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

Abstract

We consider the split special orthogonal group SO N defined over a p-adic field. We determine the structure of any L-packet of SO N containing a simple supercuspidal representation (in the sense of Gross-Reeder). We also determine its endoscopic lift to a general linear group. Combined with the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups, this leads us to get an explicit description of the L-parameter as a representation of the Weil group of F . Our result is new when p = 2 and our method provides a new proof even when p = 2. Contents 1. Introduction 2. Simple supercuspidal representations 2.1. Self-dual simple supercuspidal representations of GL N 2.2. Simple supercuspidal representations of SO 2n+1 2.3. Simple supercuspidal representations of SO 2n 3. Local Langlands correspondence for SO N 3.1. Local Langlands correspondence for SO N 3.2. Result of Moeglin and Xu 3.3. Formal degree conjecture of Hiraga-Ichino-Ikeda 4. Analysis of symmetric and exterior square L-factors 5. Twisted gamma factor for simple supercuspidal representations of SO 2n 5.1. Simple supercuspidal representations of SO 2n 5.2. The twisted γ-factors 5.3. The computation of γ(s, π × τ, ψ) 5.4. L-parameter 6. L-packets and L-parameters for simple supercuspidals of SO N 6.1. Rough form of the L-parameters 6.2. From twisted γ-factor to Swan conductor 6.3. Utilization of the formal degree conjecture 6.4. Main Theorem Appendix A. Several remarks on simple supercuspidal representations A.1. Iwahori subgroups A.2. Another parametrization of simple supercuspidal representations A.3. Comparison of our approach with others Appendix B. On lifting from classical groups to GL N Appendix C. Unramified case of Arthur's classification theorem C.1. Classical groups as twisted endoscopy of GL N C.2. Fundamental lemma of Lemaire-Moeglin-Waldspurger C.3. Arthur's local classification theorem 47 C.4. Unramified representation in an A-packet 49 References 51, On the cuspidal support of discrete series for p-adic quasisplit Sp(N ) and SO(N ), Manuscripta Math. 154 (2017), no. 3-4, 441-502.

On the Langlands parameter of a simple supercuspidal representation: even orthogonal 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.…”

“…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. Indeed the twisted γ-factors had no poles, and their computation was sufficient to determine the Langlands parameter using, among other result, the works of Moeglin [Moeg14] and Kaletha [Kal15].…”

“…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.…”

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].…”

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).…”