On the geometry and quantization of symplectic Howe pairs

TL;DR

This paper explores the structure and quantization of symplectic manifolds with commuting Hamiltonian actions satisfying Howe conditions, revealing a canonical orbit correspondence and duality in geometric quantization.

Contribution

It establishes a canonical correspondence between orbit spaces and demonstrates Howe duality in geometric quantization for symplectic Howe pairs.

Findings

Orbit spaces of the actions are canonically related.

Reduced spaces are symplectomorphic to coadjoint orbits.

In the Kähler case, a Howe duality appears in the representation theory.

Abstract

We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical correspondence between the orbit spaces of the respective moment images. Furthermore, we show that reduced spaces with respect to the action of one group are symplectomorphic to coadjoint orbits of the other group. In the Kaehler case we show that the linear representation of a pair of compact Lie groups on the geometric quantization of the manifold is then equipped with a representation-theoretic Howe duality.