Reducibility points and characteristic p local fields I- Simple supercuspidal representations of symplectic groups

2024/06/22 by Blondel, Corinne, Henniart, Guy, Stevens, Shaun
#11F70 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2406.15767

Abstract

Let F be a non-Archimedean local field with odd characteristic p. Let N be a positive integer and G=Sp_2N(F). By work of Lomelí on γ-factors of pairs and converse theorems, a generic supercuspidal representation π of G has a transfer to a smooth irreducible representation Π_π of GL_2N+1(F). In turn the Weil-Deligne representation Σ_π associated to Π_π by the Langlands correspondence determines a Langlands parameter ϕ_π for π. That process produces a Langlands correspondence for generic cuspidal representations of G. In this paper we take π to be simple in the sense of Gross and Reeder, and from the explicit construction of π we describe Π_π explicitly. The method we use is the same as in our previous paper arXiv:2310.20455, where we treated the case where F is a p-adic field, and π a simple supercuspidal representation of G=Sp_2N(F). It relies on a criterion due to Moeglin on the reducibility of representations parabolically induced from GL_M(F)× G for varying positive integers M. We extend this criterion to the case when F has any positive characteristic. The main new feature consists in relating reducibility to γ-factors for pairs.

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