## Duke Mathematical Journal

### On special representations of $p$-adic reductive groups

Elmar Grosse-Klönne

#### 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 $\begin{equation}\label{*}C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).\qquad(*)\end{equation}$ 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 $G=\operatorname{GL}_{n}(F)$) 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.

#### Article information

Source
Duke Math. J., Volume 163, Number 12 (2014), 2179-2216.

Dates
First available in Project Euclid: 15 September 2014

https://projecteuclid.org/euclid.dmj/1410789513

Digital Object Identifier
doi:10.1215/00127094-2785697

Mathematical Reviews number (MathSciNet)
MR3263032

Zentralblatt MATH identifier
1298.22018

#### Citation

Grosse-Klönne, Elmar. On special representations of $p$ -adic reductive groups. Duke Math. J. 163 (2014), no. 12, 2179--2216. doi:10.1215/00127094-2785697. https://projecteuclid.org/euclid.dmj/1410789513

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