Gamma factors of level zero supercuspidal representations
Chang Yang

	
	
	
	
	
	
	
	Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP)
 College of Mathematics and Statistics 
 Hunan Normal University 
Changsha,  410081
China
	cyang@hunnu.edu.cn
		
	
	
	
	[2010]22E50, 33C10
	
	
	
	
	
		We give an explicit formula for the twisted gamma factor for a pair of irreducible supercuspidal representations of level zero. We also obtain an explicit formula for the unramified base change of level zero supercuspidal representations.
	
	
	Gamma factors of level zero supercuspidal representations
    Chang Yang
    December 30, 2023
=========================================================



§ INTRODUCTION

	
	

 §.§ Main result and Motivation

	Throughout this note, let F be a p-adic field with residue field of q elements. Denote by 𝔬 and 𝔭 the ring of integers of F and the maximal ideal of 𝔬.
	
	 Let π and τ be irreducible smooth representations of _n(F) and _m(F). Let ψ be a nontrivial character of the additive group of F. The gamma factor γ (s,π×τ,ψ) was first introduced in <cit.>, along with an L-function L(s,π×τ) and a local constant ε(s,π×τ,ψ).  
	
	It is desirable to have explicit formulae for these gamma factors. By <cit.>, the computation can be reduced to the case of supercuspidal representations. When τ=1_F is the trivial representation of _1(F), then γ(s,π× 1_F,ψ) is nothing but the gamma factor γ(s,π,ψ) in the sense of Godement-Jacquet <cit.>. There are explicit formulae for γ(s,π,ψ) in terms of “non abelian Gauss sums” (See <cit.>, <cit.>; also <cit.> for the _2 case). On the other hand, in the twisted case, one has only an explicit formula for the conductor of the local constants ε(s,π×τ,ψ) (See <cit.>). 
	
	In this short note, we confine ourselves to irreducible supercuspidal representations of level zero. A smooth representation π of _n(F) is called of level zero if it contains a nontrivial fixed vector for the subgroup 1+𝔭M_n(𝔬) of _n(𝔬). Our main result is
     
     	Let π resp. τ be an irreducible supercuspidal representation of level zero of _n(F) resp. _m(F), with n>m. Let ψ be a character of F of levle one, that is, ψ is trivial on 𝔭 but not on 𝔬. Then
     	
    γ (s,π×τ,ψ) = (-1)^nm-(n,m)q^-nm/2∏_i=0^(n,m)-1 G(η_π^q^i∘_[n,m]:n·η_τ∘_[n,m]:m,ψ̃).

     Here
     
     	
  ∙ (n,m) is the greatest common divisor of n and m;
     	
  ∙ η_π resp. η_τ is the regular character of _q^n^× resp. _q^m^× that corresponds to π̃ resp. τ̃ via Green's parameterization (see <ref>, (<ref>)), while π̃ resp. τ̃ is the cuspidal representation of _n(_q) resp. _m(_q) which is uniquely determined by π resp. τ (see <ref>, (<ref>));
     	
  ∙ _[n,m]:n resp. _[n,m]:m is the norm map from _q^[n,m]^× to _q^n^× resp. _q^m^×;
     	
  ∙ ψ̃ is the character on κ_F ≅_q which is induced by ψ, ψ̃(x̅)=ψ(x) for all x∈𝔬. 
      
     and 
      
    G(β,φ) := ∑_a ∈_q^N^×β(a) φ ( a^-1) = ∑_a ∈_q^N^×β^-1(a) φ ( a)

     is the Gauss sum for a multiplicative character β of _q^N^× and an additive character φ of _q, where  is the trace map from _q^N_q.
     
 
	The motivation of our work arises from a question on gamma factors over finite fields that was proposed in <cit.>. A formula similar to (<ref>) for gamma factors over finite fields is given in Theorem <ref>. Let us add some words on the strategy of our method. We first use a result in <cit.> to reduce the general case to the case m=1. The main point of our work is then to determine the base change, in the sense of <cit.>, of a level zero supercuspidal representation. We use the explicit local Langlands correspondence for level zero representations, developped in <cit.>, to transfer the question to the Galois side. When m=1, the computation is a consequence of a result on gamma factors over finite fields <cit.> and a connection formula between gamma factors for level zero supercuspidal represenations and gamma factors over finite fields <cit.>.
	
   
    

 §.§ Notations and Conventions

   
    (1) For a smooth representation π of _n(F), denote by π^∨ the smooth dual of π. If π is irreducible, denote by ω_π the central character of π. For representations π_i of _n_i(F), i=1,⋯,r, denote by π_1 ×⋯×π_r the representation of _n_1+⋯+n_r(F) obtained from π_1⊗⋯⊗π_r by normalized parabolic induction. For a representation π and a character χ of _n(F), denote χπ to be the representation on the space of π given by (χπ)(g)=χ(g)π(g). Let |·| denote the normalized absolute value on F and ν denote the character |·|∘ of _n(F).
    
    (2) Denote by U the group of units in 𝔬, by κ = 𝔬/𝔭 the residue field of F. We fix a uniformizer ϖ in 𝔭.
    
    
     If E/F is a finite field extension, we use the analogous notations 𝔬_E, 𝔭_E, U_E, etc. The norm map E^× F^× is denoted _E/F, and the trace E  F is _E/F.
     
    We fix a separable algebraic closure F̅ of F. All finite extension of F are supposed to be contained in F̅. Hence an unramifed extension over F of a fixed degree is unique. For E/F a finite unramified extension, we still use ϖ as a uniformizer in 𝔭_E.
    
    For a finite field extension K/F, denote by _K the Weil group of K. We will view _K as a subgroup of _F as in <cit.>. Let a_F _F  F denote the Artin reciprocity map (<cit.>). We fix a geometric Frobenius element Φ in _F and a arithmetic Frobenius element ϕ = Φ^-1.
    
    If E/F is a finite extension, we write _E/F rather than _𝒲_E^_F for the functor of smooth induction from _E to _F. Also, if ρ is a smooth representation of _F, we put _E/Fρ = ρ |__E.
    
  
  

§ PRELIMINARIES

    

 §.§ Gamma factors for Weil representations
 
    
     Let 𝒢^ss_n(F) denote the set of isomorphism classes of semisimple smooth representations of _F of dimension n and write 𝒢^ss(F) = ⋃_n ⩾ 1𝒢^ss_n(F)[In this note, we do not need the nilpotent part in Weil-Deligne representations.]. For σ∈𝒢^ss(F) and a nontrivial character ψ_F of F, let L(s,σ) be the Artin L-function and ε(s,σ,ψ_F) be the local constant defined by Langlands and Deligne (see <cit.>). Denote by σ^∨ the contragradient representation of σ. Similar to the definition of local gamma factors in <cit.>, we define
   
    γ(s,σ,ψ_F) = ε(s,σ,ψ_F) L(1-s,σ^∨)/L(s,σ).

   
   Suppose that σ_1,σ_2 ∈𝒢^ss(F). By the additive properties of L and ε (<cit.>), we have
   
    γ(s,σ_1⊕σ_2,ψ_F)    = γ(s,σ_1,ψ_F) γ(s,σ_2,ψ_F).

   
   Let K/F be a finite field extension and ρ∈𝒢^ss_n(K). Note that (_K/Fρ)^∨≅_K/Fρ^∨. By the inductive properties of L and ε (<cit.>), we have 
   
    γ(s,_K/Fρ,ψ_F)/γ(s,ρ,ψ_K)   = γ(s,_K/F 1_K,ψ_F)^n/γ(s,1_K,ψ_K)^n,

   where 1_K stands for the trivial representation of _K and ψ_K=ψ_F∘_K/F.
   
   	The functions L and ε can be defined for Weil-Deligne representations of F (see <cit.>), so is γ by (<ref>). It turns out that γ(s,(σ,n),ψ_F) = γ (s,σ,ψ_F), where we denote by (σ,n) a Weil-Deligne representation and n is the nilpotent part.
   
   
      

 §.§ Local Langlands correspondence for _n(F)

      Let 𝒜_n(F) be the set of isomorphism classes of irreducible smooth representations of _n(F) and 𝒜^0_n(F) its subset consisting of supercuspidal representations. Let 𝒢_n(F) be the set of isomorphism classes of semisimple Weil-Deligne representations of F of dimension n. Denote by 𝒢^0_n(F) the set of isomorphism classes of irreducible smooth representations of _F of dimension n. We identify 𝒢^0_n(F) as a subset of 𝒢_n(F) by letting the nilpotent part be 0.
   
   The Langlands correspondence for _n over F, proved by Harris-Taylor <cit.> and by Henniart <cit.>, is the unique collection of bijections 
   
    _n^F 𝒜_n ( F ) 𝒢_n ( F )

   such that the map _1^F is given by the local class field theory, and that for π∈𝒜_n(F), π' ∈𝒜_n' (F), we have
   
    L ( s, π×π')     =  L (s, _n^F(π) ⊗_n'^F ( π') ), 
    ε ( s, π×π' , ψ)     =  ε (s, _n^F(π) ⊗_n'^F ( π'),ψ).

   Hence
   
    γ(s,π×π',ψ) = γ(s, _n^F(π)⊗_n^F(π'),ψ).

   
   The maps {_n^F} are first constructed for 𝒜^0_n(F) and then extended to 𝒜_n (F), using the classification of Langlands, Bernstein and Zelevinsky. We recall briefly this process (see <cit.>). According to <cit.>, each essentialy square-integrable representation π is of the form St_k(ρ), for some k ≥ 1 and ρ∈𝒜^0_n(F), where St_k(ρ) denotes the unique irreducible quotient of ρ×νρ×⋯×ν^k-1ρ. We have 
   
    _n^F (St_k (ρ)) = _n^F(ρ) ⊗Sp_k,

   where Sp_k denotes the special Weil-Deligne representation of _F of dimension k (<cit.>). Denote by 𝒜^□_n(F) the isomorphism classes of essentially square-integrable representations. For every π∈𝒜_n^□(F), there is a unique α(π ) ∈ such that ν^-απ is unitary and square-integrable. Let r ≥ 1 and π_1,⋯,π_r be elements of 𝒜^□_n(F). Suppose that
   
    α(π_1) ≥⋯≥α(π_r). ⋆

   Then the representation π_1 ×⋯×π_r has a unique irreducible quotient J(π_1,⋯,π_r). 
   By the Langlands classification (<cit.>), each π∈𝒜_n(F) is of the form J(π_1,⋯,π_r) for some r ≥ 1 and π_1,⋯,π_r ∈𝒜^□_n(F). The isomorphism class of J(π_1,⋯,π_r) does not depend on the order of π_i provided that the conditon (<ref>) is satisfied. We have
   
    _n^F (J(π_1,⋯,π_r)) = _n_1^F(π_1) ⊕⋯⊕_n_r^F(π_r).

   
    

§ BASE CHANGE AND AUTOMORPHIC INDUCTION

    
   
    A key ingredient in our method is to use a formula on the twisted local gamma factors which compares base change and automorphic induction.
   		
    Let K/F be a cyclic extension. Recall that, in <cit.>, base change associates to every π∈𝒜^0_n(F) an irreducible smooth representation π_K/F of _n(K)[The definition of π_K/F was extended to all π∈𝒜_n(F) in <cit.>, but we do not need this fact in our context.]. Via the Langlands correspondence, the base change from F to K corresponds to the restriction to _K from representations of 𝒲_F (<cit.>). So, we have
   
    _K/F(l^F_n (π)) =  l^K_n ( π_K/F ).

   
    Assume that K/F is cyclic of degree d. According to <cit.>, automorphic induction sends any generic ρ in 𝒜_n(K) to a unique ρ^K/F∈𝒜_nd(F) (see also <cit.>). The automorphic induction from K to F corresponds, again via the Langlands correspondence, to the induction to _F from representations of _K, i.e., 
    
    _K/F(_n^K (ρ)) = _nd^F( ρ^K/F),

    where ρ is generic (<cit.>).
   
   
   
    The following result can be found in <cit.>. We supply here a “proof” different than that in loc.cit.
    
    	Assume K/F to be a cyclic extension. Let π∈𝒜^0_n(F) and ρ∈𝒜^0_n'(K) with ρ generic. Let ψ_F be a nontrivial character of F. Then
    	
    	
    γ (s, π×ρ^K/F,ψ_F)/∏_∈γ (s,χ,ψ_F)^nn'  =   γ (s,π_K/F×ρ ,ψ_K)/γ (s,1_K,ψ_K)^nn',

    	where  is the group of characters of F^× that is trivial on _K/F(K^×) and ψ_K := ψ_F ∘Tr_K/F.
    
    
    
    	We omit the subscripts in the _n^F.
    	
    γ (s,π×ρ^K/F,ψ_F)     = γ (s,(π)⊗(ρ^K/F),ψ_F)   
       =γ(s,(π)⊗_K/F(ρ),ψ_F) 
       =γ (s,_K/F(Res_K/F(π) ⊗(ρ )),ψ_F).

    	In view of (<ref>), we have
    	
    γ (s,_K/F(Res_K/F(π) ⊗(ρ )),ψ_F)   
       =   γ(s,Res_K/F(π) ⊗(ρ),ψ_K) ·γ (s,_K/F1_K,ψ_F)^nn'/γ(s, 1_K,ψ_K)^nn'
       =γ (s,π_K/F×ρ ,ψ_K)γ (s,_K/F1_K,ψ_F)^nn'/γ(s, 1_K,ψ_K)^nn'.

    	Note that, by local class field theory,  corresponds to the group of characters of _F that is trivial on _K. The equality (<ref>) follows then from the local Langlands correspondence (<ref>) and the simple fact that
    	
    _H^G  1_H = ⊕_χ∈G/Hχ,

    	where H is a normal subgroup of G of finite index such that G/H is abelian, 1_H is the trivial representation of H and G/H is the group of characters of G/H.
    
    
    	Strictly speaking, this is not really a proof, as the formula (<ref>) predates and is used to establish the local Langlands correspondence for _n(F), while we use the Langlands correspondence in the “proof”. The original proof in <cit.> involves a global method.
    
    
    

§ GAMMA FACTORS OVER FINITE FIELDS

    
    

 §.§ Baisc facts

    In her thesis <cit.>, Roditty considered a finite field analogue of Rankin-Selberg convolution in the same way as in <cit.>. There she defined the gamma factor (a complex number in this case, instead of a function!) as the ratio appeared in certain functional equations. 
    
    We fix a nontrivial character ψ of _q in this subsection. Let U_n(_q) be the group of upper triangular unipotent matrices in _n(_q). Let 
    
    ψ_n(u) = ψ (∑_i=1^n-1 u_i, i+1),  for u=(u_ij)∈U_n(_q).

    If π is a generic representation of _n(_q), denote by 𝒲(π,ψ_n) the Whittaker model of π with respect to ψ_n. We refer to <cit.> for the undefined notations in this subsection.
    
    
    	Let π be an irreducible cuspidal representation of _n(_q) and τ an irreducible generic representation of _m(_q), with n > m. Then there exists a complex number γ (π×τ,ψ) such that
    	
    γ(π×τ,ψ) q^km∑_ g ∈U_m \_m(_q)∑_x ∈M_n-m-1-k,m W_π( ( 
        		 g     0     0 
     x     I_n-m-1-k    0 
     0     0    I_k+1) ) W_τ ( g )  
       =    ∑_ g ∈U_m \(_q)∑_y ∈M_m,k W_π( ( 
        		0     I_n-m-k    0 
     0     0     I_k 
     g     0     y
        		)  ) W_τ ( g ),

    	for all 0 ≤ k ≤ n-m-1, W_π∈𝒲(π, ψ_n) and W_τ∈𝒲(τ, ψ_m^-1).
    
    
    There is a distinguished element in the Whittaker model that can be used to compute the gamma factors.
    
    	Let π be an irreducible generic representation of _n(_q) with χ_π its (trace) character. The functon
    	
    J_π,ψ_n (g) = |U_n(_q)|^-1∑_ u ∈U_n(_q)χ_π(gu) ψ_n(u^-1)

    	lies in 𝒲(π,ψ_n) and satisfies
    	
    J_π,ψ_n(u_1 g u_2) = ψ_n(u_1u_2)J_π,ψ_n(g), u_i ∈U_n(_q)   and   J_π,ψ_n(I_n)=1.

      
    The function J_π,ψ_n is called the (normalized) Bessel function of π with respect to ψ_n. In terms of Bessel functions, we have the following
     
    	Let π be an irreducible cuspidal representation of _n(_q) and τ an irreducible generic representation of _m(_q), n>m. Then
    	
    γ(π×τ, ψ)  =  ∑_g ∈U_m \_m(_q) J_π,ψ_n( (  
        	0     I_n-m
     g     0
        	) ) J_τ, ψ_m^-1 (g).

    

    

 §.§ n × 1 gamma factors

    We recall Green's parameterization <cit.> for irreducible cuspidal representations of _n(_q). A character η of _q^n^× is called _q-regular if the conjugates η^s, s∈(_q^n/_q) are distinct. Two characters η_1 and η_2 are called equivalent if η_1 = η_2^q^j for some integer j, i.e., they are in the same Frobenius orbit. 
    
    
    	A character η is _q-regular if and only if η cannot factor through __q^n / _q^d for some subextension _q^d/_q.
    
   
     Let Λ_n(_q) denote the set of equivalence classes of _q-regular characters of _q^n^×. Let 𝒜_n^0(_q) denote the set of isomorphism classes of irreducible representations of _n(_q).  Green's parameterization gives a bijection
    
    𝒜_n^0(_q)    ⟷Λ_n(_q)  
    π   ⟷η_π

    The (trace) character χ_π of π is connected with η_π in an explicit way (see <cit.>). So, by (<ref>), the Bessel function J_π,ψ_n can be expressed in terms of η_π. 
    
    In view of (<ref>), Nien computes the n× 1 gamma factor as an “abelian Gauss sum" using special values of Bessel functions. Denote by _q^× the set of characters of _q^×.
    
    	Let π be an irreducible cuspidal representation of _n(_q), n ≥ 2 and τ∈_q^×. Then
    	
    γ ( π×τ ,ψ) = (-q^-1τ (-1))^n-1 G(η_π·τ∘_n:1,ψ).

    	where η_π is the regular character of _q^n^× which corresponds to π by Green's parameterization; the Gauss sum G is as defined in (<ref>).
    
    
    

  
    
    

 §.§ Connection with local gamma factors over p-adic fields

    
    We recall a result of Nien and Zhang which shows that gamma factors over finite fields and gamma factors for level zero supercuspidal representations over p-adic fields are closely related.
    
    A representation π∈𝒜^0_n(F) is called of level zero if it contains a nontrivial fixed vector for the subgroup 1+𝔭M_n(𝔬) of _n(𝔬). According to <cit.>, every level zero supercuspidal representation is of the form 
    
    π≅_F^×_n(𝔬)^_n(F)χσ,

    where σ is a representation of _n(𝔬) that is inflated from an irreducible cuspidal representation σ of _n(_q)=_n(𝔬/𝔭), χ is a character of F^× such that χ |_U_F equals to the central character of σ and  is the compact induction. 
    
    The character χ and the representation σ are uniquely determined by π. Let 𝒜_n^0(F)_0 denote the set of isomorphism classes of level zero supercuspidal representations of _n(F). Theorem 8.4.1 in <cit.> then gives a bijection 
    
    𝒜_n^0(F)_0     ⟷^××𝒜_n^0(_q)  
    π   ⟷ (ω_π(ϖ),σ).


    
    Convention: For the rest of this note, if π∈𝒜_n^0(F)_0, we shall always denote by π the second component of the image of π under the bijection (<ref>).
    
     
     	Let π∈𝒜^0_n(F)_0 and τ∈𝒜^0_m(F)_0, n > m. Suppose that ψ has level one. Then 
     	
    γ (s , π×τ, ψ )  = ω_τ(-1)^n-1   q^m(n-m-1)/2γ (π×τ, ψ),

     	where ψ is the character of κ_F ≅_q that is induced by ψ.
     
    




     
    










  

§ EXPLICIT LOCAL LANGLANDS CORRESPONDENCE FOR LEVEL ZERO SUPERCUSPIDAL REPRESENTATIONS

  To determine the base change of a representation as indicated in Section <ref>, we need an explicit knowledge of the local Langlands correspondence. The explicit correspondence for level zero representations of _n(F) was worked out by Bushnell and Henniart in <cit.>. We recall it here.
    

 §.§ Admissible tame pairs

    Recall that a tame pair over F consists of a finite unramified field extension E/F and a character θ of E^× that is trivial on U_E^1=1+𝔭_E. A tame pair (E/F,θ) is called admissible if the conjugates θ^s, s∈(E/F), are distinct. The following lemma is easy and can be found in <cit.>.
  
    Let (E/F,θ) be a tame pair. The following statements are equivalent:    
    
    	
  (i) The pair (E/F,θ) is admissible; 
    	
  (ii) The restrictions θ^s |_U_E, s∈(E/F) are distinct;
    	
  (iii)  The character θ cannot factor through _E/K for any subextension K/F.
    
  



      
     Two admissible tame pairs (E_i/F,θ_i), i=1,2, are called F-isomorphic if there is an F-isomorphism α:E_1  E_2 such that θ_1 = θ_2 ∘α. The degree of (E/F,θ) is the degree [E:F] of the extension E/F. Denote by 𝒯_n(F) the F-isomorphism classes of admissible tame pairs of degree n.
       
     Let (E/F,θ) be a tame pair of degree n. As θ is trivial on U^1_E, it reduces to a character θ̃ of U_E/U^1_E ≅κ_E^×≅_q^n^×. The pair (E/F,θ) is admissible if and only if θ̃ is _q-regular. 
     
     Recall that Λ_n(_q) is denoted as the set of equivalence classes of _q-regular characters of _q^n^×. Then we have a bijection
     
    𝒯_n(F)    ⟷^××Λ_n(_q)    
    
         (E/F,θ)    ⟷ (θ(ϖ), θ̃).
 
   
     
     

 §.§ Parameterization of level zero representations

     
     
     Firstly, composing (<ref>) with Green's parameterization (<ref>) and then the description of level zero supercuspidal representations (<ref>), we therefore have a bijective map 
      
    π_n 𝒯_n(F)    ⟶𝒜^0_n(F)_0 
    
          (E/F,θ)    ⟼π_n(θ).

     
     Secondly, let (E/F,θ) ∈𝒯_n(F). We view θ as a character of _E via Artin's reciprocity map a_E, and form the smooth induced representation  
    
    σ_n ( θ ) = __E^_Fθ=_E/Fθ.

     
     Recall that σ∈𝒢_n^0(F) is called of level zero if σ is trivial on the wild inertia subgroup of _F (see <cit.>). Deonte by 𝒢^0_n(F)_0 the subset of 𝒢^0_n(F) consisting of classes of representations of level zero. The following result is contained in <cit.>.
     
     	Let (E/F,θ) be an admissible tame pair of degree n. Then
     	
     	
     		
  (1) The representation _E/Fθ is irreducible and of level zero. The equivalence class of _E/Fθ depends only on that of (E/F,θ);
     		
  (2) The map
     		
    σ_n𝒯_n(F)    ⟶𝒢_n^0(F)_0    
    
         		(E/F,θ)    ⟼_E/Fθ

     		is a bijection.
     	
      
     
     

 §.§ Explicit correspondence for level zero supercuspidal representations

     
     It turns out that σ_n ∘π_n^-1 is not the local Langlands correspondence for level zero supercuspidal representations. Some modifications have to be made.
     
     
     Let (E/F,θ) be an admissible tame pair of degree n. Define Δ_E to be the unique unramified character of E^× of order 2. Then 
     	
    _n^F(π_n(θ)) = σ_n (Δ^n-1_E θ).

     
     
        
   

§ UNRAMIFIED BASE CHANGE

	
	We determine the unramified base change of a level zero supercuspidal representation in this section.

	 For two unramified extensions K and E over F, we will write KE as the compositum of K and E, which is still an unramified extension. Recall that ϕ is a fixed arithmetic Frobenius element in _F. For two nonnegative integers n and m, let (n,m) denote as usual the greatest common divisor of n and m.
	 
	
		Let (E/F,θ) ∈𝒯_n(F) and π=π_n(θ) as in (<ref>). Assume that K/F is an unramified field extension of degree m. Then 
		
    π_K/F = π_n/(n,m)(ς_1) ×⋯×π_n/(n,m)(ς_(n,m)),

		where, for each i, 
		
    ς_i = Δ_KE^n/(n,m)-1· (Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E

		is a character of (KE)^× and (KE/K,ς_i) is an admissible tame pair. The representation on the right hand side of (<ref>) does not depend on the order of these π_n/(n,m)(ς_i).
	
	 
	 	By (<ref>) and (<ref>), we have
	 	
    _n^K(π_K/F) = _K/F_n^F(π) =  Res_K/F_E/FΔ_E^n-1θ.

	 	As 𝒲_E is a normal subgroup of _F of finite index, we can apply Mackey's restriction formula (<ref>). Note that, as E/F, K/F are unramified extensions, the inertia groups ℐ_F, ℐ_E and ℐ_K are all the same. Recall that Φ is a fixed geometric Frobenius element in _F, then Φ^n (resp. Φ^m) is a geometric Frobenius element of _E (resp. _K). Therefore we can take {Φ,⋯,Φ^(n,m)} to be a set of representatives of double cosets _K
	 	\_F /_E. Note that EK is the unramified extension of F of degree [n,m], where [n,m] is the least common multiple of n and m, and that _E ∩_K= _KE. Applying (<ref>), we get 
	 	
    Res_K/F_E/FΔ_E^n-1θ =⊕_i=1^(n,m)_KE/K^Φ^i(Δ_E^n-1θ)|__KE.

        It follows from <cit.> that the character ^Φ^i(Δ_E^n-1θ) of _E corresponds exactly to the character (Δ_E^n-1θ) ∘ϕ^i|_E of E^× via local class field theory, where ϕ=Φ^-1 is a arithmetic Frobenius element of _F. Hence the restriction of ^Φ^i(Δ_E^n-1θ) to _KE corresponds to the character (Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E of (KE)^×. So we rewrite (<ref>) as
        
    Res_K/F_E/FΔ_E^n-1θ =⊕_i=1^(n,m)_KE/K (Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E.

        
        Claim: (KE/K,(Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E) is an admissible tame pair of degree n/(n,m). In fact, KE/K is an unramified field extension of degree [n,m]/m=n/(n,m). Set ξ=(Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E. As N_KE/E(U^1_KE) ⊂ U^1_E and θ is trivial on U^1_E, ξ is trivial on U_KE^1. So (KE/K,ξ) is by definition a tame pair of degree n/(n,m).  We are then left to show that (KE/K,ξ) is admissible. Suppose that ξ is not admissible, then it is fixed by a subgroup G', G'≠ e, of (KE/K). Note that ξ is obviously fixed by (KE/E). As the orders of (KE/K) and (KE/E) are coprime, we infer that ξ is fixed by a subgroup of (KE/F) that is strictly larger than (KE/E). By the arguments in the proof of Lemma <ref>, this means that ξ factors through _KE/L, where L/F is a subextension of E/F. By the transitivity of the norm map and the fact that _KE/E is surjective, we conclude that (Δ_E^n-1θ)∘ϕ^i|_E factors through _E/L. This implies easily that θ also factors through _E/L, which contradicts the assumption that (E/F,θ) is an admissible pair.
        
        Therefore, by (<ref>), we have
        
    _KE/K(Δ_E^n-1θ)∘ϕ^i|_E ∘_KE/E =     _n/(n,m)^K(π_n/(n,m) (ς_i))

        with ς_i defined in (<ref>).
        Note that no two of the representations π_n/(n,m)(ς_i) and π_n/(n,m)(ς_j) are linked in the sense of <cit.> and that the condition (<ref>) is satisfied, hence the representation 
        
    π_n/(n,m)(ς_1) ×⋯×π_n/(n,m)(ς_(n,m))

         is irreducible (<cit.>) and does not depend on the order of these π_n/(n,m)(ς_i). 
         
         By (<ref>), together with (<ref>), (<ref>) and (<ref>), we get
         
    _n^K(π_K/F)    = ⊕_i=1^(n,m)_n/(n,m)^K(π_n/(n,m)(ς_i) 
       =_n^K(π_n/(n,m)(ς_1) ×⋯×π_n/(n,m)(ς_(n,m))) .

        So, by the Langlands correspondence, we are done.
        







	 	
	In particular, we see that an unramified base change of π∈𝒜^0_n(F)_0 is again supercuspidal if and only if the degree of the base field extension is coprime with n.
	




	
To make use of (<ref>) later in our computation, we need one simple observation. Any level zero supercuspidal representation is automorphically induced by a character.


	Let (K/F,) be an admissible tame pair of degree m and π_m(λ) be the representation as in (<ref>). Then
	
    π_m () = (Δ_K^m-1)^K/F.



	This follows immediately from (<ref>) and (<ref>).

	
	
	
	
	
    

§ MAIN RESULT

    
     
    Before the proof of our main result, we note that
    
    	Let K/F be an unramified field extension of degree m. Let  be the group of characters of F^× that is trivial on _K/F(K^×). Suppose that ψ has level one. Then
    	
    ∏_χ∈γ(s,χ,ψ) = (-1)^m-1γ(s,1_K,ψ_K).

    
    
    	The characters in  are unramified as K/F is unramified and the norm map _K/F is surjective. Then by <cit.>, for χ∈, we have that
    	
    γ(s,χ,ψ) = q^s-1/2/χ(ϖ)·1-χ(ϖ)q^-s/1-χ(ϖ)^-1q^s-1=-q^-s-1/2q^s-χ(ϖ)/q^s-1-χ(ϖ).

    	When χ runs over , the value χ(ϖ) runs over the m-th roots of unity, as the group  is cyclic of order m. Hence
    	
    ∏_χ∈γ(s,χ,ψ) = (-1)^mq^-ms-m/2q^ms-1/q^ms-m-1=(-1)^m-1γ(s,1_K,ψ_K).

    	
    
    	Let π∈𝒜^0_n(F)_0 and τ∈𝒜^0_m(F)_0 with n > m and ψ of level one. Then
    	
    γ (s,π×τ,ψ) = (-1)^nm-(n,m)q^-nm/2∏_i=0^(n,m)-1 G(η_π^q^i∘_[n,m]:n·η_τ∘_[n,m]:m,ψ̃).

    
    where η_π resp. η_τ is the regular character of _q^n^× resp. _q^m^× that corresponds to π̃ resp. τ̃ via Green's parameterization and G is the Gauss sum as defined by (<ref>).
   
   	We split the proof in the following steps.
 

   
   	
   	 Step (1). Suppose that (K/F,η_τ) is the admissible tame pair that corresponds to τ by π_m (see (<ref>)). By Lemma <ref>, we have  
   	 
    τ = π_m(η_τ)=(Δ_K^m-1η_τ)^K/F.

   	
   	Applying (<ref>), we get
   	
    γ (s, π×τ,ψ)   =γ (s, π× (Δ_K^m-1η_τ)^K/F,ψ) 
       =γ (s,π_K/F×Δ_K^m-1η_τ ,ψ_K) ∏_χ∈γ (s,χ,ψ_F)^n/γ (s,1_K,ψ_K)^n
       =(-1)^mn-nγ (s,π_K/F×Δ_K^m-1η_τ ,ψ_K).

   	The last equality follows from Lemma <ref>.
   	
   	
   	Step (2). Suppose that (K'/F,η_π) is the admissible tame pair that corresponds to π. So π = π_n(η_π). By (<ref>), we get
   	
    π_K/F= π_n/(n,m)(ς_1) ×⋯×π_n/(n,m)(ς_(n,m)).

   	where π_n/(n,m)(ς_i)∈𝒜^0_n/(n,m)(K)_0 corresponds to the admissible tame pair (K'K/K,ς_i), and 
   	
    ς_i = Δ_KK'^n/(n,m)-1· (Δ_K'^n-1η_π)∘ϕ^i|_K'∘_KK'/K'.

   	 By the multiplicativity of local γ-factors <cit.>, we get
   	
    γ (s,π_K/F×Δ_K^m-1η_τ ,ψ_K) = ∏_i=1^(n,m)γ(s,π_n/(n,m)(ς_i)×Δ_K^m-1η_τ ,ψ_K).

   	
   	Step (3). Now we invoke the formula (<ref>). We have, for each i, that
   	
    γ(s,π_n/(n,m)(ς_i)×Δ_K^m-1η_τ ,ψ_K)
       = η_τ(-1)^n/(n,m)-1q^[n,m]/2-mγ(π_n/(n,m)(ς_i)×Δ_K^m-1η_τ,ψ_K).

   	The gamma factor on the right hand side of (<ref>) is a gamma facotr over finite fields. From (<ref>), the regular character which corresponds to π_n/(n,m)(ς_i) is 
   	
    η_π^q^i∘_[n,m]:n.

   	 In view of (<ref>), we get
   	
    γ(π_n/(n,m)(ς_i)×Δ_K^m-1η_τ,ψ_K) 
       = (-q^-mη_τ(-1))^n/(n,m)-1G(η_π^q^i∘_[n,m]:n·η_τ∘_[n,m]:m,ψ).

   	
   	Finally, combining (<ref>), (<ref>), (<ref>) and (<ref>) together, we get (<ref>).
   	
   	
   	
   	
   	
   	
   	
   	
    We can also reformulate a formula for gamma factors over finite fields.
    
    	Let π and τ be irreducible cuspidal representations of _n(_q) and _m(_q) respectively, n > m. Let η_π and η_τ be the correspoding _q-regular characters of _q^n^× and of _q^m^× via Green's parameterization. Then
    	
    γ(π×τ,ψ) = (-1)^nm-(n,m)η_τ(-1)^n-1q^-mn+m^2+m2∏_i=0^(n,m)-1 G(η_π^q^i∘_[n,m]:n·η_τ∘_[n,m]:m,ψ).

    

	
	




§ MACKEY'S RESTRICTION FORMULA



	Let G be a locally profinite group. Let H be an open subgroup, and let (σ,W) be a smooth representaton of H. Let K be a closed subgroup of G. There is a natural isomorphism
	
    ^G_K  _H^G σ≅⊕_g ∈ K \ G /H_K ∩gH^K ^gH_K ∩gHgσ

	where gH is the subgroup gHg^-1 of G and gσ is the representation of gH on W defined by gσ(ghg^-1) = σ(h).


    
	Suppose {g_i} is a set of representative of double cosets K\ G / H and {k^(i)_j} is a set of representative of cosets K/K∩ g_i H g_i^-1. Then {k^(i)_jg_i} is a set of representative of cosets G/H. For w∈ W, let f_w ∈_H^G σ such that f_w is supported in H and f_w(h) = σ(h) w. Suppose  is a -basis of W. Then, by the lemma in <cit.>, {k^(i)_jg_i f_w  | w∈} is a -basis of _H^G σ. For a fixed g_i=g, we show that {k_jgf_w  | w∈} is a -basis of a K-representation that is isomorphic to ^K_K ∩gHRes^gH_K ∩gHgσ. This latter representation has a -basis {k_jϕ_w  | w∈}, where ϕ_w is supported in K ∩gH and ϕ_w(k) = σ(g^-1kg)w. Then the map k_jgf_w ↦ k_jϕ_w extends linearly and gives the required K-isomorphism. In fact, k_jϕ_w = λ(g) k_jgf_w |_K, where λ(g) is the left translation by g, so the K-equivalence follows immediately. 
    












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