On the Howe duality conjecture in classical theta correspondence

TL;DR

This paper proves the Howe duality conjecture for almost equal rank dual pairs and establishes the irreducibility of small theta lifts for all tempered representations over nonarchimedean local fields.

Contribution

It provides a complete proof of the Howe duality conjecture in full generality for almost equal rank pairs and extends irreducibility results to all tempered representations.

Findings

Proof of Howe duality conjecture for almost equal rank dual pairs.

Irreducibility of small theta lifts for all tempered representations.

Results valid over any nonarchimedean local field of characteristic not 2.

Abstract

We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for any nonarchimedean local field of characteristic not 2 and in arbitrary residual characteristic.