Strongly positive representations of $$GSpin_{2n+1}$$ G S p i n 2 n + 1 and the Jacquet module method

Abstract

We explicitly construct the structure of Jacquet modules of parabolically induced representations of $$GSpin_{2n+1}$$ over a $$p$$ -adic field $$F$$ of any characteristic. Using this construction of the Jacquet module, we obtain a classification of strongly positive representations of $$GSpin_{2n+1}$$ over $$F$$ and describe the general discrete series representations of $$GSpin_{2n+1}$$ over $$F$$ , assuming the half-integer conjecture. One application of this paper is the proof of the equality of $$L$$ -functions from the Langlands–Shahidi method and Artin $$L$$ -functions through the local Langlands correspondence (Kim in Langlands–Shahidi $$L$$ -functions for $$GSpin$$ groups and the generic Arthur packet conjecture, preprint).