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

Abstract. We prove that for any split, almost simple, connected reductive group G over a p-adic field F , the Kottwitz homomorphism κ : G(F ) → Ω exhibits a homomorphic section Ω ֒→ G(F ). We then extend this result to certain additional split connected reductive groups.

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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.

We introduce a novel ultrametric on the set of equivalence classes of cuspidal irreducible representations of a general linear group GL N over a non-archimedean local field, based on distinguishability by twisted gamma factors. In the case that N is prime and the residual characteristic is greater than or equal to N 2 , we prove that, for any natural number i ≤ N 2 , there are pairs of cuspidal irreducible representations whose logarithmic distance in this ultrametric is precisely −i. This implies that, under the same conditions on N , the bound N 2 in the Local Converse Theorem for GL N is sharp.