An analogue of Weyl's law for quantized irreducible generalized flag manifolds

TL;DR

This paper establishes a Weyl's law analogue for quantized irreducible generalized flag manifolds by defining a zeta function with properties paralleling the classical case, applicable to compact quantum groups.

Contribution

It introduces a zeta function for quantized flag manifolds that mirrors classical Weyl's law and extends the framework to compact quantum groups.

Findings

The zeta function is proportional to the Haar state.

The first singularity of the zeta function matches the classical dimension.

The results apply to a broad class of compact quantum groups.

Abstract

We prove an analogue of Weyl's law for quantized irreducible generalized flag manifolds. By this we mean defining a zeta function, similarly to the classical setting, and showing that it satisfies the following two properties: as a functional on the quantized algebra it is proportional to the Haar state; its first singularity coincides with the classical dimension. The relevant formulae are given for the more general case of compact quantum groups.