Local theta correspondences and Langlands parameters for rigid inner twists

TL;DR

This paper proposes a conjecture linking local theta correspondences with Langlands parameters for rigid inner twists and verifies it in specific cases, advancing understanding in the representation theory of p-adic groups.

Contribution

It introduces a conjecture connecting local theta correspondences and Langlands parameters for rigid inner twists, including quaternionic dual pairs, and confirms its validity in certain instances.

Findings

Conjecture relating theta correspondences and Langlands parameters formulated.

Verification of the conjecture in specific cases.

Enhanced understanding of local Langlands correspondence for inner forms.

Abstract

In this paper, we formulate a conjecture that describes the local theta correspondences in terms of the local Langland correspondences for rigid inner twists, which contain the correspondences for quaternionic dual pairs. Moreover, we verify the conjecture holds in some specific cases.