Howe duality for a quasi-split exceptional dual pair

TL;DR

This paper establishes Howe duality for a specific p-adic dual pair involving an exceptional group, expanding the understanding of theta correspondences in the context of quasi-split groups.

Contribution

It proves Howe duality for the theta correspondence associated with a p-adic dual pair inside an exceptional quasi-split group, a novel result in the theory of automorphic forms.

Findings

Established Howe duality for the specified dual pair.

Extended the scope of theta correspondence to a new exceptional setting.

Contributed to the understanding of dual pairs in quasi-split groups.

Abstract

We prove Howe duality for the theta correspondence arising from the $p$-adic dual pair $G_2 \times (\text{PU}_3 \rtimes \mathbb{Z}/2\mathbb{Z})$ inside the adjoint quasi-split group of type $E_6$.