Distinguished representations and poles of twisted tensor 𝐿-functions

Abstract

Abstract. Let E/F be a quadratic extension of p-adic fields. If π is an admissible representation of GLn(E) that is parabolically induced from discrete series representations, then we prove that the space of Pn(F )-invariant linear functionals on π has dimension one, where Pn(F ) is the mirabolic subgroup. As a corollary, it is deduced that if π is distinguished by GLn(F ), then the twisted tensor L-function associated to π has a pole at s = 0. It follows that if π is a discrete series representation, then at most one of the representations π and π ⊗ χ is distinguished, where χ is an extension of the local class field theory character associated to E/F . This is in agreement with a conjecture of Flicker and Rallis that relates the set of distinguished representations with the image of base change from a suitable unitary group.

On -Distinguished Representations of the Quasi-Split Unitary Groups

“…We treat the special case where the standard module is induced from a Levi subgroup of the Siegel Levi subgroup of U 2n and the GL-part is irreducible. When this is the case, if in addition exp(δ i ) ∈ 1 2 Z, i = 1, . .…”

“…Thus ν −1/2 π 1,1 cannot be GL(F)-distinguished (by [7,Proposition 12]) and Claim 2 follows from Corollary 4. 1 2 ] (ρ) ). This means again that l = 1 and e( d 1 ) = ν 1/2 ρ which is a contradiction (since e( i ) ν 1+b ρ, i = 1, .…”

“…Note that V (2) is a normal subgroup of V and V (1) is a set of representatives for V /V (2) . Thus,…”

“…Since by assumption we have that 2k+1 = ν 2k+2 as well as 2k+1 = 2k+2 ∨ we deduce that exp( 2k+1 ) = 1 2 and exp( 2k+2 ) = − 1 2 . Let ρ ∈ Cusp GL be such that π is centered at ρ (see Definition 9) and a ∈ 1 2 Z such that 2k+1 = [−a, 1 + a] (ρ) . Then 2k+2 = [−(1 + a), a] (ρ) .…”

The local period integrals and essential vectors

“…The history of the essential vector goes back to at least Casselman, where he established a theory of new forms for GL 2 (F ) [10]. In this paper, by applying the essential Whittaker functions (also called newforms), we study the test vector problem and the non-vanishing of local periods for five integrals representing Rankin-Selberg models [5,15], Flicker model [1,2,13], Jacquet-Shalika model [24,30,43], Friedberg-Jacquet model [16,24,[43][44][45], and Bump-Ginzburg model [9,34,35,57]. The local L-functions of GL n (F ) that are associated to these integrals include the tensor product L-factor of GL n (F ) × GL n (F ), the Asai L-factor, the exterior square L-factor, the Bump-Friedberg L-factor, and the symmetric square L-factor.…”