d) ]
 
where P_(a,c,d) ^ ∘  :  =   ϖ^-(a,c,d)  P^∘ϖ^(a,c,d). Since 𝒴 _ϱ_1≃_2(_F ) = U_1   (via the projection 𝐏→_1, (γ_1, γ_2 ) ↦γ_1), the index on the RHS of (<ref>)  can be found by comparing the intersection 𝒴_ϱ_1∩ P^∘_(a,c,d) with the Iwahori subgroups I_1^± in U_1. One easily sees that that the RHS of (<ref>)   is congruent to 1 or 2 modulo q+1, and that the former only happens if and only if β_0(λ) = β_2(λ) = 0. This takes care of  the index calculations for all the  Hecke operators in parts (a)-(c) except for (U ϖ^(2,1,2,0) H_ϱ_1 ).  Here,  we  invoke <cit.>. More precisely, we use that  _2,3(H_ϱ_1) = 
P_ϖ^∘𝒴_ϱ_1 and the result in loc.cit. implies that  
    | P^∘\ P^∘ϖ^(2,2,0)   P_ϖ^∘𝒴 _ϱ_1 ] =   e^-1·  |  P ^  ∘\    P^∘ϖ^(2,2,0) P^∘ _ϖ | · |   (  P_ϖ ^∘∩𝒴 _ϱ_1  ) 
    \𝒴 
     _ϱ_1 |

where e =  [  𝒴_ϱ_1 P_ϖ^∘∩ P_(2,2,0)^∘ :   P_ϖ^∘∩  P_(2,2,0)^∘ ].  Now P_ϖ ^∘∩ Y_ϱ_1 is identified with I_1^±, and [U_1 : I_1∩ I_1^-] = q ( q+  1 ) and similarly |  P^∘\ P ^∘ϖ^(2,2,0) P^∘_ϖ   | = q^2.   Moreover   𝒴_ϱ_1  P_ϖ^∘∩ P_(2,2,0)^∘  =   (𝒴_ϱ_1∩ P_(2,2,0)^∘  ) ·  ( P_ϖ^∘∩ P_(2,2,0)^∘  ), which implies that 
    e =   [ 𝒴 _ϱ_1∩ P_(2,2,0)^∘   :  𝒴 _ϱ_1∩  P_ϖ^∘∩  P_(2,2,0)^∘  ] .
 
from which it is not too hard to see that e  = q.  
It follows that ξ_(2,1,2,0)≡   2  z_0·ϕ. Now  recall from (<ref>) that 
    
    𝔥_ϱ_1 = 
      - ( 1 + ρ^6 ) (U H_ϱ_1) +  ( 1 +  2 ρ^2 + ρ^4 ) 𝔞_ϱ_1 - ( 1 + ρ^2 ) 𝔟_ϱ_1    +  𝔠_ϱ_1
 
  So we see that

    𝔥_ϱ_1, * ( ϕ  )     ≡ (  - ( 1 + z_0^3)  ( 1 + z_0 ) + (1 + z_0)^2  
    ( 2 + 6z_0 +  2 z_0 ^2 )   - ( 1 + z_0  ) ( 1 + 10 z_0 + 10 z_0 ^2 + z_0 ^3 )  
            + 2  z_0 ( 1 + z_0  ) ^  2  )
 
which is zero since  the  polynomial expression in z_0  vanishes.   

    
Let 𝐏,  be as in Notation <ref>,   _ϱ_2   : 𝐏↪𝐇 be the given by (γ _1, γ _2) 
 ↦ ( 
γ _1 
,   γ _2 ,   γ_2  ) and 𝒳_ϱ_2 =  _ϱ_2(P^∘ ).   
Let pr_2,3 : →𝐏 be the projection as before and let 𝒴_ϱ_2,  P_ϖ^2^∘ denote respectively  the projections of  𝒳_ϱ_0, U_ϖ^2  under _2,3.     


 H _ ϱ_2 = 𝒳_ϱ_2   U_ϖ^2. 
 
 If h ∈ H is written as in Notation <ref>, then 

    ϱ_2^-1 h ϱ_2 =   
    [                a                                                  b                                  ;                               a_1             -c_2                    b_1 -  c_2 ϖ^2  a_1 -  d_2 ϖ ^2;                              -c_1              a_2                  a_2  -  d_1 ϖ ^2  b_2  - c_1 ϖ ^2;                c                                                  d                                  ;                            c_1  ϖ                                                d_1              c_1;                                             c_2  ϖ                               c_2              d_2 ] .
 
Now an argument similar to Lemma  <ref> yields the desired factorization. 
  
    Modulo q - 1, 
𝔥_ϱ_2,*(ϕ)  ≡ 0.  
 
 Recall from (<ref>)   that 
    𝔥_ϱ_2 =  ( 1 + ρ^2 + ρ^4 ) (U H _ϱ_2 )  -  ( 1 +  ρ ^2  )  (  U   ϖ^(1,1,0,1)  H_ϱ_2 )   + (  U ϖ^( 2,2,1,1) H_ϱ_2 ) +  (   U  ϖ^(2,2,1,0) H_ϱ_2 ) .
 
If λ  = ( a , b, c, d )   ∈Λ has depth at most 2, then Uϖ^λ H_ϱ_2 =  U ϖ^λ𝒳 _ϱ_2 by  Lemma  <ref> and so 
    [U ϖ^λ  H_ϱ_2]_*( ϕ )  =  | P^∘\  P^∘ϖ^(a,c,d)𝒴_ϱ_2 | ·𝒯_b,a-b,*(ϕ) .

Now 𝒯_b, a-b,*(ϕ) is computed (modulo q-1) by Corollary <ref>. As 𝒴_ϱ_2≃_2(_F ),  | P^∘\  P^∘ϖ^(a,c,d)𝒴_ϱ_2 | ≡ 1  or  2 q  - 1 depending on whether c = d = 2a or not. So one finds that 
 
    [U H_ϱ_2]_*(ϕ)     = ϕ, 
     
     [U ϖ^(1,1,0,1) H_ϱ_2  ]_*( ϕ)     ≡2  ( 1 + z_0  ) ϕ ,  
    
     [U ϖ^(2,2,1,1) H_ϱ_2]_*(ϕ)     ≡ ( 1 + z_0^2) ϕ  , 
    
     [ U ϖ^(2,1,2,0) H_ϱ_2]_*(ϕ)    ≡  2   z_0·ϕ .

 From these, the claim easily  follows.  
 


 §.§ Convolutions with restrictions of  𝔥_1
    In this subsection, we compute the convolution  𝔥_ς_i,*(ϕ) for i = 0 , 1 ,2 , 3. Recall that ς_i = σ_iτ_1. Explicitly, 

    ς_0 =  
    1.1([ ϖ       1;   ϖ   1;     ϖ  ;       1;         1;           1 ])
    ,   ς_1 =  1.1([ ϖ       1;     ϖ  ;   ϖ   1;       1;           1;         1 ]),   ς_2 =   1.1( [ ϖ       1;   ϖ   1   1;     ϖ   1;       1;         1;           1 ] ) ,    ς_3 =   1.1( [ ϖ ^2                   ϖ;       ϖ^2         ϖ         1;            ϖ^2         1;                   1;                        1;                             1 ] )
 

   Let 𝐏,  and ∂ be as in <ref> and define  embeddings 
 _ς_0   : 𝐏       ↪𝐇 ,          _ς_0 :  𝐏      ↪𝐇  
 
(γ_1, γ_2)     ↦(γ_1, γ_1  ,  ∂γ_2  ∂^-1   )               (γ_1,  γ_2)        ↦(γ_1 ,  γ_1, 
∂γ_2   ∂^-1   ) 
 
We denote 𝒳_ς_0, the common images _ς_0(P^∘ ) = _ς_0( P^∘ ). 
We let M_ς_0 (resp., M_ς_0 ')  denote the subgroup of U_ϖ in which the first and third (resp., second and third) components are identity.
We also let _1,2 : →𝐏 denote the projection ( h_1, h_2, h_3) ↦  (h_1, h_2 ).  Finally, we let 𝒴_ς_0, L_ς_0, L'_ς_0, P_ϖ^∘⊂ P the projections of 𝒳_ς_0, M_ς_0, M'_ς_0,  U_ϖ   respectively. 
 

     H_ς_0 =   𝒳_ς_0  M_ς_0    =  𝒳_ς_0  M _ ς_0 '.   
 

  Writing h ∈ H as in Notation <ref>, we see that 
    ς_0 ^-1 h ς_0 =   
    [          a       -c_1             b - c_1 ϖ  a - d_1 ϖ           ;        - c        a_1            a_1 - d  ϖ b_1 -  c ϖ           ;                              a_2                            b_2 ϖ;        c ϖ                                d          c           ;                c_1  ϖ                   c_1        d_1           ;                           c_2  ϖ                              d_2 ]
  
Then one easily verifies that 𝒳_ς_0, N_ς_0,  M_ς_0, M'_ς_0  are contained in H_ς_0. On the other hand if h = (h_1, h_2, h_3) ∈ H_ς_0, the above matrix is in K which implies that  h_1, h_2 and ∂ ^-1 h_3∂∈_2(_F ). It follows that  η : =  _ς_0  ( h_1, ∂ h_3∂  )    )  ,
γ :=  _ς_0 (h_2, ∂^-1h_3∂)   ∈𝒳_ς_0 and η  ^-1  h  ∈  M_ς_0, γ^-1 h ∈  M'_ς_0.    



    Modulo q - 1, 
 

  * 𝔞_ς_0,*(ϕ) ≡  5  (1+z_0)  ϕ,   

  * 𝔟_ς_0,*(ϕ)  ≡   ( 4  +  14 z_0 +  4 z_0 ^2 )  ϕ   

  * 𝔠_ς_0,*  (ϕ) ≡  ( 1 +  z_0)^3·ϕ 
 
and 𝔥_ς_0,*(ϕ)   =   
𝔥_ς_1,*(ϕ)    ≡ 0. 
 
   Let λ = (a,b,c,d ) ∈Λ and ξ_λ denote [H_ς_0ϖ^λ U ](ϕ). Let   Q^∘ : = _2(_F) and Q^♢⊂_2(F) the  conjugate of Q^∘  by ∂ =    [   ϖ; 1 ]. 
Lemma  <ref> implies that 

    H_ς_0ϖ^λ U / U     →_1,2 ( H_ς_0ϖ^λ U / U   )  × 
    Q^♢ϖ^(a,d) Q^∘ / Q^∘  
     
    (γ_1, γ_2, γ_3 ) U     ↦ ( (γ_1, γ_2) _1,2(U) ,  γ_3 Q^∘ )
 
is a bijection. Now |Q^♢ϖ^(a,d)  Q^∘ / Q^∘ |  =  | Q^∘ϖ^(a-1,d)  Q^∘ / Q^∘ | which equals q^|a-1-2d|(q+1) if a -1 ≠ 2d and 1 otherwise. It remains to describe pr_1,2 ( H_ς_0ϖ^λ U / U    )   ⊂ P / P^∘.  By  Lemma  <ref>,   _1,2(  H_ς_0 )  = 𝒴_ς_0 L_ς_0 =  𝒴_ς_0 L'_ς_0. If | β_0(λ) | ≤ 1 (resp., |α_0(λ)| ≤ 1),  then  the conjugate of L_ς_0 (resp., L_ς_0') by ϖ^λ is contained in U. So if min{ |  α_0(λ) |  , |  β_0(λ) |  }∈{ 0 ,1 }, we have 
    _1, 2 (  H_ς_0ϖ^λ  U / U     )    = 𝒴 _ς_0ϖ^(a,b,c) P^∘ /  P^∘
  
where we write ϖ^(a,b,c) for _1,2(ϖ^λ ).  
To describe a system of representatives for  𝒴_ς_0ϖ^(a,b,c) P^∘ / P^∘, it suffices to describe one for 𝒴 _ς_0 /  ( 𝒴 _ς_0∩ P^∘_(a,b,c) ) where 
    P^∘_(a,b,c)  : = ϖ^(a,b,c) P^∘ϖ^-(a,b,c)
 denotes the conjugate of P^∘ by ϖ^(a,b,c).      Since 𝒴 _ς_0 is isomorphic to _2(_F) (via the projection 𝐏→𝐇_1, (γ_1, γ_2  ) ↦γ_1),    this can be done by viewing  intersection 𝒴_ς_0∩ P^∘_(a,b,c) as a subgroup of U_1 =  _2(_F ) and comparing it with the  Iwahori subgroups I_1^±. For this purpose, it will be convenient to introduce the quantities

    u_λ  = max{  0, α_0(λ) ,    -  β_0(λ) } ,         v_λ  =  max{  0 ,  - α_0(λ  ) , β_0(λ)  } .
 
These describe the valuations of the upper right and lower left entries of a matrix in  𝒴_ς_0∩ P^∘_(a,b,c). 

The case where min{ |a_0(λ) |  , |  β_0(λ)  | }≥ 2 requires a little more work (though it will only occur once in this proof).  Here  
we invoke <cit.> for the product _1,2(H_ς_0  )  =  𝒴_ς_0  P_ϖ^∘. Thus  

    (  _1,2   ( H_ς_0ϖ^λ U  )  ) =  e^-1∑  _γ (  γ  P_ϖ^∘ϖ^(a,b,c) P^∘ )
 and  where γ runs over  (the finite set) 𝒴_ς_0 / 𝒴_ς_0∩ P_ϖ^∘ and  
e =  e_(a,b,c)   : =    [ _1,2 (   H _ ς_0 
 )   ∩  P_(a,b,c) ^∘  :     P^∘_ϖ∩ P^∘_(a,b,c)   ].  So the function ξ_λ can be computed by first computing (P^∘_ϖϖ^(a,b,c) P^∘ ) ·ϕ, then summing the translates of the result by representatives of 𝒴_ς_0 /  ( Y_ς_0∩ P_ϖ^∘ ) and dividing the coefficients by e. 
  


(a) Recall that 𝔞_ς_0 =   (U ϖ^(1,1,1,0) H_ς_0) + 
(Uϖ^(1,1,0,1)H_ς_0)+ 2 (Uϖ^(1,1,0,0)H_ς_0).  Let 
    λ_1 :  = (1,0,0,1) ,   λ_2 :  = 
    (1,0,1,0) ,   λ_3 : =  (1,0,1,1) .
    Then  𝔞_ς_0, * (ϕ )  =  z_0· ( ξ_λ_1 + ξ_λ_2 +  2 ξ_λ_3 ). For each λ_i, the formula  (<ref>)  applies. For λ  =  (a,b,c,d) ∈{λ_2, λ_3}, u_λ = 0 and v_λ = 1, so 𝒴_ς_0∩ϖ^(a,b,c) P^∘ϖ^-(a,b,c) is identified with I_1^+ and one easily sees that 
    ξ_λ_2    =  (q+1)   𝒯_0,1(ϕ) ≡ 2 (ϕ + ϕ_(1,1))   
    ξ_λ_3     = 𝒯_0,1 (ϕ ) ≡ϕ + ϕ_(1,1)
 
modulo q - 1.  
For λ = λ_1, u_λ =  v_λ = 1 and we see that 𝒴_ς_0∩ϖ^(1,0,0) P^∘ϖ^-(1,0,0) is identified with I_1^+∩  I_1^-. Thus a system of representatives for 𝒴 _ς_0 /  ( 𝒴 _ς_0∩ϖ^(1,0,0) P^∘ϖ^-(1,0,0)   ) is obtained by multiplying a system of representatives for U_1 / I_1^+ with that for I_1^+ / I_1^+∩ I_1^-. So  
    ξ_λ_1 = 
    ∑_γ∈ U_1    / I_1^+γ∑_η∈ I_1^+/(I_1^+∩ I_1^-) ηϖ^λ_1·ϕ   .

Now ηϖ^λ_1·ϕ = ϖ^λ_1·ϕ for any η∈ I_1^+. So  the inner sum  equals q ϕ. The outer sum then evaluates to q ( ϕ + q ϕ_(1,1)). 
Thus  ξ_λ_1≡   (   ϕ + ϕ_(1,1)  ).
Putting everything together 
gives part (a).    


(b) Recall that  
    𝔟_ς_0  =  (U   ϖ^(2,2,1,1)   H _ς_0 ) +  (U ϖ^(2,1,2,1)   H_ς_0) + (Uϖ^(2,2,0,1) H _ς_0) + (U ϖ^(2,1,1,2) H_ς_0) + 4 (Uϖ^(2,1,1,1)H_ς_0) .
  For μ∈{  (2,1,1,2), (2,1,1,1) },  it is easy to see that   
    [  U ϖ^μ H_ς_0]_*(ϕ) ≡ 2  z_0·ϕ
 
For μ_1 =  (2,2,1,1) and μ_2 = (2,1,2,1),  arguments similar to part (a)  reveal that 
    [U ϖ^μ_1 
    H_ς_0 ]_*  ( ϕ ) ≡  2 ( 1 + z_0^2)  ϕ   ,       
     [U ϖ^μ_2 H_ς_0 ]_* ( ϕ ) ≡ 4 z_0·ϕ .
 
This leaves μ = (2,2,0,1). Denote λ = (4,2,2,2) - μ = ( 2,0,2,1) and let e denote e_(2,0,2).  It is easy to see that _1,2(H_ς_0) ∩ P^∘_(2,0,2) is equal to the product of 𝒴_ς_0∩ P_(2,0,2)^∘ with P_ϖ ^∘∩ P_(2,0,2)^∘ and therefore e  =  [  𝒴_ς_0∩ P_(2,0,2)^∘  :  𝒴_ς_0∩ P_ϖ∩  P_(2,0,2)^∘  ].  
From this, one finds that e =  q.  
Next we  compute that 
    ( P _ ϖ  ^∘ϖ^(2,0,2)  P^∘ ) ·ϕ   =    q  (  ϕ_(0,1)  -  ϕ_(1,1) + q ϕ_(1,2)   )  .

Since 𝒴_ς_0∩ P_ϖ^∘⊂𝒴_ς_0 is identified with I_1^+∩ I_1^-⊂_2(_F ), 
the  expression (<ref>)  reads   
    ξ_λ     =   ∑ _  h ∈ U_1    / I _1 ^   + ∩  I_1 ^-  
       h ( ϕ_(0,1) - ϕ_(1,1) + q ϕ_(1,2) )  
        = 
    ∑_ γ∈   U_1 / I_1^+γ∑ _ η∈ I_1^+ / ( I_1^+∩ I_1^-) η  ( ϕ_(0,1) - ϕ_(1,1) + q ϕ_(1,2) )

Then the inner sum is just multiplication by q. The outer sum then evaluates to 
    q ( ϕ +  q ϕ_(1,1)  
    )  -  q  ( q + 1 ) ϕ_(1,1)   + q  ^  2    ( ϕ_(1,1) +  q ϕ_(2,2) )   =  q  ϕ  + q  ( q- 1 )  ϕ_(1,1) +  q^3ϕ_(2,2)
 So  
we see that ξ_λ  
= q ϕ + q ( q - 1) ϕ _(1,1)   + q^3ϕ_(2,2)  and  therefore  
    [ U ϖ^(2,2,0,1) H_ς_0]_*(ϕ) =  (q+1)  z_0^2·ξ_λ≡ 2   ( 1   + 
     z_0^2 ) ϕ.
   Putting everything together, we find that  
    𝔟_ς_0,*( ϕ)     ≡ 2(1+z_0^2)ϕ +  4 z_0·ϕ + 2 ( 1 + z_0^2) ϕ +  2 z_0·ϕ +  4 ( 2 z_0·ϕ ) 
        = ( 4 +  4z_0^2 + 14 z_0 )   ϕ .




(c) 
We have 𝔠_ς_0 = 
(Uϖ^(3,2,2,2)H_ς_0)+
(Uϖ^(3,3,1,1)H_ς_0)+  (Uϖ^(3,2,0,1)H_ς_0). For each of the three Hecke operators, the formula (<ref>) applies and we find that 
    [ U ϖ^(3,2,2,2)H_ς_0 ]_*  ( ϕ )    ≡ 2( z_0 + z_0^2 ) ϕ , 
     
    [U ϖ^(3,3,1,1) H_ς_0]_* (ϕ)    ≡  ( 1 + z_0^3) ϕ ,  
     
    [ U ϖ^(3,2,0,1)  H_ς_2 ] _*  ( ϕ )      ≡ (z_0   + z_0^2 ) ϕ
 
from which (c) follows. 


Now recall that  𝔥_ς_0 = -(1+ ρ^6) (U H_ς_0)  +  ( 1+  2 ρ^2 + ρ^4 ) 𝔞_ς_0  - ( 1 + ρ^2 )   𝔟_ς_0 +   𝔠_ς_0. It is easy to see that [U H_ς_0]_*(ϕ) = ( q + 1  ) ϕ. So  by parts (a)-(c), we see  that 

    𝔥_ς_0, *    ( ϕ  )     ≡    - 2  ( 1 +   z_0  ^3 )   ϕ      + ( 1 + z_0)^2 (  5 ( 1 + z_0)  ϕ  )    -    ( 1 + z_0)  ( 4 + 14 z_0 +  4 z_0  ^2 ) ϕ  +  (1 + z_0)^3ϕ
        =  ( 1 +z_0 )  (  - 2 + 2z_0 - 2z_0^2  + 5 ( 1+ z_0)^2 - 4 - 14 z_0 - 4z_0^2 + ( 1 + z_0  )^2  )  ϕ
          = 0
 
Finally since  𝔥_ς_1 = w_2𝔥_ς_0 w_2 (<ref>) and w_2 only swaps the second and third components of H and w_2 normalizes U, we see that 𝔥_ς_1,*(ϕ) = 𝔥_ς_0,*(ϕ). 
 





   We let A_ς_2 denote the intersection A ∩ς_2 K ς_2^-1 and   J_ς_2⊂ U denote the Iwahori subgroups of triples (h_1, h_2, h_3 ) ∈ U such that h_1 , h_3 reduce modulo ϖ to lower triangular matrices and h_2 reduces to an upper triangular matrix.  We denote by M_ς_2 the three parameter additive  subgroup of all triples h = (h_1,h_2,h_3) ∈ U such that  
    h_1 = [ 1; x 1 ],    h_2 = [ 1 y;   1 ] ,     h_3  =   [           1; y - x + ϖ z           1 ]
 
where x , y , z ∈_F  are  arbitrary and by N_ς_2 the three  parameter  subgroup of all triples (h_1, h_2,  h_3) of the form 

    h_1 = 
    [   1 x ϖ;       1 ] , 
          h_2 = 
    [   1; y ϖ   1 ],        h_3 =   [   1 z ϖ;       1 ]
  
where x , y , z ∈_F are arbitrary. 
Finally, we let L_ς_2 the one-parameter subgroup of U all triples of the form (1,1, [ 1; z 1 ]  ) where z ∈_F.   
 


       H_ς_2 is the product of A_ς_2, M_ς_2, N_ς_2 and J_ς_2 is the product of A^∘, H_ς_2, L_ς_2 where these products can be taken in any order.  
 

  It is easily verified that ς_2^-1 M_ς_2ς_2, ς_2^-1 N ς_2 are contained in K, so that M_ς_2 , N_ς_2 are subgroups of H_ς_2. 
Let h∈ H_ς_2 and write h as in Notation <ref>. Then 
    ς_2 ^-1 h  ς_2   =    
    [               a            -c_1                         b-c_1 ϖ      a -  d_1 ϖ          -c_1 ϖ;              -c             a_1            -c_2       a_1 -  dϖ b_1 - c - c_2 ϖ    a_1 -  d_2 ϖ;                            -c_1             a_2          -c_1 ϖ    a_2 - d _1 ϖ     b_2 -c_1  ϖ;             c ϖ                                               d               c                ;                          c_1  ϖ                             c_1             d_1             c_1;                                          c_2  ϖ                             c_2             d_2 ] .

It follows that h ∈ U and   c_1, b_2 , b  ∈ϖ_F. In particular, H_ς_2⊂ J_ς_2 and  a, a_1, a_2, d , d_1, d_2∈_F   ^ ×.     Let m ∈ M_ς_2 be defined with parameters x =  - c/a, y = -b_1/d_1   and   z =  -(c_2/a_2 +y-x)/ϖ (see Notation <ref>).  Write h' = m h as in Notation <ref> and  let n ∈ N_ς_2 be defined  with  paramaters x = -b'/d'ϖ, -c_1'/a_1'ϖ, z  =  - c_2'/ a_2' ϖ (see Notation <ref>). Then  nmh lies in A, and hence in A _ς_2.   Thus H_ς_2 =  M_ς_2 N_ς_2 A_ς_2. Similarly we can show H _ς_2 =  N _ς_2 M_ς_2   A_ς_2. Since A_ς_2 normalizes both M_ς_2, N_ς_2, the product  holds in  all possible  orders. This establishes the first claim. The second is established in completely analogous way.         
  
    If λ∈Λ satisfies β_2  (λ)   ≤  0, then 
U ϖ^λ H_ς_2 = U ϖ^λ J_ς_2.  
 
 This follows by Lemma <ref> since ϖ^λ L_ς_2ϖ^-λ⊂ U if β_2(λ)  ≤ 0.  
  

Corollary <ref> reduces the computation of [Uϖ^λ H_ς_2]_*(ϕ) to [Uϖ^λ J_ς_2]_*(ϕ) for almost all Hecke operators appearing  in 𝔥_ς_2,*, which we  can be calculated   efficiently  using  Lemma  <ref>.   The few exceptions are handled below. 
     Modulo q - 1, we have 


  * [U ϖ^(1,1,0,1) H_ς_2 ]_*(ϕ) ≡   ( 1 + z_0 ) ϕ -  z_0·ϕ_(1,0)   , 

  * [U ϖ^(1,0,1,1) H_ς_2]_*(ϕ) ≡ z_0·ϕ_(1,0)  , 

  * [Uϖ^(2,1,1,2) H_ς_2]_* (ϕ) ≡  z_0·ϕ 

  * [U ϖ^(3,2,1,2)ψ H_ς_2]_*( ϕ) ≡ 0 

  
   
For λ∈Λ, we will denote ξ_λ := [H_ς_2ϖ^λ U ](ϕ).  


(a) This equals z_0·ξ_λ where λ = (1,0,1,0). Since λ has depth one, we have   H_ς_2ϖ^λ U = M_ς_2ϖ^λ U. Now  

    M_ς_2ϖ^λ U / U = { (  [ 1; x ϖ ] 
    , 
     [ ϖ y;   1 ] ,  [     1; y - x     ϖ ] ) U   |   x, y ∈_F}.
 
and it is easy to see that a system of representatives for  M_ς_2ϖ^λ U/U is obtained by allowing the parameters x, y in the set above to run over []. Using this system, one calculates that ξ_λ  =   ϕ - ϕ_(1,0) - ϕ_(1,1). 
    


(b)  This equals z_0·ξ_λ where λ = (1,1,0,0). As in part (a), we have H_ς_2ϖ^λ U  = M_ς_2ϖ^λ U and it is easy to see that 
    M _ς_2ϖ^λ U / U = { (1 , 1 ,  [ 1; t 1 ] )  ϖ^λ U   |   t ∈_F} .

A set of representatives is obtained by allowing the parameter t to run over elements of []. Thus ξ_λ = q ϕ_(1,0).  
  

(c) This  expression equals z_0^2·ξ_λ where λ  =   (2,1,1,0). As the first two components of ϖ^λ are central and β_2(λ) ≥ 0, we have ϖ^-λ N_ς_2ϖ^λ⊂ U and so H_ς_2ϖ^λ U = M _ς_2ϖ^λ U / U.  Using the centrality of the first two componetns again, we see that 
    M_ς_2ϖ^λ U / U  = { ( 1 , 1 ,  [ 1  ; u 1 ] )  ϖ^λ U    |   u ∈_F} .

From this, we see that a system of representatives is given by letting the parameter u run over [_2]. Thus ξ_λ = q^2ϕ_(1,1). 


(d)  It suffices to  show that [ H_ς_2ψ^-1ϖ^(1,0,1,0) U ](ϕ) ≡ 0. Let us denote ψ^-1ϖ^(1,0,1,0) by η. It is straightforward to verify that η ^-1 N_ς_2η⊂ U, so that  H_ς_2η U / U = A_ς_2  M_ς_2η U / U.  Elementary manipulations show that 
 
    A_ς_2 M_ς_2η U / U = {  (  [ 1; s ϖ ] , [ ϖ t;   1 ]  ,  [         1          ; u + s - t         ϖ ] ) U      |     s, t ∈_F ,  u ∈_F^×}
 
where we used that (a, a_1, a_1, d, d_1 , d_2 ) ∈ A_ς_2 if and only if a , d ∈_F^× with a ≡ d_1≡  a_2ϖ and d ≡ a_1≡ d_2ϖ. Let 
    C(s,t,u) : =     (  [ 1; s ϖ ] , [ ϖ t;   1 ]  ,  [         1          ; u + t - s         ϖ ] )
 
where s, t  ∈_F, u ∈_F^×.   Then C(s,t,u) U = C(s',t',u') U if and only if s ≡ s', t ≡ t', u ≡ u' modulo ϖ. Thus  a system of representatives for H_ς_2η U / U    is given by C(s,t,u) where s, t run over [] and u runs over []^×. Thus for each fixed s , t, there are q - 1 choices of u from which it easily follows that the function [H_ς_2η U ](ϕ) vanishes modulo q - 1.  
  


    Modulo q - 1,  we have 
 

  * 𝔞_ς_2,*(ϕ) ≡  2(1+z_0)  ϕ + 2z_0·ϕ_(1,0),   

  * 𝔟_ς_2,*(ϕ)  ≡   ( 1 + 6z_0 + z_0^2)  ϕ + 3z_0(1+z_0)ϕ_(1,0) 

  * 𝔠_ς_2,*  (ϕ) ≡   z_0(1+z_0)  ϕ + z_0(1+z_0)^2ϕ _(1,0) 
 
and 𝔥_ς_2,*(ϕ)  ≡ 0. 
 


  For λ = (a,b,c,d) ∈Λ, let ξ _λ denote [U ϖ^λ H_ς_2]_*(ϕ).  If β_2(λ) = 2d - a ≤ 0, then ξ_λ = [U ϖ^λ J_ς_2]_*(ϕ). It is easily seen from Lemma <ref> and the decompositions given therein  that 
    [U ϖ^λ J_ς_2]_*(ϕ) ≡ℐ_-b,a-b^-(ϕ)    q-1

This  formula in conjunction with Lemma <ref> can be used to calculate all Hecke operators. 
For instance, 
we have 𝔞_ς_2 =     ( U ϖ^(1,1,1,0) H_ς_2)   +  ( U ϖ^(1,0,0,0) H_ς_2) +  (U ϖ^(1,1,0,1) H_ς_2) + ( U ϖ^(1,0,1,1) H_ς_2) + 2( U ϖ^(1,0,1,0) H_ς_2) and we compute    
2 
 

  * [ U ϖ^(1,1,1,0) H_ς_2]_*(ϕ)  ≡  ( 1 + z_0 ) ϕ  -  z_0·ϕ_(1,0), 

  * [U ϖ^(1,0,0,0)H_ς_2]_*(ϕ) ≡ z_0·ϕ_(1,0), 

  * [U ϖ^(1,1,0,1) H_ς_2]_*(ϕ )   ≡ ( 1 + z_0 ) ϕ -  z_0·ϕ_(1,0), 

  * [U ϖ^(1,0,1,1) H_ς_2]_*(ϕ)  ≡  z_0·ϕ_(1,0), 

  * 2 [U ϖ^(1,0,1,0) H_ς_2 ] (ϕ) ≡  2  z_0·ϕ_(1,0).  
 
  
Now adding all these retrieves the  expression in part (a). Similarly for parts (b) and (c).  
 

Now   𝔥_ς_2 = - ( 1 + ρ^6) (U H_ς_2) + ( 1 +  2 ρ^2 + ρ^4) 𝔞_ς_2  - ( 1 +  ρ^2)   𝔟_ς_2 +    𝔠_ς_2 from (<ref>). 
Therefore    

    𝔥_ς_2,*( ϕ   )     ≡ (- 1 - z_0^3) ϕ +  (1+z_0)^2  ( 2(1 + z_0 )ϕ  + 2z_0·ϕ_(1,0)  )    - 
              (1+z_0)    ( (1+6z_0 + z_0^2)ϕ + 3z_0(1+z_0)ϕ_(1,0)  )  + z_0(1+z_0) ϕ + z_0(1+z_0)^2ϕ_(1,0)
        =   (  -1-z_0^3 + 2(1+z_0)^3 - (1+z_0)(1+6z_0 + z_0^2)  + z_0(1+z_0)  )  ϕ   +  
           (  2 z_0 ( 1+  z_0 )  ^2  - 3z_0  ( 1 + z_0 ) ^2  + z_0 ( 1+ z_0)^2 ) ϕ_(1,0)  = 0
 
 


   As usual, we let A_ς_3 denote the intersection A ∩ς_3 K ς_3^-1. 
We denote by M_ς_3 the three parameter additive  subgroup of all triples h = (h_1,h_2,h_3) ∈ U such that 

    h_1 = [   1; x/ϖ   1 ],    h_2 = [ 1 y;   1 ] ,     h_3  =   [                1; y - x ϖ  +  zϖ^2                1 ]
 
where x , y , z ∈_F  are  arbitrary and by N_ς_2 the three  parameter  subgroup of all triples (h_1, h_2,  h_3) of the form 

    h_1 = 
    [     1 x ϖ^2;           1 ] , 
          h_2 = 
    [   1    ; y ϖ   1 ],        h_3 =   [             1 y ϖ +   z ϖ^2;                           1 ]
  
where x , y , z ∈_F are arbitrary. 
 

       H_ς_3 = M_ς_3 N_ς_3 A_ς_3 = N_ς_3  M_ς_3  A_ς_3.  



  Writing h ∈ H as in Notation <ref>, we find that 
    ς_3 ^-1 h ς_3 =  
    [                    a              -c_1  ϖ                                 bϖ^2 -c_1            a - d_1 ϖ              -c_1 /ϖ;                 -c ϖ                  a_1                 -c_2             a_1 - dϖ b_1 - c_2 - c ϖ^2ϖ^2       a_1  - d_2 ϖ^2;                                      -c_1                  a_2               -c_1 ϖ         a_2- d_1 ϖ^2      b_2 
     - c_1 ϖ^2;                c ϖ^2                                                              d                  c ϖ                     ;                                  c_1  ϖ^2                                    c_1  ϖ                  d_1                  c_1;                                                       c_2  ϖ^2                                       c_2                  d_2 ] .

From this matrix, we easily see that H_ς_3 contains M_ς_3 and N_ς_3. We also see that if h ∈ H_ς_3, then  all entries of h except for c are integral. Moreover,  c_1, b_2∈ϖ_F,  b ∈ϖ^2_F and c ∈ϖ^-1_F.   Thus a, a_1 ,a_2, d, d_1, d_2∈_F^×.  An argument analogous to Lemma  <ref> applies to yield the desired decompositions.   
   





     Modulo q - 1, we have 


  * 𝔞_ς_3,*(ϕ)  =  z_0·ϕ_(2,0) 

  * 𝔟_ς_3,*(ϕ) ≡  ( z_0^2 + z_0 )·ϕ_(2,0)  + z_0·ϕ_(1,0) 

  * 𝔠_ς_3,* (ϕ) ≡   (z_0^2 + z_0) ·ϕ_(1,0) 



and 𝔥_ς_3,*  (ϕ) ≡     0 
   



  For λ∈Λ, let ξ_λ denote [H_ς_2ϖ^λ U ] ( ϕ ).   


(a) This equals ξ_f_1.   From Lemma <ref>, we find that  H_ς_3ϖ^f_1  U / U = {ϖ^f_1 U }, so that  ξ_f_1 = ϖ^f_1·ϕ  =  ϕ_(1,-1).   
   

(b)  Recall that 𝔟_ς_3 =     (U  ϖ^(1,0,1,0) H_ς_3) +  (U ϖ^(1,-1,1,0)H_ς_3 )  +  (U ϖ^(1,0,0,1)H_ς_3). Let λ_1 =  (1,1,0,1), λ_2 = (3,3,1,2) and λ_3 =  (1,1,1,0). This 𝔟_ς_3,*(ϕ) = z_0·ξ_λ_1 + z_0^2·ξ_λ_2 + z_0·ξ_λ_3. From Lemma <ref>, we find that 
    H_ς_3ϖ^λ_1 U / U      = {ϖ^λ_1 U }
    
    H_ς_3ϖ^λ_2 U / U      =   {  (  [ ϖ^2 x ϖ;       1 ] , [ 1  ;   ϖ ]  ,  [ ϖ  ;   1 ] ) U        |    x ∈_F}
     
    H_ς_3ϖ^λ_3 U / U      =  {  (  [ ϖ  ;   1 ] , 
    [ ϖ y;   1 ]  ,
    [ 1  ; y ϖ ] ) U      |     y ∈_F} .
  
So H_ς_3ϖ^λ_i U / U is a singleton for i = 1 and a complete system of representatives for i = 2 (resp., i = 3) is given by  letting the parameter x (resp., y) run over []. One then  easily finds that ξ_λ_1 =  ϕ_(1,0), ξ_λ_2 =  ϕ_(2,0) - ϕ_(2,1) + ϕ_(3,1) and ξ_λ_3 =  q ϕ_(1,0). 



(c) Recall that 𝔠_ς_3 =  
(U  ϖ^(2,1,1,1) H_ς_3 ) + (U 
 ϖ^(2,0,2,0) H_ς_3). Let λ_1 = (0,0,0,0) and λ_2 =  (2,2,0,2). Then 𝔠_ς_3,*(ϕ) =    z_0·ξ_λ_1 + z_0 ^2·ξ_λ_2. Using Lemma <ref>,    we find that 
    H_ς_3ϖ^λ_1 U / U      = { ( [     1; x / ϖ     1 ] , 1 , 1  )  U }
    
    H_ς_3ϖ^λ_2 U / U      =   {  (  [ ϖ^2    ;       1 ] , [     1      ;   y ϖ ϖ  ^2 ]  ,  [ ϖ ^2  y ϖ;         1 ] ) U        |    x ∈_F}
 
So a system for representative cosets for H_ς_3ϖ^λ_1 U/ U (resp., H_ς_3ϖ^λ_2U/U is obtained by letting x (resp., y) run over [].  Using this, we compute that ξ_λ_1  = ϕ_(0,-1) - ϕ_(1,-1) + ϕ_(1,0)  and ξ_λ_2 =   ϕ_(2,0). 


Finally, we have 𝔥_ς_3 =    ( 1 + 2 ρ^2  +  ρ^4 )  𝔞_ς_3  - ( 1 + ρ^2 ) 𝔟_ς_3  + 𝔠_ς_3, so 

    𝔥_ς_3,  *(ϕ)      ≡  ( 1 + z_0)^2 ( z_0·ϕ_(2,0))  )  - ( 1 +  z_0 )  (  z_0^2 + z_0 ) ϕ_(2,0)  + z_0·ϕ_(1,0) )  +  (z_0^2+ z_0)  ϕ_(1,0)
        =   ( z_0 ( 1+ z_0)^2  - (1+z_0) (z_0^2 + z_0 )  ) ϕ_(2,0)  +   ( z_0^2 + z_0 - (1+z_0) z_0 )  ϕ_(1,0) 
        = 0
 
  







 §.§ Convolutions with restrictions of 𝔥_2

In this subsection, we compute the convolution  𝔥_ϑ,*(ϕ) for ϑ∈{ϑ_0, ϑ_1, 
ϑ_2, ϑ_3} ∪{ϑ̃_k  |   k ∈  []^∘}. These matrices are as follows: 

    ϑ_0 =  
    0.85(ϖ              1ϖ
       ϖ        1ϖ
            1      
             1ϖ
                1ϖ
                    1  
    )
    ,   ϑ_1 =  0.85(ϖ              1ϖ
            1      
       ϖ        1ϖ
             1ϖ
                    1  
                1ϖ
    ),   ϑ_2 =   0.85[  ϖ          1ϖ;     ϖ    1ϖ     1;        1    1ϖ;          1ϖ;             1ϖ;                 1 ]


    ϑ_3 =    0.85[    ϖ                  1ϖ;         ϖ        1ϖ        1ϖ;              1      1ϖ^2;                  1ϖ;                       1ϖ;                             1 ] ,    ϑ̃_k =    0.85[      ϖ                          1ϖ;            kϖ      1     kϖ              ;        (k+1)ϖ      1   k+1ϖ;                          1ϖ       ;                             -   1ϖ  k + 1;                                 1ϖ     -k ]

where k ∈ []^∘  = [] ∖{ -1 }. Recall that H _ϑ denotes the intersection H ∩ϑ K ϑ ^-1.    
     H_ϑ is a subgroup of U  for ϑ∈{ϑ_0, ϑ_1, ϑ_2, ϑ̃_k  |   k ∈ []^∘}. 

  Since θ =  ϑτ_2^-1∈ U and H_τ_2' ⊂ U by Lemma <ref>, we see that H_ϑ = H ∩θ H_τ_2' θ^-1⊂ U. 
 



 Let 𝒳_ϑ_0⊂ U denote the subgroup of all triples (h_1, h_2, h_3 )  where h_2=   [   1; 1   ] h_1[   1; 1   ].    

   





      H_ϑ_0 equals the product 𝒳_ϑ_0  U_ϖ^2. 
   
 Let h ∈ U and write h as in Notation <ref>. Then h∈ H_ϑ_4 if and only if 
    ϑ_0^-1 h ϑ_0 =  [           a        -c_1             b - c_1 ϖ^2 a - d_1 ϖ^2            ;          -c         a_1              a_1 - dϖ^2 b_1  - cϖ^2            ;                                 a_2                                 b_2;       c ϖ^2                                   d           c            ;                c_1  ϖ^2                     c_1         d_1            ;                                 c_2                                 d_2 ]∈ K.
 
It follows that 𝒳_ϑ_0, U_ϖ^2 are both contained in H_ϑ_0 and hence so is their product. If  h = (h_1, h_2, h_3 )  ∈ H_ϑ_0 is arbitrary, let γ = ( h_1^-1, h_2' , h_3) where h_2' =     [   1; 1   ] h_1^-1[   1; 1   ]. Then γ∈𝒳_ϑ_0 and γ h = ( 1, h_2'h_2 , 1 )  ∈ H_ϑ_0 and it is easily seen from the matrix formula above (applied to γ h in place of h) that γ h ∈ U_ϖ^2.  Thus h = γ^-1·γ h ∈𝒳_ϑ_0  U_ϖ^2 which  establishes the reverse   inclusion.  
  




  
Modulo q - 1, we have


 

  * [U H_ϑ_0]_*(ϕ) =  ϕ,


  * [U ϖ^(3,2,1,2)  H_ϑ_0]_* (ϕ) 
≡ 2 (z_0^2 + z_0)   ϕ , 

  * [U ϖ^(4,2,2,3)  H_ϑ_0]_* (ϕ) ≡ 2 z_0^2·ϕ,  

  * [U ϖ^(4,3,1,2)     H_ϑ_0]_* (ϕ) ≡  (z_0 ^3 + z_0 )  ·ϕ.  
 


and 𝔥_ϑ_0, *(ϕ) = 𝔥_ϑ_1,*(ϕ)  ≡ 0.  
    
 Part (a) is clear since H_ϑ_0⊂ U. Let λ∈Λ be such that dep(λ) ≤ 2. Then  Lemma <ref> implies that  H_ϑ_0ϖ^λ U = 𝒳_ϑ_0ϖ^λ U. Let us denote  𝐏  : = _2(F) × _F^×_2(F) and let  P, P^∘ denote  the  groups of F, _F-points of 𝐏 respectively.   Consider the embedding 
    : 𝐏↪𝐇 ,         (h_1, h_2) ↦ (h_1,  h_1, h_2 )
 where = [   1; 1   ]. Then  identifies  P^∘ with 𝒳_ϑ_0. If λ = (a,b,c,d) satisfies b = a - c, then we also have ϖ^λ∈(P) and we write ϖ^(a,b,d)∈ P for the pre-image.  Then  
    P^∘ϖ^(a,b,d) P^∘ / P^∘→𝒳_ϑ_0ϖ^λ   U / U ,       γ P ↦(γ) 
     U
 
is a bijection.  It follows  λ = (a,b,c,d) satisfying b = a - c and with dep(λ) ≤ 2, we have   
    [  H_ϑ_0ϖ^λ U ]  (ϕ)  =     |  U_1\  U_1ϖ^(a,d) U_1 |  ·    T_b,a-b,*(ϕ) .
 
Parts (b), (c), (d)  are then easily obtained using  Corollary <ref> and the formula above.   
Now recall that 

    𝔥_ϑ_0     =   ρ^2(1 + 2ρ^2 +  ρ^4)  (U H_ϑ_0 ) -  ( 1 + ρ^2)     (U  ϖ^(3,2,1,2)  H_ϑ_0 )  +   ( U ϖ^(4,2,2,3) H_ϑ_0)  + ( U ϖ^(4,3,1,2)H_ϑ_0) .
 
So putting everything together, we have 
    𝔥_ϑ_0,*(ϕ)    ≡ (   z_0(1+2z_0+z_0^2) - (1 + z_0)(2z_0^2+ 2z_0)   + 2z_0^2 + ( z_0^3 + z_0)   ) ϕ = 0
 
Since 𝔥_ϑ_1 =  w_2𝔥_ϑ_0 w_2 and conjugation by w_2 only swaps the second and third components of H, we obtain the equality 𝔥_ϑ_0, * (ϕ) = 𝔥_ϑ_1,*(ϕ).  
  


    Let A_ϑ_2 = A ∩ϑ_2 K ϑ_2^-1 and U_ϖ^2 the subgroup of all elements in U that reduce to identity   modulo ϖ^2. We let M_ϑ_2 be the subgroup of all triples h = (h_1,h_2,h_3) ∈ U such that  
    h_1 = [ 1; x 1 ],    h_2 = [ 1 y;   1 ] ,     h_3  =   [     1; y - x     1 ]
 
where x , y ∈_F satisfy x - y ∈ϖ_F. We define N_ϑ_2 to be the two parameter  subgroup of triples (h_1, h_2,  h_3) given by 
  
    h_1 = [   1 x ϖ;       1 ] ,       h_2 = 
    [    1; x  ϖ    1 ],        h_3 =   [ 1 y;   1 ]
  
where x , y ∈_F are arbitrary. 
 



        H_ϑ_2  =M_ϑ_2 N_ϑ_2  A _ϑ_2  U_ϖ^2 =  N_ϑ_2 M_ϑ_2 A _ϑ_2 U_ϖ^2.  



 That M_ϑ_2, N_ϑ_2 , U_ϖ^2 are subgroups of H_ϑ_2 is easily verified by checking that their conjugates by ϑ_2^-1 are in K , so H_ϑ_2 contains the product. If  h  ∈ H_ϑ_2⊂ U is arbitrary, then  
    ϑ^-1_2 h ϑ_2 =   [                 a             - c_1                            b-c_1ϖ^2         a-d_1 ϖ^2            -c_1 ϖ;                -c               a_1            -c_2 ϖ         a_1-d ϖ^2 b_1 - c - c_2 ϖ^2       a_1 - d_2 ϖ;                            -   c_1ϖ               a_2            - c_1ϖ     a_2   - d_1 ϖ          b_2 -c_1;             c ϖ^2                                                     d                 c                  ;                             c_1 ϖ^2                                 c_1               d_1              c_1ϖ;                                                   c_2                                c_2ϖ               d_2 ]∈   K.

From the matrix, we see that b, c_1, c_2, b_1 - c, a - d_1∈ϖ_F. In particular, a, a_1, a_2,  d , d_1,  d_2∈_F^×. Let m ∈ M_ϑ_2  be defined with x = -c/a, y = -b_1/d_1 (see Notation <ref>). Then h' = mh satisfies b_1' =  c' = 0. Then c_2' ∈ϖ^2_F. If we define n ∈ N_ϑ_2 
with x = -b'/d'ϖ, y = -b'_2/d', we find that h” satisfies  b_1” =  c” = 0 (inherited from h') and  b” = b_2” = 0. The latter condition   forces  c_1”∈ϖ^2_F. Now h”  clearly lies in the product A_ϑ_2 U_ϖ^2 which proves the first equality. The second follows similarly by first using  N_ϑ_2 to make the entries b, b_2 in h zero.      
  




  Modulo q - 1,  we have 


2  

  * [ U H   _ ϑ_2 ] _*( ϕ ) = ϕ, 
 

  * [U ϖ^(3,2,1,2)  H_ϑ_2]_* (ϕ) ≡      (z_0^2 + z_0 )   ϕ    -  ϕ̅_(1,2), 

  * [U ϖ^(3,1,2,1)  H_ϑ_2]_* (ϕ) ≡ϕ̅_(1, 2 ),  

  * [U ϖ^(3,1,2,2)     H_ϑ_2]_* (ϕ)  =  ϕ̅ _  ( 1, 2 ),  

  * [U ϖ^(4,2,2,3)     H_ϑ_2]_* (ϕ) ≡    z_2^2·ϕ,

  * [U ϖ^(4,1,3,2)     H_ϑ_2]_* (ϕ) ≡    ( z_0 + 1 )·ϕ_(1, 2)  -  z_0^2·ϕ 
  


and 𝔥_ϑ_2, * (ϕ)  = 𝔥_ϑ̃_0, * ( ϕ )   ≡  0. 
    
 
 Part (a) is immediate since H_ϑ_2⊂ U.  For λ∈Λ, let ξ_λ = [H_ϑ_2ϖ^λ U](ϕ). If λ depth at most 2, then  H _ϑ_2ϖ^λ U / U = M_ϑ_2 N_ϑ_2ϖ^λ U/  U  by Lemma  <ref>. If moreover λ  has depth one and  β_2(λ) ≤ 0, then we also have  M_ϑ_2 N_ϑ_2ϖ^λ U = M _ϑ_2ϖ^λ U. Similarly if α_0(λ), β_2(λ) ≥ 0 and β_0(λ) ≤ 0,  then   H_ϑ_2ϖ^λ U / U = N_ϑ_2ϖ^λ U / U. 
 

(b) We need to compute  z_0^2·ξ_λ where λ = (1,0,1,0). Then dep(λ) = 1 and β_2(λ) = -1, so   H_ϑ_2ϖ^λ U/U = M_ϑ_2ϖ^λ U /  U.  It is then easily seen that the quotient M_ϑ_2 / M_ϑ_2∩ϖ^λ U ϖ^-λ has cardinality q with representatives given by elements with parameters x = y running over [] (see Notation <ref>). From this,  one finds that  ξ_λ = ϕ - ϕ_(1,0) +  q  ϕ_(1,1).    
  








(c) We need to compute z_0^2·ξ_λ where  λ = (1,1,0,1).  Here α_0(λ) = β_2(λ) = 1 and β_0(λ) = - 1, so H_ϑ_2ϖ^λ U / U = N_ϑ_2ϖ^λ U /  U. This  coset space has  cardinality q and a set of representatives is γϖ^λ where γ∈  N_ϑ_2 runs over elements defined with x = 0 and y ∈ [] (see Notation <ref>). So ξ_λ = q ϖ^λ·ϕ =     q  ϕ_(1,0).   


(d) If λ = -(3,1,2,2), then H_ϑ_2ϖ^λ U / U = M_ϑ_2ϖ^λ U / U as in part (b) and its easy to see that this equals ϖ^λ U/U. So ξ_λ = ϖ^λ·ϕ = ϕ̅_(1,2).   
     


(e)  We need to compute z_0^3·ξ_λ where λ =  (2,1,1,0).  As the first and second  components of ϖ^λ are central and β_2(λ) = - 2 < 0, we see that H_ϑ_2ϖ^λ U / U =  M_ϑ_2ϖ^λ U / U. From the structure of M, we see that a set of 
 representatives is given by γϖ^λ where γ =   (1,1, [   1; ϖ z   1 ] ) and z running over []. So ξ_λ = q ϖ^λ·ϕ = z_0^-1·ϕ.  



(f) This equals z_0^3·ξ_λ where λ = ( 2,2,0,1).   Then  H _ϑ_2ϖ^λ U / U = N_ϑ_2ϖ^λ U / U.  A set of representatives for this quotient is γϖ^λ where γ runs over elements of N_ϑ_2  defined with y = 0 and x ∈ []. From this, one calculates that ξ_λ vanishes on ( X ∖ X_1,0 ) ∪ ( X_1,1∖  X_2,1), takes value one on X_1,0∖ X_1,1 and q on X_2,1. So ξ_λ = ϕ_(1,0) - ϕ_(1,1)   +  q ϕ_(2,1) and z_0^3·ξ_λ = ϕ̅_(2,3) - z_0^2ϕ + q ϕ̅_(1,2).  
 

Now recall that 
    𝔥_ϑ_2    = ρ^2( 1  + 2 ρ^2 +  ρ^4 )  (  U H_ϑ_2  )   - ( 1 + ρ^2 ) (     (  U    ϖ^(3,2,1,2)   H _ϑ_2  )  +   ( U   ϖ^(3,1,2,1)   H _ϑ_2  )  +    ( U    ϖ^(3,1,2,2)   H _ϑ_2  )  )  
         +  ( U ϖ^(4,2,2,3)  H_ϑ_2 )   + ( U ϖ^(4,1,3,2)  H _ ϑ_2  )

By  parts (a)-(f), we see that 

    𝔥_ϑ_2,*(ϕ)    ≡   z_0(1+z_0)^2·ϕ - (1+z_0)  (  (z_0^2 + z_0) ·ϕ - ϕ̅_(1,2)  + ϕ̅_(1,2)  +  ϕ̅_(1,2)  ) +     
             z_0^2·ϕ + ( z_0+1) ·ϕ_(1,2) - z_0^2·ϕ
         =   (  z_0(1 + z_0)^2 - ( 1 + z_0) (z_0^2 + z_0 )   ) ·ϕ   - ( 1 + z_0) ϕ̅_(1,2) + (1 + z_0) ·ϕ̅_(1,2)
         = 0

modulo q -  1.  Since 𝔥_ϑ̃_0 is the conjugate of 𝔥_ϑ_2 by w_2 w_3 and this only affects the second and third components of H, we see that 𝔥_ϑ̃_0 (ϕ) = 𝔥_ϑ_2 (ϕ). This completes the proof.  
   






         Let I_ϑ_3⊂ U denote the subgroup of triples (h_1, h_2, h_3 ) such that modulo ϖ^2, h_1 reduces  to   a  lower triangular matrix and h_2, h_3 reduce to upper triangular matrices. Then H_ϑ_3⊂  I_ϑ_3.  
 
 Write h ∈ H_ϑ_3 as in Notation <ref>. Then 
    ϑ_3 ^-1 hϑ_3  =      
    [                  a                  *                           -b - c_1ϖ^2                  *            -c_1ϖ^2;                  *                a_1            -c_2ϖ^2                  * b_1- cϖ^2 - c_2ϖ^4                  *;                                     *                a_2                  *                  *      b_2 -  c_1ϖ^2;                  *                                                        d                  c                   ;                                     *                                     *                d_1                  *;                                                        *                                     *                d_2 ]∈ K

Since all entries of this matrix must be integral, it is easily seen that  h ∈ U and  that  b , c_1, c_2∈ϖ^2_F. 
 


          We have 
 

  * [U ϖ^(3,1,2,2) H_ϑ_3]_* (ϕ)  = ϕ̅_(1,2) 

  * [U ϖ^(4,2,2,3)  H_ϑ_3]_* (ϕ)  = ϕ̅_(2, 2)  

  * [U ϖ^(4,1,3,2)  H_ϑ_3]_* (ϕ)  =  ϕ̅_(1, 3)  
    
and 𝔥_ϑ_3, * ( ϕ )  = -  [ ϖ ^-1_F^×;  ϖ^-2_F^× ]. 
  
 For λ∈Λ, let ξ_λ denote [ H  _   ϑ_3ϖ^λ U ]  ( ϕ ).  If  each of α_0(λ) , -β_0(λ) , -β_2(λ) lies in { 0 ,1 ,2 }, then ϖ^-λ I_ϑ_3ϖ^λ⊂ U.  So for such λ,  H _ ϑ_3ϖ^λ U  = ϖ^λ U and so ξ_λ = ϖ^λ·ϕ.  Parts (a), (b), (c) then follow immediately.  Now recall that 

    𝔥_ϑ_3  =  - ( 1 + ρ^2 ) (U ϖ^(3,1,2,2) H_ϑ_3 )  +    ( U ϖ^(4,2,2,3)   H _ϑ_3 )   +  (U ϖ^(4,1,3,2)   H _ϑ_3 )  .
 
Using parts (a)-(c), we find that   
Therefore   
    𝔥_ϑ_3, * ( ϕ )  
         = - ( 1 + z_0  ) ϕ̅ _(1,2)  
     + ϕ̅_(2,2)  + ϕ̅_(1,3)
        = ϕ̅_(2,2) - ϕ̅_(1,2)  + ϕ̅_(1,3) - ϕ̅_(2,3)
        =   -  [ ϖ ^-1_F^×;    ϖ^-2_F ]  +  [ ϖ ^-1_F^×;    ϖ^-3_F ]
          =  -  [ ϖ ^-1_F^×;  ϖ^-2_F^× ]
  
    
 For k ∈ []  ∖{ 0 , -  1 }, let 𝒳̃_k⊂ U denote the subgroup of all triples (h_1, h_2, h_3 ) where 
    h_1 = [ a b; c d ] ,    h_2 = [        d    - c k; -  b / k        a ] ,     h_3 =  [              d    c ( k + 1 ); b / ( k +  1 )              a ] .

That is, h_1∈_2(_F) is arbitrary and h_2, h_3 are certain conjugates of h_1 by anti-diagonal matrices.    Recall that U_ϖ denotes the subgroup of U which reduces to the trivial group  modulo ϖ.  
    
 For k ∈ [] ∖{ 0 , -1 }, H_ϑ̃_k is   equal to  the product of 𝒳̃_k with U _ϖ∩ H_ϑ̃_k.   
 

 
It is  straightforward to verify that 𝒳̃_k⊂ H_ϑ̃_k by checking that the matrix ϑ̃_k^-1𝒳̃_kϑ̃_k has all its entries integral. This implies that the reduction of H_ϑ̃_k modulo ϖ contains the reduction of  𝒳̃_k modulo ϖ. Thus  H_ϑ̃_k contains the product 𝒳̃_k·   ( U _ϖ∩ H_ϑ̃_k).   For the reverse inclusion,  write h ∈ H_ϑ̃_k as in Notation <ref>. 
Then 
    ϑ̃_k^-1  h ϑ̃_k  =    
    [                             a                             *                             *   b - c_1 k^2 - c_2(k+1)^2ϖ^2 a + d_1 k - d_2   ( k+1)  ϖ^2                             *;                            -c                             *                      a_2-a_1ϖ a_2 ( k + 1 ) - a_1 k  -d ϖ^2               b_1 + b_2 - c ϖ   -   b_2 k  + b_1 ( k+1)   ϖ;                                                           *                             *                             *                             *                             *;                                                                                                                       d                             c                                                            ;                                                           *                             *                             *                             *                             *;                                                           *                             *             c_1k + c_2 (k+1)ϖ                    d_2 - d_1ϖ                             * ]∈ K
  
As the displayed entries must be integral (and the entries of h are also integral by Lemma <ref>), one easily deduces all the congruence conditions on entries of h for its reduction to lie in the reduction of 𝒳̃_k.  For instance, we have b_2 k ≡ -b_1 (k+1) and b_1 + b_2≡ c modulo ϖ, which implies  that b_1≡ - ck. 
    

 Modulo q - 1,  
  

  * [U H_ϑ̃ _k ]_*(ϕ ) =  ϕ,  

  * [ U ϖ^(3,2,1,1) H_ϑ̃_k]_*(ϕ)  ≡ (z_0^2 + z_0 ) ·ϕ 
 and 
𝔥_ϑ̃_k,*(ϕ)  ≡ 0 for all k ∈ [] ∖{ 0 , -1  }.  
   

 Part (a) is trivial since H_ϑ̃_̃k̃⊂ U. For part (b), let λ = -(3,2,1,1). Then dep(λ) = 1 and so H_ϑ_2ϖ^λ U = 𝒳̃_kϖ^λ U. An   argument analogous to Proposition  <ref> shows that there is a bijection  
    U_1ϖ^-(3,2)   U_1  /  U_1→𝒳̃_kϖ^λ U / U
  
(where U_1 = _2(_F))  using which one obtains the equality [H_ϑ̃_kϖ^λ U](ϕ)  =  𝒯_2,1,*(ϕ). Corollary  <ref> then implies the claim. Now recall that 

    𝔥_ϑ̃_k       =  ρ^2 (1 + 2ρ^2 + ρ^4 ) ( U H_ϑ̃_k  ) - ( 1 + ρ^2 )   (   U ϖ^(3,2,1,1) H_ϑ̃_k)  .

So   
𝔥_ϑ̃_k,*(ϕ) ≡ (   z_0(1+z_0)^2ϕ - (1+z_0) ( z_0^2+ z_0)  )  ϕ   = 0. 
    
 
amsalpha    
  
