Local Theta correspondence of Tempered Representations and Langlands parameters

TL;DR

This paper explicitly determines the theta lifting process for certain dual pairs over nonarchimedean fields, linking it to the local Langlands correspondence and identifying when theta lifts of tempered representations are nonzero.

Contribution

It provides an explicit description of theta lifts for symplectic-orthogonal and unitary dual pairs, connecting them with local Langlands parameters and clarifying nonvanishing conditions.

Findings

Explicit determination of theta lifts for dual pairs.

Criteria for nonvanishing of theta lifts of tempered representations.

Connection established between theta lifts and local Langlands correspondence.

Abstract

In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.