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