On the unicity of types for tame toral supercuspidal representations

TL;DR

This paper investigates the uniqueness of types within tame toral supercuspidal representations of p-adic groups, providing conditions for unicity, counterexamples to strong conjectures, and confirming the conjecture under certain depth conditions.

Contribution

It characterizes when components of restrictions are unique types, presents counterexamples to the strong unicity conjecture, and proves the conjecture holds under mild depth conditions.

Findings

Identifies conditions for unique types in tame toral supercuspidal representations.

Provides counterexamples to the strong unicity conjecture.

Proves the unicity conjecture under mild depth assumptions.

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.