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

Monica Nevins

Vol. 329 (2024), No. 2, 259–301

DOI: 10.2140/pjm.2024.329.259

Abstract

We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}<!--FUNCTION\; APPLICATION-->(2,F)$, $p\ne 2$, for each nilpotent coadjoint orbit, such that every irreducible admissible (complex) 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.

Keywords

representation theory, nilpotent orbits, local character expansion, p-adic groups

Mathematical Subject Classification

Primary: 22E50

Milestones

Received: 10 November 2023

Revised: 20 May 2024

Accepted: 1 June 2024

Published: 6 July 2024