Test vectors for local periods

Abstract

Abstract. Let E/F be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation π of GL(n, E) is said to be distinguished with respect to GL(n, F) if it admits a non-trivial linear form that is invariant under the action of GL(n, F). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

Gamma Factors, Root Numbers, and Distinction

“…This led to the current collaboration, aiming to correct the mistakes and explore further the relation between distinction and special values of local factors. We remark that in addition, the proof of [Off11, Corollary 7.2] is not valid, although the statement is correct and a proof can be found in the recent work of Anandavardhanan and Matringe, [AM17].…”

Test vectors for finite periods and base change

“…Let N, P, and ψ have similar meanings as in §1. For details we refer to [AM17] and the references therein.…”

“…Since π is squareintegrable there are two natural candidates for GL n (k)-invariant linear forms on the Whittaker model W (π, ψ). One is the analogue of the linear form in Theorem 1.1 given by the convergent integral [Kab04] λ(W) = k × N(k)\GL n (k) W(h)dh, which is obviously GL n (k)-invariant but non-vanishing precisely when π is distinguished, and the other, for which the finite field analogue is not very useful for the reasons detailed in §4 and §5, is given by the convergent integral [AKT04,AM17] ℓ(W) = N(k)\P(k) W(p)dp, which is known to be always non-vanishing but GL n (k)-invariant precisely when π is distinguished (cf. §4).…”

“…For the purposes of comparison we restrict ourselves to the former symmetric pair in which case the unique invariant linear form can be explicitly realized on the Whittaker model for any irreducible generic distinguished representation and there exists an explicit test vector for this invariant form. For more details, we refer to [AM17]. Suffices to say here that the mirabolic subgroup P(k) of GL n (k) plays a crucial role in the p-adic setting whereas it is of not much use in the finite field case.…”

Distinguished representations for $\mathrm{SL}(n)$

“…In a recent work of the first author with Matringe [AM17], it has been proved that the integral representation for the invariant linear form ℓ(W) = N n (F)\P n (F) W(p)dp can be defined on the Whittaker space W ( π, ψ) (absolutely convergent integral for π unitary [Fli88, Lemma 4], and defined by regularization in general [AM17, §7]), associated to an irreducible generic representation π of GL n (E), and up to multiplication by scalars, is the unique non-zero element in Hom GL n (F) ( π, 1), which allows one to conclude as in [AP03] that any irreducible generic representation of SL n (E) which is distinguished by SL n (F) has a Whittaker model for a non-degenerate character ψ :…”