q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra U_q(u(n,1))

TL;DR

This paper develops a q-analog of the Gelfand-Graev basis for the noncompact quantum algebra U_q(u(n,1)), providing explicit constructions of Hermitian irreducible representations and a Gelfand-Graev basis.

Contribution

It introduces an explicit Mickelsson-Zhelobenko reduction for U_q(gl(n+1)), enabling the construction of a Gelfand-Graev basis for U_q(u(n,1)).

Findings

Explicit description of Z_q(gl(n+1),gl(n)) in terms of generators

Construction of Hermitian irreducible representations

Explicit orthonormal Gelfand-Graev basis for U_q(u(n,1))

Abstract

For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.