Positive Casimir and Central Characters of Split Real Quantum Groups

TL;DR

This paper explores the structure of positive representations of split real quantum groups, introducing generalized Casimir operators, virtual weights, and analyzing their central characters and associated semi-algebraic regions.

Contribution

It introduces the notion of virtual highest and lowest weights and characterizes the central characters' images for positive representations of split real quantum groups.

Findings

Central characters are positive for all parameters.

The image of central characters forms a semi-algebraic region.

Explicit examples provided for lower rank cases.

Abstract

We describe the generalized Casimir operators and their actions on the positive representations $P_{\lambda}$ of the modular double of split real quantum groups $U_{q\tilde{q}}(g_R)$. We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters $\lambda$. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of $q$, and we discuss explicit examples in the lower rank cases.