Exterior square gamma factors for cuspidal representations of GL n : Finite field analogs and level-zero representations
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EXTERIOR SQUARE GAMMA FACTORS FOR CUSPIDAL REPRESENTATIONS OF GLn : FINITE FIELD ANALOGS AND LEVEL-ZERO REPRESENTATIONS
BY
Rongqing Ye Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA e-mail: [email protected]
AND
Elad Zelingher Department of Mathematics, Yale University, New Haven, CT 06510, USA e-mail: [email protected]
ABSTRACT
We follow Jacquet–Shalika [7], Matringe [12] and Cogdell–Matringe [3] to define exterior square gamma factors for irreducible cuspidal representations of GLn (Fq ). These exterior square gamma factors are expressed in terms of Bessel functions associated to the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level-zero representations.
Received October 21, 2018 and in revised form April 19, 2020
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R. YE AND E. ZELINGHER
Isr. J. Math.
1. Introduction Let F be a p-adic local field of characteristic zero with residue field f. Fix a non-trivial additive character ψ of F . In their work [7], Jacquet and Shalika define important integrals which we call local Jacquet–Shalika integrals, see [7, Sections 7, 9.3]. These Jacquet–Shalika integrals enable them to introduce integral representations for the exterior square L-function L(s, π, ∧2 ) of a (generic) representation π of GLn (F ). Later, Matringe [12] and Cogdell–Matringe [3] prove local functional equations for these local exterior square L-functions, in which local factors γ(s, π, ∧2 , ψ) and (s, π, ∧2 , ψ) play an important role. These local factors are related via the following equation: (1.1)
γ(s, π, ∧2 , ψ) =
, ∧2 ) (s, π, ∧2 , ψ)L(1 − s, π . 2 L(s, π, ∧ )
If π is an irreducible supercuspidal representation of GLn (F ), then it is generic, and the local factors γ(s, π, ∧2 , ψ) and (s, π, ∧2 , ψ) are defined. By type theory of Bushnell and Kutzko [2], π is constructed from some maximal simple type (J, λ); see [2, Section 6] for details. Partially, λ comes from some irreducible cuspidal representation π0 of GLm (e), where e is some finite extension of f. In this paper, we are interested in defining the exterior square gamma factor γ(π0 , ∧2 , ψ0 ) for some non-trivial additive character ψ0 of e, and we relate π to π0 via their exterior square gamma factors, in the case where π is of level-zero. To be concrete, the main result of this paper is that if π is a level-zero representation constructed from π0 , a cuspidal representation of GLn (f) which does not admit a Shalika vector, then γ(s, π, ∧2 , ψ) = γ(π0 , ∧2 , ψ0 ). In Section 2, we work with a general finite field F and a non-trivial additive character ψ. For a generic representation π of GLn (F) where n = 2m or n = 2m + 1, we follow [7] to define the Jacquet–Shalika integral Jπ,ψ (W, φ) and its dual J˜π,ψ (W, φ), where W is a Whittaker function of π and φ is a complex valued function on Fm . Using ideas from [12, 3], we define the exterior square gamma factor γ(π, ∧2 , ψ) of π in Theorem: Let π be an irreducible cuspidal representation of GLn (F) that does not adm
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