We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. 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.

Keywords:

principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groups

MSC Classifications:

20G25, 20G05 show english descriptions Linear algebraic groups over local fields and their integers
Representation theory 20G25 - Linear algebraic groups over local fields and their integers
20G05 - Representation theory

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