Arithmetic Branching Law and generic $L$-packets

Abstract

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. For any irreducible admissible representation $\pi$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean, we extend the study of spectral decomposition of local descents in [JZ18] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field $F$. In particular, if $\pi$ has a generic local $L$-parameter, we introduce the spectral first occurrence index $\mathfrak{f}_{\mathfrak{s}}(\pi)$ and the arithmetic first occurrence index $\mathfrak{f}_{\mathfrak{a}}(\pi)$ of $\pi$ and prove in Theorem 1.4 that $\mathfrak{f}_{\mathfrak{s}}(\pi) = \mathfrak{f}_{\mathfrak{a}}(\pi)$. Based on the theory of consecutive descents of enhanced $L$-parameters developed in [JLZ22], we are able to show in Theorem 1.5 that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result ([JZ18, Theorem 1.7]) to the great generality.