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Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. (English) Zbl 0874.33011

In this impressive paper the author introduces quantum analogues of the homogeneous spaces \(\text{GL}(n)/\text{SO}(n)\) and \(\text{GL}(2n)/\text{Sp}(2n)\), for general \(n\), and studies the zonal spherical functions associated with finite dimensional representations. The quantum groups are introduced via the \(R\)-matrix formalism. The quantum analogy of \(\text{GL}(n)/\text{SO}(n)\) is essentially that studied by K. Ueno and T. Takebayashi [Quantum groups, Proc. Workshops, Euler Int. Math. Inst. Leningrad/USSR 1990, Lect. Notes Math. 1510, 142–147 (1992; Zbl 0743.33012)] but the computation of the spherical functions is new. The author shows that the zonal spherical functions can be identified with Macdonald’s symmetric functions \(P_\mu(x_1,\cdots,x_n;q,t)\) with \((q,t)\) replaced by \((q^4,q^2)\) in the case \(\text{SO}\), and by \((q^2,q^4)\) in the case \(Sp\).

MSC:

33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations

Citations:

Zbl 0743.33012
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