Geometric quantization and quantum moment maps on coadjoint orbits and K\"ahler-Einstein manifolds

TL;DR

This paper explores the compatibility of symmetries in geometric and deformation quantization on K"ahler manifolds, focusing on coadjoint orbits and K"ahler-Einstein manifolds, establishing exact symmetry correspondence.

Contribution

It demonstrates that symmetries in geometric and deformation quantization are exactly compatible on certain K"ahler manifolds, not just asymptotically.

Findings

Symmetries are strictly compatible on coadjoint orbits.

Symmetries are strictly compatible on K"ahler-Einstein manifolds.

Compatibility extends beyond asymptotic approximation.

Abstract

Deformation quantization and geometric quantization on K\"ahler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a K\"ahler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and K\"ahler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).