On Branching Rules of Depth-Zero Representations

TL;DR

This paper investigates how depth-zero representations of semisimple p-adic groups behave when restricted to maximal compact subgroups, revealing coincidences and intricate intertwining patterns among different classes of representations.

Contribution

It establishes the coincidence of branching rules for Deligne-Lusztig supercuspidal representations and explores their intertwining with depth-zero principal series representations.

Findings

Coincidence of branching rules within Deligne-Lusztig supercuspidal classes

Intertwining of restrictions with principal series representations in infinitely many components

Quantitative analysis and illustrative example provided

Abstract

Using Bruhat-Tits theory, we analyse the restriction of depth-zero representations of a semisimple simply connected $p$-adic group $G$ to a maximal compact subgroup $K$. We prove the coincidence of branching rules within classes of Deligne-Lusztig supercuspidal representations. Furthermore, we show that under obvious compatibility conditions, the restriction to $K$ of a Deligne-Lusztig supercuspidal representation of $G$ intertwines with the restriction of a depth-zero principal series representation in infinitely many distinct components of arbitrarily large depth. Several qualitative and quantitative results are obtained, and their use is illustrated in an example.