On special representations of p-adic reductive groups

Abstract

Let $F$ be a non-Archimedean locally compact field, and let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$, we consider the $G$-representation on the $L$-module $$C^{\infty }(G/Q,L)/{\sum }_{Q\text{'}⊋Q}{C}^{\infty }(G/Q\text{'},L).(\ast )$$ Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank-$L$ module $\mathfrak{M}$ (depending on $Q$; if $Q$ is a Borel subgroup, then (∗) is the Steinberg representation and $\mathfrak{M}$ is of rank $1$) and construct an $I$-equivariant embedding of (∗) into $C^{\infty }(I,\mathfrak{M})$. This allows the computation of the $I$-invariants in (∗). We then prove that if $L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for $L=\mathbb{C}$); it had been conjectured by Vignéras.

Herzig (for $G={GL}_n\left(F\right)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.