Branching Rules for $n$-fold Covering Groups of $\mathrm{SL}_2$ over a Non-Archimedean Local Field

TL;DR

This paper investigates how principal series representations of n-fold covering groups of SL_2 over non-Archimedean local fields decompose when restricted to maximal compact subgroups, providing explicit branching rules.

Contribution

It explicitly determines the irreducible decomposition of these principal series representations upon restriction, a novel result for covering groups of SL_2.

Findings

Explicit branching rules for principal series representations

Decomposition into irreducible components characterized

Provides tools for further harmonic analysis on covering groups

Abstract

Let $\widetilde{G}$ be the $n$-fold covering group of the special linear group of degree two, over a non-Archimedean local field. We determine the decomposition into irreducibles of the restriction of the principal series representations of $\widetilde{G}$ to a maximal compact subgroup of $\widetilde{G}$.