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

TL;DR

This paper analyzes how ramified principal series representations of GL(3) over a p-adic field decompose when restricted to a maximal compact subgroup, revealing dependence on ramification degree and providing explicit irreducibility results.

Contribution

It introduces a detailed decomposition framework for ramified principal series representations of GL(3) over p-adic fields, based on ramification and Iwahori subgroup filtrations.

Findings

Decomposition depends on ramification degree

Established several irreducibility criteria

Provided explicit examples of decompositions

Abstract

We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,\mathscr{R})$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples.