Harmonic analysis on quantum spheres associated with representations of U q (𝔰𝔬 N ) and q-Jacobi polynomials
Compositio Mathematica, Tome 99 (1995) no. 3, pp. 249-281. 
@article{CM_1995__99_3_249_0,
     author = {Sugitani, Tetsuya},
     title = {Harmonic analysis on quantum spheres associated with representations of $U_q (\mathfrak {so}_N)$ and $q${-Jacobi} polynomials},
     journal = {Compositio Mathematica},
     pages = {249--281},
     publisher = {Kluwer Academic Publishers},
     volume = {99},
     number = {3},
     year = {1995},
     mrnumber = {1361741},
     zbl = {0872.17013},
     language = {en},
     url = {http://www.numdam.org/item/CM_1995__99_3_249_0/}
}
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Sugitani, Tetsuya. Harmonic analysis on quantum spheres associated with representations of $U_q (\mathfrak {so}_N)$ and $q$-Jacobi polynomials. Compositio Mathematica, Tome 99 (1995) no. 3, pp. 249-281. http://www.numdam.org/item/CM_1995__99_3_249_0/

[A] Abe, E.: Hopf algebras, Cambridge University Press, 1980. | MR | Zbl

[AW] Askey, E. and Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials 54 (1985), Memoirs Amer. Math. Soc., No. 319. | MR | Zbl

[B] Bergman, G.M.: The Diamond lemma for ring theory, Adv. in Math. 29 (1978), 178-218. | MR | Zbl

[D1] Drinfel'D, V.G.: Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk. SSSR 283 (1985), 1060-1064. | MR | Zbl

[D2] Drinfel'D, V.G.: Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, California, USA, 1986, pp. 798-820 | MR | Zbl

[GR] Gasper, G. and Rahman, M.: Basic hypergeometric series, Encyclopedia of Mathematics and its applications, Cambridge University Press, 1990. | MR | Zbl

[H] Howe, R.: Remarks on classical invariant theory, Trans. Amer. Math. 318 (1989), 539-570; Erratum, ibid. 318 (1990) p. 823. | MR | Zbl

[HU] Howe, R. and Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 565-619. | MR | Zbl

[J1] Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. | MR | Zbl

[J2] Jimbo, M.: Quantum R matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), 537-548. | MR | Zbl

[K1] Koornwinder, T.H.: Continuous q-Legendre polynomials as spherical matrix elements of irreducible representations of the quantum SU(2) group, CWI Quarterly 2 (1989), 171-173. | Zbl

[K2] Koornwinder, T.H.: Orthogonal polynomials in connection with quantum groups, in: Orthogonal Polynomials: Theory and Practice (P. Nevai, ed.), NATO ASI Series, Kluwer Academic Publishers, 1990, pp. 257-292. | MR | Zbl

[L] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), 237-249. | MR | Zbl

[Na] Nakashima, T.: Crystal base and a generalization of the Littlewood-Richardson Rule for the classical Lie algebras, Comm. Math. Phys. 154 (1993), 215-243. | MR | Zbl

[N1] Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. in Math. (to appear). | Zbl

[N2] Noumi, M.: A realization of Macdonald's symmetric polynomials on quantum homogeneous spaces, Int. J. Mod. Phys. A (Proc. Suppl.) 3A, World Scientific, 1993, pp. 218-223.

[N3] Noumi, M.: Quantum groups and q-orthogonal polynomials - Towards a realization of Askey-Wilson polynomials on SUq(2), in "Special Functions" (M. Kashiwara and T. Miwa, eds.), ICM-90 Satellite Conference Proceedings, Springer-Verlag, 1991, pp. 260-288. | MR | Zbl

[NM1] Noumi, M. and Mimachi, K.: Quantum 2-spheres and big q-Jacobi polynomials, Comm. Math. Phys. 128 (1990), 521-531. | MR | Zbl

[NM2] Noumi, M. and Mimachi, K.: Spherical functions on a family of quantum 3-spheres, Compositio Mathematica 83 (1992), 19-42. | Numdam | MR | Zbl

[NM3] Noumi, M. and Mimachi, K.: Rogers's q-ultraspherical polynomials on a quantum 2-sphere, Duke. Math. J. 63 (1991), 65-80. | MR | Zbl

[NM4] Noumi, M. and Mimachi, K.: Askey-Wilson polynomials and the quantum group SUq(2), Proc. Japan Acad. 66 (1990), 146-149. | MR | Zbl

[NUW1] Noumi, M., Umeda, T. and Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculus on GLq(n), preprint J-Tokyo-Math 91-16 (1991). | Zbl

[NUW2] Noumi, M., Umeda, T. and Wakayama, M.: A quantum dual pair (sl2, On) and the associated Capelli identity, preprint (1993).

[NUW3] Noumi, M., Umeda, T. and Wakayama, M.: Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, preprint (1993). | Numdam | Zbl

[NYM] Noumi, M., Yamada, H. and Mimachi, K.: Finite dimensional representations of the quantum group GLq(n;C) and the zonal spherical functions on U q(n - 1)\Uq(n), Japanese J. Math. 19 (1993), 31-80. | MR | Zbl

[P1] Podles, P.: Differential calculus on quantum spheres, Lett. Math. Phys.18 (1989), 107-119. | MR | Zbl

[P2] Podles, P.: Quantum spheres, Lett. Math. Phys. 14 (1987), 193-202. | MR | Zbl

[R] Rosso, M.: Finite representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581-593. | MR | Zbl

[RTF] Reshetikhin, N. Yu., Takhtajan, L.A. and Faddeev, L.D.: Quantization of Lie groups and Lie algebras, Algebra and Analysis 1 (1989), 178-206, English transl., in Leningrad Math. J. 1 (1990), 193-225. | MR | Zbl

[T] Tanisaki, T.: Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras, Int. J. Mod. Phys. A vol. 7 suppl. 1B (1992), 142-147. | MR | Zbl

[V] Vilenkin, N.J.: Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs vol. 22, 1968. | MR | Zbl

[W1] Woronowicz, S.L.: Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665. | MR | Zbl

[W2] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1980), 125-170. | MR | Zbl

[WSW] Watamura, U.C., Schlieker, M. and Watamura, S.: SOq(N) covariant differential calculus on quantum spaces and quantum deformation of Schrödinger equation, Zeitschrift für Physik C - Particles and Fields 49 (1991), 439-446. | MR