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

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.


Introduction
In 1950 I.M. Gelfand and M.L. Tsetlin [1] proposed a formal description of finite-dimensional irreducible representations (IR) for the compact Lie algebra u(n). This description is a generalization of results for u(2) and u(3) on the case u(n). It is the following. In the IR space of u(n) there is a orthonormalized basis which is numerated by the following formal schemes where all numbers m ij (1 ≤ i < j ≤ n) are nonnegative integers and they satisfy the standard inequalities, "between conditions": m ij+1 ≥ m ij ≥ m i+1j+1 for 1 ≤ i ≤ j ≤ n − 1. (1. 2) The first line of this scheme is defined by the components of the highest weight of u(n) IR, the second line is defined by the components of the highest weight of u(n − 1) IR and so on. Later this basis was constructed in many papers (for example, see [2,3,4]) by using one-step lowering and raising operators.
In 1965 I.M. Gelfand and M.I. Graev [5] using analytic continuation of the results for u(n) obtained some results for non-compact Lie algebra u(n, m). They shown that some class of Hermitian IR of u(n, m) is characterized by a "extremal weight" parametrized by a set of integers m N = (m 1N , . . . , m N N ) (N = n + m) such that m 1N ≥ m 1N ≥ · · · ≥ m N N , and by a representation type which is defined by a partition of n in the sum of two nonnegative integers α and β, n = α + β (also see [6]).
For simplicity we consider the case u (2,1). In this case we have three type of scheme The numbers m ij of the second scheme satisfy the following inequalities The numbers of the third scheme satisfy the following inequalities Construction of Gelfand-Graev basis for u(n, m) in terms of one-step lowering and raising operators is more complicated then in the compact case u(n + m).
In 1975 T.J. Enright and V.S. Varadarajan [7] obtained some classification of discrete series of non-compact Lie algebras. Later A. Molev [8] shown that for the case u(n, m) Gelfand-Graev modules are some part of Enright-Varadarajan modules and Molev constructed Gelfand-Graev basis for u(n, m) in terms of Mickelsson S-algebra [9].
A goal of this work to obtain analogous results for the non-compact quantum algebra U q (u(n, m)). Because the general case is very complicated we at first consider the case U q (u(n, 1)). It should be noted that the special case U q (u(2, 1)) was considered in [10,11].
2 Quantum algebra U q (gl(N )) and its noncompact real forms U q (u(n, m)) (n + m = N ) The quantum algebra U q (gl(N )) is generated by the Chevalley elements q ±e ii (i = 1, . . . , N ), e i,i+1 , e i+1,i (i = 1, 2, . . . , N − 1) with the defining relations: where [e β , e γ ] q denotes the q-commutator: The definition of a quantum algebra also includes operations of a comultiplication ∆ q , an antipode S q , and a co-unit ε q . Because explicit formulas of these operations will not used in our later calculations, they are not given here. For construction of the composite root vectors e ij for |i − j| ≥ 2 we fix the following normal (convex) ordering of the positive root system ∆ + (see [12]) (2.10) According to this ordering we set Using these explicit constructions and the defining relations (2.1)-(2.8) for the Chevalley basis it is not hard to calculate the following relations between the Cartan-Weyl generators e ij (i, j = 1, 2, . . . , N ): If we apply the Cartan involution (e * ij = e ji ) the formulas above, we will get all relations between elements of the Cartan-Weyl basis.
The explicit formula for the extremal projector for U q (gl(N ) has the form [13] where the elements p ij (1 ≤ i < j ≤ N ) are given by The extremal projector p := p(U q (gl(N )) satisfies the relations: The extremal projector p belongs to the Taylor extension T U q (gl(N ) of the universal enveloping algebras U q (gl(N ). The Taylor extension T U q (gl(N ) is an associative algebra generated by formal Taylor series of the form provided that nonnegative integers r 12 , r 13 , r 23 , . . . , r N −1,N and r 12 , r 13 , r 23 , . . . , r N −1,N are subject to the constraints for each formal series and the coefficients C {r },{r} (q e 11 , . . . , q e N N ) are rational functions of the q-Cartan elements q e ii . The quantum universal enveloping algebra U q (gl(N )) is a subalgebra of the Taylor extension T U q (gl(N )), U q (gl(N )) ⊂ T U q (gl(N )). The noncompact quantum algebra U q (u(n, m)) can be considered as the quantum algebra U q (gl(N )) (N = n + m) endowed with the additional Cartan involution ( * ): Below we will consider the real form U q (u(n, 1)), i.e. the case N = n + 1.
3 The reduction algebra Z q (gl(n + 1), gl(n)) In the linear space T U q (gl(n + 1)) we separate out a subspace of "two-sided highest vectors" with respect to the subalgebra U q (gl(n)) ⊂ U q (gl(n + 1)), i.e.
It is evident that if x ∈Z q (gl(n + 1), gl(n)) then where p := p(U q (gl(n)). Again, using the annihilation properties of the projection operator p we have that any vector x ∈Z q (gl(n + 1), gl(n)) presents a formal Taylor series on the following monomials ,n e rn n,n+1 · · · e r 1 1,n+1 p .

(4.2)
This involution can be considered as generalization of the Cartan involution in U q (gl(n + 1)) to the Taylor extension, T U q (gl(n + 1)). The Z-algebra Z q (gl(n + 1), gl(n)) with this involution is called the compact real form and denoted by the symbol Z (c) q (gl(n + 1), gl(n)). The noncompact real form on Z q (gl(n + 1), gl(n)) is defined by the involution * which is given as follows e * ii = e ii (i = 1, 2, . . . , n + 1) .

Summary
Thus we obtain the explicit description of the Hermitian irreducible representations for the noncompact quantum algebra U q (u(n, 1)) by the reduction Z-algebras for description of which we used the standard extremal projectors. Next step: to obtain an analogous results for the case U q (u(n, 2)). For this aim we need construct extremal projector p (α) which is expressed in terms of the Z-algebra Z q (gl(n + 1), gl(n)).
Final aim: to consider the general case U q (u(n, m)). In this case extremal projectors of new type will be used.