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Finite dimensional representations of the quantum group \(GL_ q(n;\mathbb{C})\) and the zonal spherical functions on \(U_ q(n-1)\setminus U_ q(n)\). (English) Zbl 0806.17016

The authors study the quantum groups \(\text{GL}_ q (n;\mathbb{C})\), \(U_ q(n)\), their finite dimensional representations and zonal spherical functions. For each dominant integral weight \(\lambda\) the authors realize the finite dimensional irreducible representations of \(\text{GL}_ q (n;\mathbb{C})\) with highest weight \(\lambda\) as a space of relative invariants in the algebra \(A(\text{GL}_ q (n;\mathbb{C}))\) of functions on \(\text{GL}_ q(n; \mathbb{C})\) with respect to the Borel subgroup \(B^ -_ n\). The basis of this space consists of standard monomials. The irreducible decomposition of \(A(\text{GL}_ q (n;\mathbb{C}))\) as a two-sided \(A(\text{GL}_ q (n;\mathbb{C}))\)-comodule is given by using the complete irreducibility theorem for finite dimensional representations. This leads to the existence of an invariant functional (an invariant integral) on \(A(\text{GL}_ q (n;\mathbb{C}))\). Two Hermitian forms on \(A(U_ q (n))\) are introduced by means of the invariant functional. The irreducible decomposition of \(A(U_ q (n))\) is orthogonal with respect to these Hermitian forms.
The quantum homogeneous space \(U_ q(n)/ U_ q(n-1)\) is introduced. It is a quantum \((2n-1)\)-sphere. The algebra of functions on this quantum sphere is defined. Its irreducible decomposition as a \(A(U_ q (n))\)- comodule is given. The zonal spherical functions on \(U_ q(n)/ U_ q(n- 1)\) are explicitly calculated in terms of little \(q\)-Jacobi polynomials. The invariant functional of \(U_ q(n)/ U_ q(n-1)\) is expressed by an iterated Jackson integral.
Reviewer: A.Klimyk (Kiev)

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
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