Restriction to \(GL_ 2({\mathcal O})\) of supercuspidal representations of \(GL_ 2(F)\) (Q1112190)

Based on results of Kutzko, this paper deals with the decomposition of the restriction to \(K:=GL_ 2(R)\) of any irreducible supercuspidal representation \(\pi\) of \(G:=GL_ 2(F)\), for any p-field F with ring R of integers and finite residue field k of arbitrary characteristic. While it is known that any representation such as \(\pi\) is admissible and consequently a direct sum of irreducible K-types each of finite multiplicity, the author shows actually that each such irreducible component occurs which multiplicity 1 and further describes the components explicitly (A similar result for unitary representations, of \(PGL_ 2(F)\) with k of characteristic \(\neq 2\) was obtained earlier by Silberger).