$p$-adic Banach space representations of $S{L}_2({\mathbb{Q}}_p)$

Abstract

We consider the restriction to $S{L}_2({\mathbb{Q}}_p)$ of an irreducible $p$-adic unitary Banach space representation $\Pi$ of $G{L}_2({\mathbb{Q}}_p)$. If $\Pi$ is associated, via the $p$-adic local Langlands correspondence, to an absolutely irreducible 2-dimensional Galois representation $\psi$, then the restriction of $\Pi$ decomposes as a direct sum of $r\le 2$ irreducible representations. The main result of this paper is that $r$ is equal to the cardinality $s$ of the centralizer in $PG{L}_2$ of the projective Galois representation $\overline{\psi }$ associated to $\psi$, and the restriction is multiplicity-free, except if $\psi$ is triply-imprimitive, in which case the restriction of $\Pi$ is a direct sum of two equivalent representations. From this result we derive a classification of absolutely irreducible $p$-adic unitary Banach space representations of $S{L}_2({\mathbb{Q}}_p)$.

33 pages; v2 is a substantially revised version and now treats the restriction of all unitary absolutely irreducible Banach representations of $G{L}_2({\mathbb{Q}}_p)$

References in corpus (1)

• Une étude des représentations modulo $p$ de SL(2,F)