Local L-Functions for Split Spinor Groups

Abstract

Abstract. We study the local L-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for GL 2 × GL 3 .

The local Langlands correspondence for GSp 4 over local function fields

“…Let w t = w l,∆t w l,θt and note that w t (θ t ) = θ t that is, the parabolic subgroup P t is self-associate. Let r t denote the adjoint action of L M t on L n t , where L n t denotes the Lie algebra of the complex dual group L N t of N t (Check Section 2 of [1] for the structure theory). Let sim denote the similitude character and Spin denote the standard 4-dimensional representation of GSp 4 (C) and let Std t denote the standard representation of GL t (C).…”

Generic transfer for general spin groups

“…Here, i = 1, 2, and the first L-function gives the Rankin-Selberg product while the second is either the twisted symmetric or the twisted exterior square L-function. When n = 0, we get only the second L-function (see [3,Proposition 5.6]). …”

“…Moreover, v (z) = φ v (z), considered as a diagonal matrix in GL 2n (C), actually lies, up to conjugation, in GSp 2n (C) or GSO 2n (C) as in (3). Therefore, there exist one-dimensional representations φ 0 , φ 1 , .…”

Theta distinguished representations, inflation and the symmetric square L-function

“…Proof If τ is distinguished, as in the proof of Corollary 4.4 we see that I(τ, 1/2, 1) is reducible, then according to Casselman and Shahidi [16] (Proposition 5.3, their Conjecture 1.1 was proved for G n in [2]) L(s, τ, Sym 2 ) has a pole at s = 0.…”