Quantization of function algebras on semisimple orbits in $\g^*$

TL;DR

This paper presents a detailed algebraic construction for quantizing function algebras on semisimple coadjoint orbits, extending previous work to more general orbit types with proven flatness across parameters.

Contribution

It introduces a multiparameter deformation of function algebras on semisimple orbits using representation theory, providing explicit algebraic methods and extending prior cohomological approaches.

Findings

Constructed a multiparameter deformation quantizing semisimple coadjoint orbits.

Extended the deformation to the category of representations of the quantized enveloping algebra.

Proved the deformation is flat in all parameters.

Abstract

In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson brackets on any symmetric coadjoint orbit. This paper gives a more explicit algebraic construction which includes more general orbit types and which we prove to be flat in all parameters.