Representations of $S{L}_2(F)$

Abstract

Let $p$ be a prime number, $F$ a non-archimedean local field with residue characteristic $p$, and $R$ an algebraically closed field of characteristic different from $p$. We thoroughly investigate the irreducible smooth $R$-representations of $S{L}_2(F)$. The components of an irreducible smooth $R$-representation $\Pi$ of $G{L}_2(F)$ restricted to $S{L}_2(F)$ form an $L$-packet $L(\Pi )$. We use the classification of such $\Pi$ to determine the cardinality of $L(\Pi )$, which is $1,2$ or $4$. When $p=2$ we have to use the Langlands correspondence for $G{L}_2(F)$. When $\ell$ is a prime number distinct from $p$ and $R={\mathbb{Q}}_{\ell }^{ac}$, we establish the behaviour of an integral $L$-packet under reduction modulo $\ell$. We prove a Langlands correspondence for $S{L}_2(F)$, and even an enhanced one when the characteristic of $R$ is not $2$. Finally, pursuing a theme of \cite{HV23}, which studied the case of inner forms of $G{L}_n(F)$, we show that near identity an irreducible smooth R-representation of $S{L}_2(F)$ 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).

44 pages This is the final version,

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• Representations of a reductive $p$-adic group in characteristic distinct from $p$
• The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$