The Langlands parameter of a simple supercuspidal representation: symplectic groups

Abstract

In this work, we explicitly compute a certain family of twisted gamma factors of a simple supercuspidal representation π of a p-adic odd orthogonal group. These computations, together with analogous computations for general linear groups carried out in previous work with Liu [AL14], allow us to give a prediction for the Langlands parameter of π. If we assume the "depth-preserving conjecture", we prove that our prediction is correct if p is sufficiently large. Recently, Gross, Reeder, and Yu [GR10, RY13] have constructed a class

The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic

“…The partial Bessel function as a form of Howe Whittaker functions (also referred to Howe vectors) was first introduced by R. Howe [20]. In the mid 1990's, this partial Bessel function was explored by Baruch [6] to establish the stability of gamma factors and the local converse theorem for U (2,1). A slightly different modification of Howe vectors was pursued by Cogdell and Piatetski-Shapiro, while they treated the stability of gamma factors for SO 2r+1 (F ) [11] defined by identical Rankin-Selberg integrals in Theorem A as a part of their program to establish functorial transfer from generic forms on SO 2r+1 (A k ) to GL 2r (A k ), where A k is the ring of adeles of a number field k. Afterwords, Howe vectors has been implemented in a flurry of work on the stability of numerous local factors and local converse theorems on lower rank groups via the Rankin-Selberg method [27,28,56,57].…”

“…Subsequently, R. Ye and Zelingher [55] propose the local converse theorem for twisted exterior power γ-factors, and verify it for the twisted exterior square γ-factor of simple supercuspidal representations on small rank general linear groups. What is intriguing us in this direction is that Rankin-Selberg γ-factors for simple supercuspidal representations of G defined in the way of Theorem A was explicitly computed in several manuscripts by Adrian and Kaplan [1,2], and thankfully Liu and Q. Zhang [34] very recently tackled with finite field analogue converse theorem for all classical groups. In this regard, it would be of interest to reduce Theorem A to the local converse theorem for simple or level zero supercuspidal representations of G, and we aim to do so in an upcoming paper.…”

Endoscopic lifting of simple supercuspidal representations of SO2n+1 TO GL2n

“…We remark that the liftings of simple supercuspidal representations of SO 2n+1 (F ) to GL 2n (F ) were determined in [Adr16] under the assumption that p ≥ (2+e)(2n+ 1), where e is the ramification index of F over Q p . Hence our results are new for odd primes less than (2 + e)(2n + 1).…”

On the Langlands parameter of a simple supercuspidal representation: even orthogonal groups

“…It can be shown that our results therefore give the Langlands parameter up to its restriction to the wild inertia subgroup, subject to such an analogue. The present work is the follow-up to [Adr16,AK], where the analogous computations were carried out and the theory of Rankin-Selberg integrals was applied, in order to determine the Langlands parameter of odd orthogonal groups and symplectic groups.…”

“…As mentioned above, this work is a follow-up to [Adr16,AK]. The case of odd orthogonal groups [Adr16] was a bit different in the sense that the lift Π of π was already expected to be simple supercuspidal.…”