Dual pairs, spherical harmonics and a Capelli identity in quantum group theory

Masatoshi Noumi; Tôru Umeda; Masato Wakayama

Compositio Mathematica (1996)

  • Volume: 104, Issue: 3, page 227-277
  • ISSN: 0010-437X

How to cite


Noumi, Masatoshi, Umeda, Tôru, and Wakayama, Masato. "Dual pairs, spherical harmonics and a Capelli identity in quantum group theory." Compositio Mathematica 104.3 (1996): 227-277. <http://eudml.org/doc/90489>.

author = {Noumi, Masatoshi, Umeda, Tôru, Wakayama, Masato},
journal = {Compositio Mathematica},
keywords = {quantum group; Capelli identity; dual pair; quantum spherical harmonics; Casimir element; double commutant theorem},
language = {eng},
number = {3},
pages = {227-277},
publisher = {Kluwer Academic Publishers},
title = {Dual pairs, spherical harmonics and a Capelli identity in quantum group theory},
url = {http://eudml.org/doc/90489},
volume = {104},
year = {1996},

AU - Noumi, Masatoshi
AU - Umeda, Tôru
AU - Wakayama, Masato
TI - Dual pairs, spherical harmonics and a Capelli identity in quantum group theory
JO - Compositio Mathematica
PY - 1996
PB - Kluwer Academic Publishers
VL - 104
IS - 3
SP - 227
EP - 277
LA - eng
KW - quantum group; Capelli identity; dual pair; quantum spherical harmonics; Casimir element; double commutant theorem
UR - http://eudml.org/doc/90489
ER -


  1. [Ab] Abe, E.: HopfAlgebras, Cambridge Math. Tracts 74, Cambridge Univ. Press, 1980. Zbl0476.16008MR594432
  2. [B] Bergman, G.M.: The diamond lemma for ring theory, Adv. Math.29 (1978) 178-218. Zbl0326.16019MR506890
  3. [D] Drinfel'd, V.G.: Quantum groups, in Proc. Int. Cong. Math. Berkeley, 1986, pp. 798-820. Zbl0667.16003MR934283
  4. [F] Fischer, E.: Über algebraische Modulsysteme und lineare homogene partielle Differentialgleichungen mit konstaten Koeffizienten, J. Reine Angew. Math.140 (1911) 48-81. Zbl42.0148.01JFM42.0148.01
  5. [GK] Gavrilik, A.M. and Klimyk, A.U.: q-Deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys.21 (1991) 215-220. Zbl0735.17020MR1102131
  6. [Ha] Hayashi, T.: Q-analogues of Clifford and Weyl algebras - spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys.127 (1990) 129-144. Zbl0701.17008MR1036118
  7. [HiW] Hibi, T. and Wakayama, M.: A q-analogue of Capelli's identity for GL q (2), Adv. in Math. (to appear). Zbl0937.17009
  8. [H1] Howe, R.: Remarks on classical invariant theory, Trans. Amer. Math. Soc.313 (1989) 539-570; Erratum, Trans. Amer. Math. Soc.318 (1990) p. 823. Zbl0726.15025MR986027
  9. [H2] Howe, R.: Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, in Lectures in Applied Math. vol. 21, 1985, pp. 179-207. Zbl0558.22018MR789290
  10. [HU] Howe, R. and Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann.290 (1991) 565-619. Zbl0733.20019MR1116239
  11. [Ja] Jackson, F.H.: On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh46 (1908) 253-281. 
  12. [J1] Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63-69. Zbl0587.17004MR797001
  13. [J2] Jimbo, M.: A q-analogue of U(gt(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys.11 (1986) 247-252. Zbl0602.17005MR841713
  14. [Na1] Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys.21 (1991) 123-131. Zbl0722.17004MR1093523
  15. [Na2] Nazarov, M.L.: private communication. 
  16. [N1] Noumi, M.: A remark on semisimple elements in Uq (sl(2, C)), Combinatorial Aspects in Representation Theory and Geometry, RIMS Kôkyûroku765 (1991) 71-78. 
  17. [N2] Noumi, M.: A realization of Macdonald's symmetric polynomials on quantum homogeneous spaces, in Proceedings of the 21st International Conference on Differential Geometric Methods in Theoretical Physics, Tianjin China, 1992. 
  18. [N3] Noumi, M.: Quantum Grassmannians and q-hypergeometric series, CWI Quarterly5 (1992) 293-307. Zbl0782.33015MR1213744
  19. [N4] Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, to appear, Adv. Math.. Zbl0874.33011MR1413836
  20. [NUW1] Noumi, M., Umeda, T. and Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculas on GLq (n)Duke Math. J.76 (1994) 567-594. Zbl0835.17013MR1302325
  21. [NUW2] Noumi, M., Umeda, T. and Wakayama, M.: A quantum dual pair (sl2, On) and the associated Capelli identity, Lett. Math. Phys.34 (1995) 1-8. Zbl0842.17020MR1334029
  22. [O] Olshanski, G.I.: Twisted Yangians and infinite-dimensional classical Lie algebras, in Quantum Groups, Lecture Notes in Math. 1510, Springer Verlag (1992), pp. 103-120. Zbl0780.17025MR1183482
  23. [RTF] Reshetikhin, N. Yu., Takhtadzyan, L.A. and Faddeev. L.D.: Quantization of the Lie groups and Lie algebras, Leningrad Math. J.1 (1990) 193-225. Zbl0715.17015MR1015339
  24. [U] Umeda, T.: Notes on almost homogeneity, preprint (1993). 
  25. [UW] Umeda, T. and Wakayama, M.: Another look at the differential operators on quantum matrix spaces and its applications, in preparation. Zbl0957.17023
  26. [Wy] Weyl, H.: The Classical Groups, their Invariants and Representations, Princeton Univ. Press, 1946. Zbl1024.20501MR1488158

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