Arithmetic Branching Law and generic $L$-packets

TL;DR

This paper extends the spectral analysis of local descents and branching laws for classical groups over any local field, establishing the equality of spectral and arithmetic first occurrence indices and characterizing the first descent spectrum.

Contribution

It generalizes previous results to all classical groups over any local field and introduces new indices that unify spectral and arithmetic perspectives.

Findings

Spectral and arithmetic first occurrence indices are equal.

First descent spectrum includes all discrete series representations.

Explicit branching decomposition formulas are provided.

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.