On representations of quantum conjugacy classes of GL(n)

TL;DR

This paper constructs and analyzes quantum representations of conjugacy classes in GL(n), showing that various quantizations are isomorphic and providing a framework for understanding their algebraic structure.

Contribution

It introduces a unified approach to quantizing conjugacy classes of GL(n) using highest weight modules and demonstrates the isomorphism of different quantizations.

Findings

All quantizations are isomorphic.

Construction of highest weight modules for each class.

Application to semisimple adjoint orbits.

Abstract

Let $O$ be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by $T$ the maximal torus of diagonal matrices in GL(n). With every $a\in O\cap T$ we associate a highest weight module $M_a$ over the quantum group $U_q(gl(n))$ and an equivariant quantization $C_{h,a}[O]$ of the polynomial ring $C[O]$ realized by operators on $M_a$. All quantizations $C_{h,a}[O]$ are isomorphic and can be regarded as different exact representations of the same algebra, $C_{h}[O]$. Similar results are obtained for semisimple adjoint orbits in $gl(n)$ equipped with the canonical GL(n)-invariant Poisson structure.