On the unicity of types for tame toral supercuspidal representations

Abstract

For tame arbitrary-length toral, also called positive regular, supercuspidal representations of a simply connected and semisimple $p$-adic group $G$, constructed as per Adler-Yu, we determine which components of their restriction to a maximal compact subgroup are types. We give conditions under which there is a unique such component, and then present a class of examples for which there is not, disproving the strong version of the conjecture of unicity of types on maximal compact open subgroups. We restate the unicity conjecture, and prove it holds for the groups and representations under consideration under a mild condition on depth.

16 pages; submitted to the Proceedings of the Conference Representation theory of p-adic groups, July 2017 at IISER Pune, India. Version 2: revised title; clearer statement of main theorem; one section added; figure added; references updated