On the types for supercuspidal representations of inner forms of \(\mathrm{GL}_N\) (Q2161345)

In this paper, the author studies smooth, complex representations of the group \(GL_N(D)\) where \(D\) is a division algebra over a non-archimedean field \(F\) via theory of types. Specifically, he is mostly concerned with the irreducible supercuspidal representations. The theory evolved, excluding some earlier contributions (e.g. Bernstein and Zelevinsky) mainly by Bushnell-Kutzko's work on the classification of types for \(GL_N(F)\) and then by Secherre's work on \(GL_N(D)\) with some other contributions (e.g. Paškunas). By Secherre's work, to each irreducible supercuspidal representation \(\pi\) one can attach a corresponding simple type and its extension such that \(\pi\) is compactly induced by this extension. The author attaches to \(\pi\) another type, this time on a maximal compact subgroup og \(GL_N(D).\) Since all the maximal compact subgroups of \(GL_N(D)\) are mutually conjugated, this gives us conjugated class of (maximal compact) types for this \(\pi,\) which is called an archetype. A natural question arises: is there, for a cuspidal representation \(\pi\), a unique archetype? The author gives a condition (of certain unramified sort) when the answer is positive, but also supplies a counterexample-a supercuspidal representation (without this unramified condition) is constructed-which does not have a unique archetype.