Patterns in Branching Rules for Irreducible Representations of SL_2(k), for k a p-adic field

TL;DR

This paper investigates how irreducible representations of SL_2(k), with k a p-adic field, decompose when restricted to maximal compact subgroups, revealing patterns linked to different types of representations and applications.

Contribution

It provides a detailed analysis of the branching rules for irreducible representations of SL_2(k) over p-adic fields, highlighting the variation in decomposition patterns across representation types.

Findings

Decomposition patterns differ between principal series and supercuspidal representations.

Tail-end K-representations match those in depth-zero supercuspidal decompositions.

Applications include insights into representation theory of p-adic groups.

Abstract

Building on prior work, we analyze the decomposition of the restriction of an irreducible representation of SL_2(k), for k a p-adic field of odd residual characteristic, to a maximal compact subgroup K. The pattern of the decomposition varies between principal series and different supercuspidal representations, whereas the K-representations which occur in the "tail end" of these decompositions are precisely those occurring in the decomposition of depth-zero supercuspidal representations. Various applications are considered.