Representations of $\mathrm{SL}_2(F)$

Guy Henniart and Marie-France Vignéras

Vol. 335 (2025), No. 2, 229–286

DOI: 10.2140/pjm.2025.335.229

Abstract

Let $p$ be a prime number, $F$ a nonarchimedean local field with residue field $k_F$ of characteristic $p$, and $R$ an algebraically closed field of characteristic different from $p$. We investigate the irreducible smooth $R$-representations of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$. The components of an irreducible smooth $R$-representation $\mathrm{\Pi }$ of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2(F)$ restricted to ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$ form an $L$-packet $L(\mathrm{\Pi })$. We use the classification of such $\mathrm{\Pi }$ to determine the cardinality of $L(\mathrm{\Pi })$, which is $1,2$ or $4$. When $p=2$ we have to use the Langlands correspondence for ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2(F)$. When $\ell$ is a prime number distinct from $p$ and $R={\mathbb{Q}}_{\ell }^{ac}$, we determine the behaviour of an integral $L$-packet under reduction modulo $\ell$. We prove a Langlands correspondence for ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$, and an enhanced one when the characteristic of $R$ is not $2$. Finally, pursuing a theme of Henniart and Vignéras (2024), which studied the case of inner forms of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(F)$, we show that near identity a nontrivial irreducible smooth $R$-representation $\pi$ of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$ is, up to a finite-dimensional representation, isomorphic to a sum of $1,2$ or $4$ representations in an $L$-packet of size $4$ (when $p$ is odd there is only one such $L$-packet). We show that for $\pi$ in an $L$-packet of size $r_{\pi }$ and a sufficiently large integer $j$, the dimension of the invariants of $\pi$ by the $j$-th congruence subgroup of an Iwahori or a pro-$p$ Iwahori subgroup of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$ is equal to $a_{\pi }+2r_{\pi }^{-1}|k_F{|}^j$, with $a_{\pi }=-\frac{1}{2}$ if $p$ is odd and $r_{\pi }=4$, otherwise $a_{\pi }$ is an integer. We also study the fixed points by the $j$-th congruence subgroups of the maximal compact subgroups of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(F)$ where the answer depends on the parity of $j$.

Keywords

modular irreducible representations, $L$-packets, Whittaker spaces, local Langlands correspondence

Mathematical Subject Classification

Primary: 22E50

Secondary: 11F70

Milestones

Received: 9 April 2024

Revised: 3 January 2025

Accepted: 6 February 2025

Published: 18 April 2025