Tame cuspidal representations in non-defining characteristics

TL;DR

This paper extends Yu's construction to produce all smooth, irreducible, cuspidal representations over fields of characteristic not equal to p for certain reductive groups, broadening understanding in non-defining characteristic settings.

Contribution

It demonstrates that Yu's construction yields all such representations in non-defining characteristic when p does not divide the Weyl group's order.

Findings

Construction produces all cuspidal representations under specified conditions.

Extends Yu's method to arbitrary algebraically closed fields of characteristic not p.

Provides a comprehensive classification in the non-defining characteristic case.

Abstract

Let F be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. We show that a construction analogous to Yu's construction of complex supercuspidal representations yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p. Moreover, we prove that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.