Previous volume | This volume | Most recent volume | All volumes | Next volume | Previous article | Next article
Chevalley restriction theorem for vector-valued functions on quantum groups
Author: Martina Balagović
Journal: Represent. Theory 15 (2011), 617-645
MSC (2010): Primary 17B37, 20G42
DOI: https://doi.org/10.1090/S1088-4165-2011-00408-8
Published electronically: September 8, 2011
Abstract: We generalize Chevalley's theorem about restriction $\operatorname {Res}: \mathbb {C}[\mathfrak {g}]^{\mathfrak {g}} \to \mathbb {C}[\mathfrak {h}]^W$ to the case when a semisimple Lie algebra $\mathfrak {g}$ is replaced by a quantum group and the target space $\mathbb {C}$ of the polynomial maps is replaced by a finite dimensional representation $V$ of this quantum group. We prove that the restriction map $\operatorname {Res}:(O_{q}(G)\otimes V)^{U_{q}(\mathfrak {g})}\to O(H)\otimes V$ is injective and describe the image.
Abraham Broer, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants, Indag. Math. (N.S.) 6 (1995), no. 4, 385–396. MR 1365182, DOI https://doi.org/10.1016/0019-3577%2896%2981754-X
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
P. Etingof and A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74–127. MR 1901247, DOI https://doi.org/10.1006/aima.2001.2034
Pavel Etingof and Alexander Varchenko, Traces of intertwiners for quantum groups and difference equations. I, Duke Math. J. 104 (2000), no. 3, 391–432. MR 1781477, DOI https://doi.org/10.1215/S0012-7094-00-10433-4
Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr., Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), no. 1, 59–90. MR 1360611, DOI https://doi.org/10.1215/S0012-7094-95-08003-X
Sergey Khoroshkin, Maxim Nazarov, and Ernest Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), no. 2, 1168–1180. MR 2737780, DOI https://doi.org/10.1016/j.aim.2010.08.001
Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups. Part I, Mathematical Surveys and Monographs, vol. 56, American Mathematical Society, Providence, RI, 1998. MR 1614943
Toshiyuki Tanisaki, Harish-Chandra isomorphisms for quantum algebras, Comm. Math. Phys. 127 (1990), no. 3, 555–571. MR 1040894
A. Broer, The sum of generalized exponents and Chevalley's restriction theorem for modules of covariants, Indag. Math. (N.S.) 6 (1995), no. 4, 385–396. MR 1365182 (96j:20058)
A. Borel, Linear algebraic groups, Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
P. Etingof, A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74–127. MR 1901247 (2003d:17004)
P. Etingof, A. Varchenko, Traces of intertwiners for quantum groups and difference equations, I, Duke Math. J. 104 (2000), no. 3, 391–432. MR 1781477 (2001k:17021)
P. Etingof, I. Frenkel, A. Kirillov Jr, Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), no. 1, 59–90. MR 1360611 (97e:22018)
S. Khoroshkin, M. Nazarov, E. Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), no. 2, 1168–1180. MR 2737780
L. Korogodski, Y. Soibelman, Algebras of Functions on Quantum Groups: Part I, Mathematical Surveys and Monographs, 56. American Mathematical Society, Providence, RI, 1998. MR 1614943 (99a:17022)
T. Tanisaki, Harish-Chandra isomorphisms for quantum algebras, Comm. Math. Phys. 127 (1990), no. 3, 555–571. MR 1040894 (92e:17022)
Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 17B37, 20G42
Retrieve articles in all journals with MSC (2010): 17B37, 20G42
Martina Balagović
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: martinab@math.mit.edu
Received by editor(s): April 2, 2010
Additional Notes: This work was partially supported by the NSF grant DMS-0504847.
The copyright for this article reverts to public domain 28 years after publication.