Representations of quantum conjugacy classes of orthosymplectic groups

TL;DR

This paper constructs quantum analogs of conjugacy classes in orthosymplectic groups using highest weight modules over quantum groups, revealing orbit-based isomorphisms and diverse representations within the same class.

Contribution

It introduces a method to associate quantum conjugacy class representations with points on the maximal torus, highlighting orbit-based isomorphisms and representation diversity.

Findings

Quantizations are isomorphic for points on the same Weyl group orbit.

Different points on the same orbit support different representations.

Provides a framework for quantum class representations in orthosymplectic groups.

Abstract

Let $G$ be the complex symplectic or special orthogonal group and $\g$ its Lie algebra. With every point $x$ of the maximal torus $T\subset G$ we associate a highest weight module $M_x$ over the Drinfeld-Jimbo quantum group $U_q(\g)$ and a quantization of the conjugacy class of $x$ by operators in $\End(M_x)$. These quantizations are isomorphic for $x$ lying on the same orbit of the Weyl group, and $M_x$ support different representations of the same quantum conjugacy class.