A `relative' local Langlands correspondence

TL;DR

This paper proposes a conjecture linking invariant linear forms on representations of a group over a quadratic extension to Langlands parameters, exploring parameter spaces and their morphisms.

Contribution

It formulates a new conjecture relating invariant forms to Langlands parameters and studies the structure of parameter spaces and their morphisms.

Findings

Conjecture classifies irreducible admissible representations with invariant forms.

Analysis of parameter spaces and morphisms between them.

Proposed finite map relates to the dimension of invariant form spaces.

Abstract

For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the dimension of the space of these invariant forms, in terms of the Langlands parameter of the representation. The paper studies parameter spaces of Langlands parameters, and morphisms between them associated to morphisms of $L$-groups. The conjectural answer to the question on the space of $G(F)$-invariant linear forms is in terms of fibers of a particular finite map between parameter spaces.