Branching Rules for Supercuspidal Representations of SL_2(k)

TL;DR

This paper explicitly describes how supercuspidal representations of SL_2(k) decompose when restricted to maximal compact subgroups, revealing the structure and parameter dependence of this decomposition in the odd residual characteristic case.

Contribution

It provides an explicit description of the multiplicity-free decomposition of supercuspidal representations of SL_2(k) upon restriction to maximal compact subgroups for odd residual characteristic.

Findings

Decomposition is multiplicity-free.

Explicit formulas for the spectrum variation.

Parameter dependence of the spectrum described.

Abstract

The restriction of a supercuspidal representation of SL_2(k), for k a local nonarchimedean field, to a maximal compact subgroup decomposes as a multiplicity-free direct sum of irreducible representations. We explicitly describe this decomposition in the case that the residual characteristic is odd, and determine how the spectrum of this decomposition varies as a function of the parameters describing the supercuspidal representation.