On multiplicity in restriction of tempered representations of $p$-adic groups

TL;DR

This paper proves a formula equating multiplicities in the restriction of tempered representations of p-adic groups with dimensions of representations of their Langlands dual component groups, under certain hypotheses.

Contribution

It establishes a new equality linking restriction multiplicities of tempered representations to their Langlands dual $ ext{S}$-groups, providing a way to compute multiplicities via $ ext{S}$-group representations.

Findings

Equality between restriction multiplicities and $ ext{S}$-group representation dimensions.

Provides a formula for multiplicities in terms of $ ext{S}$-group irreducible representations.

Assumes hypotheses related to the tempered local Langlands conjecture.

Abstract

We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal{S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered $L$-packets. This provides a formula of the multiplicity for $p$-adic groups by means of dimensions of irreducible representations of their $\mathcal{S}$-groups.