Howe duality and dichotomy for exceptional theta correspondences

TL;DR

This paper investigates exceptional theta correspondences involving the G2 group over p-adic fields, proving the Howe duality conjecture, establishing a dichotomy theorem, and explicitly describing theta lifts of non-cuspidal representations.

Contribution

It proves the Howe duality conjecture and a dichotomy theorem for exceptional dual pairs involving G2, and explicitly determines theta lifts of non-cuspidal representations.

Findings

Proved Howe duality conjecture for exceptional G2 dual pairs

Established a dichotomy theorem for these dual pairs

Explicitly determined theta lifts of all non-cuspidal representations

Abstract

We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.