q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra 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.

Keywords

• quantum algebra
• extremal projector
• reduction algebra
• Shapovalov form
• noncompact quantum algebra
• discrete series of representations
• Gelfand-Graev basis