Function Theory on a q-Analog of Complex Hyperbolic Space

TL;DR

This paper develops function theory on a quantum analog of complex hyperbolic space, providing explicit formulas for invariant integrals and analyzing principal series modules related to the quantum isotropic cone.

Contribution

It introduces a new quantum hyperbolic space framework, deriving explicit invariant integral formulas and classifying principal series modules, advancing quantum geometric analysis.

Findings

Explicit formulas for invariant integrals on quantum hyperbolic space

Classification criteria for principal series modules

Analysis of quantum isotropic cone structure

Abstract

This work deals with function theory on quantum complex hyperbolic spaces. The principal notions are expounded. We obtain explicit formulas for invariant integrals on `finite' functions on a quantum hyperbolic space and on the associated quantum isotropic cone. Also we establish principal series of $U_q \mathfrak{su}_{n,m}$-modules related to this cone, and obtain the necessary conditions for those modules to be equivalent.