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

For the quantum algebra $U_q(\mathfrak{gl}(n+1))$ in its reduction on the subalgebra $U_q(\mathfrak{gl}(n))$ $Z_q(\mathfrak{gl}(n+1),\mathfrak{gl}(n))$ 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 $U_q(\mathfrak{gl}(n+1))$, namely, an orthonormal Gelfand–Graev basis is constructed in an explicit form.