Typical representations, parabolic induction and the inertial local Langlands correspondence

TL;DR

This paper establishes a connection between parabolic induction and typical representations, constructing an inertial Langlands correspondence that respects monodromy, with applications in parametrizing representations via Bruhat--Tits structures.

Contribution

It introduces a new link between parabolic induction and typical representations, and constructs an inertial Langlands correspondence respecting monodromy actions.

Findings

Proves a link between parabolic induction and typical representations.

Constructs a well-defined inertial Langlands correspondence.

Provides a parametrization of typical representations using Bruhat--Tits building.

Abstract

We prove a result which provides a link between the decomposition of parabolically induced representations and the Bushnell--Kutzko theory of typical representations. As an application, we show that there exists a well-defined inertial Langlands correspondence which respects the monodromy action of L-parameters, under some standard conjectures regarding the local Langlands correspondence. To allow for potential applications of this inertial Langlands correspondence, we also provide a complete construction of the set of typical representations, giving a parametrization of these in terms of the structure of the Bruhat--Tits building of $G$.