The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair
Annales de l'Institut Fourier, Volume 49 (1999) no. 4, p. 1179-1214

Let G/K a noncompact symmetric space with Iwasawa decomposition KAN. The Harish-Chandra homomorphism is an explicit homomorphism between the algebra of invariant differential operators on G/K and the algebra of polynomials on A that are invariant under the Weyl group action of the pair (G,A). The main result of this paper is a generalization to the quantum setting of the Harish-Chandra homomorphism in the case of G/K being an hermitian (classical) symmetric space

Soit G/K un espace symétrique non compact avec décomposition d’Iwasawa KAN. L’homomorphisme d’Harish-Chandra est un homomorphisme explicite entre l’algèbre des opérateurs différentiels sur G/K et l’algèbre des polynômes sur A invariante par rapport à l’action du groupe de Weyl de la paire (G,A). Le résultat principal de cet article est une généralisation dans le cas quantique de l’homomorphisme d’Harish-Chandra pour G/K symétrique hermitien (classique).

@article{AIF_1999__49_4_1179_0,
     author = {Baldoni, Welleda and Frajria, Pierluigi M\"oseneder},
     title = {The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {4},
     year = {1999},
     pages = {1179-1214},
     doi = {10.5802/aif.1713},
     zbl = {0932.17014},
     mrnumber = {2001d:17010},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_4_1179_0}
}
Baldoni, Welleda; Frajria, Pierluigi Möseneder. The Harish-Chandra homomorphism for a quantized classical hermitian symmetric pair. Annales de l'Institut Fourier, Volume 49 (1999) no. 4, pp. 1179-1214. doi : 10.5802/aif.1713. http://www.numdam.org/item/AIF_1999__49_4_1179_0/

[1] M. Baldoni, D. Barbach and P. Möseneder Frajria, Parabolic subalgebras of quantized enveloping algebras, Preprint.

[2] N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968.

[3] P. Möseneder Frajria, The annihilator of invariant vectors for a quantized parabolic subalgebra, Preprint.

[4] P. Möseneder Frajria, A guide to L-operators, Rend. Mat., (7) 18 (1998), 65-85. | MR 2000a:17019 | Zbl 0914.17007

[5] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Academic Press, 1978. | Zbl 0451.53038

[6] S. Helgason, Groups and geometric analysis, Academic Press, 1984.

[7] J. C. Jantzen, Lectures on quantum groups, Graduate studies in Mathematics, vol. 6, A.M.S., 1995. | MR 96m:17029 | Zbl 0842.17012

[8] A. Joseph, Quantum groups and their primitive ideals, Springer, Berlin, 1995. | MR 96d:17015 | Zbl 0808.17004

[9] A. Joseph and G. Letzter, Local finiteness of the adjoint action for quantized enveloping algebras, J. Algebra, 153 (1992), 289-317. | MR 94b:17023 | Zbl 0779.17012

[10] B. Kostant and S. Sahi, The Capelli identity, tube domains and the generalized Laplace transform, Adv. in Math., 87 (1991), 71-92. | MR 92h:22033 | Zbl 0748.22008

[11] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math., 70 (1988), 237-249. | MR 89k:17029 | Zbl 0651.17007

[12] G. Lusztig, Quantum groups at root of 1, Geom. Dedicata, 35 (1990), 89-114. | MR 91j:17018 | Zbl 0714.17013

[13] N. Yu. Reshetikhin, L.A. Takhtadzhyan, and L.D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J., 1 (1990), 193-225. | MR 90j:17039 | Zbl 0715.17015

[14] N. R. Wallach, The analytic continuation of the discrete series II, Trans. Amer. Math. Soc., 251 (1979), 19-37. | MR 81a:22009 | Zbl 0419.22018