Representations of unipotent reduction for SO(2n+1), I: an involution

TL;DR

This paper studies unipotent reduction representations of SO(2n+1) over p-adic fields, introduces a new involution related to Lusztig's construction, and proves its compatibility with Jacquet functor.

Contribution

It provides a new definition of an involution on elliptic representations and demonstrates its commutation properties with Jacquet functor, extending previous work with Moeglin.

Findings

New involution definition for elliptic representations

Proved involution commutes with Jacquet functor

Connected involution to Lusztig's finite group involution

Abstract

We consider a group SO(2n+1) over a p-adic field, and tempered irreducible representations of this group, of unipotent reduction. We use the construction due to Lusztig of these representations. In an old paper with Moeglin, we have defined an involution in the complex vector space generated by those representations which are elliptic. It is strongly related to another involution defined by Lusztig for finite groups. We give a new definition of our involution and we prove it commutes, in some sense, with Jacquet functor.