On Branching Rules of Depth-Zero Representations

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.

Final version: applications to bounds on GK-dimension added; deeper discussion conditions for coincidence of full branching rules of supercuspidals, final example corrected to include all three conjugacy classes of maximal tori

References in corpus (3)

• A note on Bruhat decomposition of GL(n) over local principal ideal rings
• Restriction de la représentation de Weil à un sous-groupe compact maximal ou à un tore maximal elliptique
• On the unramified principal series of GL(3) over non-archimedean local fields