Let F be a p-field with ring
of integers 𝒪 whose maximal prime ideal is p = ω𝒪, and with finite residue field
k = 𝒪∕p. Let G =GL2(F) and let K be the subgroup GL2(𝒪) of G. In this
paper we obtain the decomposition of the restriction to K of any irreducible
supercuspidal representation of G. (The corresponding result for unitary
representations, G =PGL2, and k of characteristic ≠2 was found by Silberger. Here
we make no assumption on the characteristic of k.) It is well-known that any
irreducible supercuspidal representation of G is admissible and hence decomposes
as a direct sum of irreducible K-types, each of which appears with finite
multiplicity. Here we show that, in fact, each of these irreducible components
occurs with multiplicity one, and we give an explicit description of each
component.
This work is based upon results of Kutzko, who proved that any irreducible
supercuspidal representation of G is twist-equivalent to another such representation
which, in turn, may be compactly induced from one of two compact-modulo-center
subgroups of G.
