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

[摘要] 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.

[关键词] quantum algebra;extremal projector;reduction algebra;Shapovalov form;noncompact quantum algebra;discrete series of representations;Gelfand–Graev basis