Howe duality for the dual pair $SL_2(\mathbb R) \times F_{4,1}$: a ping-pong of $K$-types

TL;DR

This paper establishes Howe duality for an exceptional theta correspondence involving the dual pair $SL_2(R) imes F_{4,1}$, using see-saw identities to relate $K$-types of representations.

Contribution

It introduces a proof of Howe duality for an exceptional dual pair, utilizing see-saw identities to analyze $K$-types in the correspondence.

Findings

Proves Howe duality for the pair $SL_2(R) imes F_{4,1}$.

Relates $K$-types of representations via see-saw identities.

Advances understanding of exceptional theta correspondences.

Abstract

We prove Howe duality for an exceptional theta correspondence. To that end we exploit a pair of see-saw identities and relate the $K$-types of corresponding representations.