Dual pairs, spherical harmonics and a Capelli identity in quantum group theory
Compositio Mathematica, Tome 104 (1996) no. 3, pp. 227-277. 
@article{CM_1996__104_3_227_0,
     author = {Noumi, Masatoshi and Umeda, T\^oru and Wakayama, Masato},
     title = {Dual pairs, spherical harmonics and a {Capelli} identity in quantum group theory},
     journal = {Compositio Mathematica},
     pages = {227--277},
     publisher = {Kluwer Academic Publishers},
     volume = {104},
     number = {3},
     year = {1996},
     mrnumber = {1424556},
     zbl = {0930.17012},
     language = {en},
     url = {http://archive.numdam.org/item/CM_1996__104_3_227_0/}
}
TY  - JOUR
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
SP  - 227
EP  - 277
VL  - 104
IS  - 3
PB  - Kluwer Academic Publishers
UR  - http://archive.numdam.org/item/CM_1996__104_3_227_0/
LA  - en
ID  - CM_1996__104_3_227_0
ER  - 
%0 Journal Article
%A Noumi, Masatoshi
%A Umeda, Tôru
%A Wakayama, Masato
%T Dual pairs, spherical harmonics and a Capelli identity in quantum group theory
%J Compositio Mathematica
%D 1996
%P 227-277
%V 104
%N 3
%I Kluwer Academic Publishers
%U http://archive.numdam.org/item/CM_1996__104_3_227_0/
%G en
%F CM_1996__104_3_227_0
Noumi, Masatoshi; Umeda, Tôru; Wakayama, Masato. Dual pairs, spherical harmonics and a Capelli identity in quantum group theory. Compositio Mathematica, Tome 104 (1996) no. 3, pp. 227-277. http://archive.numdam.org/item/CM_1996__104_3_227_0/

[Ab] Abe, E.: HopfAlgebras, Cambridge Math. Tracts 74, Cambridge Univ. Press, 1980. | MR | Zbl

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

[D] Drinfel'D, V.G.: Quantum groups, in Proc. Int. Cong. Math. Berkeley, 1986, pp. 798-820. | MR | Zbl

[F] Fischer, E.: Über algebraische Modulsysteme und lineare homogene partielle Differentialgleichungen mit konstaten Koeffizienten, J. Reine Angew. Math. 140 (1911) 48-81. | EuDML | JFM | MR

[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. | MR | Zbl

[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. | MR | Zbl

[HiW] Hibi, T. and Wakayama, M.: A q-analogue of Capelli's identity for GL q (2), Adv. in Math. (to appear). | MR | Zbl

[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. | MR | Zbl

[H2] Howe, R.: Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, in Lectures in Applied Math. vol. 21, 1985, pp. 179-207. | 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

[Ja] Jackson, F.H.: On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253-281.

[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.: A q-analogue of U(gt(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986) 247-252. | MR | Zbl

[Na1] Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991) 123-131. | MR | Zbl

[Na2] Nazarov, M.L.: private communication.

[N1] Noumi, M.: A remark on semisimple elements in Uq (sl(2, C)), Combinatorial Aspects in Representation Theory and Geometry, RIMS Kôkyûroku 765 (1991) 71-78.

[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.

[N3] Noumi, M.: Quantum Grassmannians and q-hypergeometric series, CWI Quarterly 5 (1992) 293-307. | MR | Zbl

[N4] Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, to appear, Adv. Math.. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[U] Umeda, T.: Notes on almost homogeneity, preprint (1993).

[UW] Umeda, T. and Wakayama, M.: Another look at the differential operators on quantum matrix spaces and its applications, in preparation. | Zbl

[Wy] Weyl, H.: The Classical Groups, their Invariants and Representations, Princeton Univ. Press, 1946. | MR | Zbl