Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A family of quantum projective spaces
and related $q$-hypergeometric
orthogonal polynomials


Authors: Mathijs S. Dijkhuizen and Masatoshi Noumi
Journal: Trans. Amer. Math. Soc. 350 (1998), 3269-3296
MSC (1991): Primary 33D80, 81R50, 17B37, 33D45
DOI: https://doi.org/10.1090/S0002-9947-98-01971-0
MathSciNet review: 1432197
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A one-parameter family of two-sided coideals in $\mathcal{U}_{q} (\mathfrak{g}\mathfrak{l}(n))$ is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group $U_{q}(n)$ are studied. The Plancherel decomposition of these algebras with respect to the natural transitive $U_{q}(n)$-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little $q$-Jacobi polynomials depending on one discrete parameter.


References [Enhancements On Off] (What's this?)

  • [AA1] G.E. Andrews, R. Askey, Enumeration of partitions: the role of Eulerian series and $q$-orthogonal polynomials, Higher Combinatorics (M. Aigner, ed.), Reidel, Boston, MA, 1977, pp. 3-26. MR 80b:10021
  • [AA2] -, Classical orthogonal polynomials, Polynômes Orthogonaux et Applications (C. Brezinski, A. Draux, A.P. Magnus, P. Maroni, A. Ronveaux, eds.), Lecture Notes in Math. 1171, Springer-Verlag, Berlin, 1985, pp. 36-62. MR 88c:33015b
  • [AW] R. Askey, J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (319) (1985), 1-55. MR 87a:05023
  • [Dz] M.S. Dijkhuizen, Some remarks on the construction of quantum symmetric spaces, Acta Appl. Math. 44 (1996), 59-80. CMP 96:17
  • [DK1] M.S. Dijkhuizen, T.H. Koornwinder, Quantum homogeneous spaces, duality and quantum 2-spheres, Geom. Dedicata 52 (1994), 291-315. MR 95i:16037
  • [DK2] -, CQG algebras: a direct algebraic approach to compact quantum groups, Lett. Math. Phys. 32 (1994), 315-330. MR 95m:16029
  • [Dr1] V.G. Drinfel'd, Quantum groups, Proceedings ICM Berkeley (1986) (A.M. Gleason, ed.), Amer. Math. Soc., Providence, RI, 1986, pp. 798-820. MR 89f:17017
  • [Dr2] -, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (2) (1990), 321-342. MR 91b:16046
  • [GR] G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications 35, Cambridge University Press, London, 1990. MR 91d:33034
  • [J] M. Jimbo, A $q$-analogue of $U(\mathfrak{g}\mathfrak{l}(n))$, Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252. MR 87k:17011
  • [Kk] H.T. Koelink, The addition formula for continuous $q$-Legendre polynomials and associated spherical elements on the $SU(2)$ quantum group related to Askey-Wilson polynomials, SIAM J. Math. Anal. 25 (1) (1994), 197-217. MR 95f:33023
  • [K1] T.H. Koornwinder, Representations of the twisted $SU(2)$ quantum group and some $q$-hypergeometric orthogonal polynomials, Indag. Math. 51 (1989), 97-117. MR 90h:17016
  • [K2] -, Orthogonal polynomials in connection with quantum groups, Orthogonal polynomials: Theory and Practice (P. Nevai, ed.), NATO-ASI Series C 294, Kluwer, Norwell, MA, 1990, pp. 257-292. MR 91i:33018
  • [K3] -, Askey-Wilson polynomials as zonal spherical functions on the $SU(2)$ quantum group, SIAM J. Math. Anal. 24 (3) (1993), 795-813. MR 94k:33042
  • [KV] L.I. Korogodsky, L.L. Vaksman, Quantum $G$-spaces and Heisenberg algebra, Quantum Groups (P.P. Kulish, ed.), Lecture Notes in Math. 1510, Springer-Verlag, Berlin, 1992, pp. 56-66. MR 93k:17031
  • [L] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249. MR 89k:17029
  • [M] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, K. Ueno, Representations of quantum groups and a $q$-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 559-564. MR 90a:17013
  • [N] M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), 16-77. MR 98a:33004
  • [NM1] M. Noumi, K. Mimachi, Quantum 2-spheres and big $q$-Jacobi polynomials, Comm. Math. Phys. 128 (1990), 521-531. MR 91f:33010
  • [NM2] -, Askey-Wilson polynomials and the quantum group $SU_{q}(2)$, Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), 146-149. MR 91k:33015
  • [NM3] -, Spherical functions on a family of quantum 3-spheres, Compositio Math. 83 (1992), 19-42. MR 93f:33010
  • [NS1] M. Noumi, T. Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, Group Theoretical Methods in Physics (A. Arima et al., eds.), Proceedings XX ICGTMP, Toyonaka (Japan), 1994, World Scientific, Singapore, 1995, pp. 28-40. MR 97h:33033
  • [NS2] -, Quantum symmetric spaces and multivariable orthogonal polynomials, preprint (1996).
  • [NYM] M. Noumi, H. Yamada, K. Mimachi, Finite-dimensional representations of the quantum group $GL_{q}(n,{\mathbb C})$ and the zonal spherical functions on $U_{q}(n-1)\backslash U_{q}(n)$, Japanese J. Math. 19 (1) (1993), 31-80. MR 94i:17023
  • [P] P. Podles, Quantum spheres, Lett. Math. Phys. 14 (1987), 193-202. MR 89b:46081
  • [RTF] N. Reshetikhin, L.A. Takhtadzhyan, L.D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225. MR 90j:17039
  • [R] M. Rosso, Finite-dimensional representations of the quantum analog of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), 581-593. MR 90c:17019
  • [So] Ya. S. Soibel'man, Irreducible representations of the function algebra on the quantum group $SU(n)$, and Schubert cells, Soviet Math. Dokl. 40 (1) (1990), 34-38. MR 91d:17019
  • [SV] Ya. S. Soibel'man, L.L. Vaksman, On some problems in the theory of quantum groups, Representation Theory and Dynamical Systems (A.M. Vershik, ed.), Adv. Sov. Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 3-55. MR 93h:17043
  • [S1] T. Sugitani, Harmonic analysis on quantum spheres associated with representations of $\mathcal U_q(\mathfrak{s}\mathfrak{o}_{N})$ and $q$-Jacobi polynomials, Compositio Math. 99 (1995), 249-281. MR 96k:33029
  • [S2] -, Zonal spherical functions on quantum Grassmann manifolds, preprint (1996).
  • [VS1] L.L. Vaksman, Y.S. Soibel'man, Algebra of functions on the quantum group $SU(2)$, Funct. Anal. Appl. 22 (1988), 170-181. MR 90f:17019
  • [VS2] -, The algebra of functions on the quantum group $SU(n+1)$ and odd-dimensional quantum spheres, Leningrad Math. J. 2 (5), 1023-1042. MR 92e:58021
  • [W] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665. MR 88m:46079

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 33D80, 81R50, 17B37, 33D45

Retrieve articles in all journals with MSC (1991): 33D80, 81R50, 17B37, 33D45


Additional Information

Mathijs S. Dijkhuizen
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan
Email: msdz@math.s.kobe-u.ac.jp

Masatoshi Noumi
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe 657, Japan
Email: noumi@math.s.kobe-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-98-01971-0
Keywords: Quantum unitary group, quantum projective space, two-sided coideal, zonal spherical function, Casimir operator, radial part, second-order $q$-difference operator, Askey-Wilson polynomials, big and little $q$-Jacobi polynomials
Received by editor(s): April 28, 1996
Received by editor(s) in revised form: October 1, 1996
Additional Notes: The first author acknowledges financial support by the Japan Society for the Promotion of Science (JSPS) and the Netherlands Organization for Scientific Research (NWO)
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society