The local character expansion as branching rules: nilpotent cones and the case of $\mathrm{SL}(2)$

TL;DR

This paper establishes a connection between nilpotent coadjoint orbits and representations of maximal compact subgroups in p-adic groups, providing a new perspective on local character expansions and wave front sets.

Contribution

It introduces a representation-theoretic analogue of local character expansion for SL(2) and relates wave front sets to nilpotent support of unrefined minimal K-types in reductive groups.

Findings

Existence of specific representations attached to nilpotent orbits.

Decomposition of irreducible representations upon restriction to subgroups.

Wave front sets determined by nilpotent support in many cases.

Abstract

We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a suitable subgroup of $K$, is a sum of these five representations in the Grothendieck group. This is a representation-theoretic analogue of the analytic local character expansion due to Harish-Chandra and Howe. Moreover, we show for general connected reductive groups that the wave front set of many irreducible positive-depth representations of $G$ are completely determined by the nilpotent support of their unrefined minimal $K$-types.