Branching rules for unramified principal series representations of GL(3) over a p-adic field

TL;DR

This paper investigates how unramified principal series representations of GL(3) over a p-adic field decompose when restricted to the maximal compact subgroup, revealing a structured decomposition into components that reflect equivalences among irreducible parts.

Contribution

It introduces a new decomposition framework for these representations, highlighting the relationships between irreducible constituents upon restriction.

Findings

Decomposition into components capturing equivalences

Finite multiplicities in the restriction

Structured understanding of irreducible constituents

Abstract

On restriction to the maximal compact subgroup $\mathrm{GL}(3,\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each appearing with finite multiplicity. We describe a coarser decomposition into components which, although reducible in general, capture the equivalences between the irreducible constituents.