Representations of $SL_2(F)$

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 $SL_2(F)$. The components of an irreducible smooth $R$-representation $Π$ of $GL_2(F)$ restricted to $SL_2(F)$ form an $L$-packet $L(Π)$. We use the classification of such $Π$ to determine the cardinality of $L(Π)$, which is $1,2$ or $4$. When $p=2$ we have to use the Langlands correspondence for $GL_2(F)$. 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 $SL_2(F)$, 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 $GL_n(F)$, we show that near identity an irreducible smooth R-representation of $SL_2(F)$ 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).