On special representations of $p$-adic reductive groups

Abstract

Let $F$ be a non-Archimedean locally compact field, 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$$(\ast )C^{\infty }(G/Q,L)/\sum _{Q^{\prime }⊋Q}{C}^{\infty }(G/Q^{\prime },L).$$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 $(\ast )$ is the Steinberg representation and $\mathfrak{M}$ is of rank one) and construct an $I$-equivariant embedding of $(\ast )$ into $C^{\infty }(I,\mathfrak{M})$. This allows the computation of the $I$-invariants in $(\ast )$. 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 $(\ast )$ is irreducible. This is the analog of a theorem of Casselman (which says the same for $L=\mathbb{C}$); it had been conjectured by Vignéras. Herzig (for $G={\mathrm{GL}}_n(F)$) 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.

References in corpus (1)

• Coefficient systems and supersingular representations of $G{L}_2(F)$