Equivariant deformation quantization and coadjoint orbit method

TL;DR

This paper develops a method for applying deformation quantization to coadjoint orbits of real reductive groups, leading to new geometric constructions of representations, including for exceptional Lie groups.

Contribution

It proves the existence of equivariant deformation quantization on Lagrangian subvarieties and constructs new irreducible Harish-Chandra modules for specific coadjoint orbits.

Findings

Established existence of equivariant deformation quantization in relevant settings

Constructed new irreducible Harish-Chandra modules from coadjoint orbits

Provided geometric realizations of representations for real exceptional Lie groups

Abstract

The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to orbit method and construct nontrivial irreducible Harish-Chandra modules for certain coadjoint orbits. Our examples include new geometric construction of representations associated to certain orbits of real exceptional Lie groups.