Endo-parameters for p-adic classical groups

TL;DR

This paper establishes a criterion for intertwining and conjugacy of cuspidal types in p-adic classical groups, introduces endo-parameters, and conjectures their relation to wild Langlands parameters, advancing the local Langlands program.

Contribution

It proves intertwining implies conjugacy for cuspidal types, generalizes endo-equivalence, and introduces endo-parameters for classical groups, linking them to Langlands parameters.

Findings

Intertwining of cuspidal types implies conjugacy.

Introduction of (self-dual) endo-parameters for classical groups.

Conjecture relating endo-parameters to wild Langlands parameters.

Abstract

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal C-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart's notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.