Multivariable orthogonal polynomials and quantum Grassmannians (Q2761014)

This is a reprint of the author's PhD dissertation from 1998. Although most results have been published elsewhere (cf. \textit{J. F. van Diejen} and \textit{J. V. Stokman} [Duke Math. J. 91, 89-136 (1998; Zbl 0951.33010)], \textit{M. S. Dijkhuizen} and \textit{J. V. Stokman} [Publ. Res. Inst. Math. Sci. 35, 451-500 (1999; Zbl 0960.33010)], [\textit{J. V. Stokman}, SIAM J. Math. Anal. 28, No. 2, 453-480 (1997; Zbl 0892.33010), Trans. Am. Math. Soc. 352, 1527-1579 (2000; Zbl 0936.33008)], \textit{J. V. Stokman} and \textit{T. H. Koornwinder} [Can. J. Math. 49, 373-404 (1997; Zbl 0881.33026)], \textit{J. V. Stokman} and \textit{M. S. Dikhuizen} [Commun. Math. Phys. 203, 297-324 (1999; Zbl 0969.58003)]), it is valuable that this well-written thesis has been made more generally available. The thesis may be divided into three closely related parts. The first part is devoted to an analytic study of the Koornwinder polynomials (a simultaneous generalization of Askey-Wilson and Macdonald polynomials) and their limit cases. The parameter range of these polynomials is extended to the case when the orthogonality measure is partially discrete. Limit transitions then lead to polynomials with purely discrete orthogonality measures, which are multivariable analogues of \(q\)-Racah, big \(q\)-Jacobi and little \(q\)-Jacobi polynomials. The constant term identities for the two latter classes are discrete \(q\)-Selberg integrals of a type going back to \textit{R. Askey} [SIAM J. Math. Anal. 11, 938-951 (1980; Zbl 0458.33002)] and studied extensively since then (\textit{L. Habsieger} [SIAM J. Math. Anal. 19, 1475-1489 (1988; Zbl 0664.33001)], \textit{K. W. J. Kadell} [SIAM J. Math. Anal. 19, 969-986 (1988; Zbl 0643.33004)], \textit{R. J. Evans} [Contemp. Math. 166, 341-357 (1994; Zbl 0820.33001)], \textit{V. Tarasov} and \textit{A. Varchenko} [Astérisque 246 (1997; Zbl 0938.17012)], \textit{K. Aomoto} [J. Algebr. Comb. 8, 115-126 (1998; Zbl 0918.33013)]). The author's approach yields new proofs of these identities in their most general form.NEWLINENEWLINENEWLINEIn the second part the author interprets his multivariable big and little \(q\)-Jacobi polynomials as zonal spherical functions on quantum Grassmannians. Formally, these results are limit cases of a similar interpretation of certain Koornwinder polynomials due to \textit{M. Noumi, M. S. Dijkhuizen} and \textit{T. Sugitani} [Fields Inst. Commun. 14, 167-177 (1997; Zbl 0877.33012)]. The author provides detailed proofs of results announced in the paper mentioned and gives a rigorous interpretation of the limit procedures involved. The last part of the thesis deals with algebraic properties of quantum flag manifolds. A factorization theorem, generalizing the description of a complex Grassmannian using holomorphic and antiholomorphic Plücker coordinates, is obtained for a large class of quantized function algebras on flag manifolds. This makes it possible to apply the quantum orbit method to these algebras, leading to a classification of their unitary irreducible representations; these are parametrized by the Schubert cells of the underlying flag manifold.