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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
Published electronically: September 8, 2011
MathSciNet review: 2833470
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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.

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

  • [Br] 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)
  • [Bo] A. Borel, Linear algebraic groups, Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
  • [EV1] P. Etingof, A. Varchenko, Dynamical Weyl groups and applications, Adv. Math. 167 (2002), no. 1, 74-127. MR 1901247 (2003d:17004)
  • [EV2] 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)
  • [EFK] 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)
  • [KNV] S. Khoroshkin, M. Nazarov, E. Vinberg, A generalized Harish-Chandra isomorphism, Adv. Math. 226 (2011), no. 2, 1168-1180. MR 2737780
  • [KS] 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] T. Tanisaki, Harish-Chandra isomorphisms for quantum algebras, Comm. Math. Phys. 127 (1990), no. 3, 555-571. MR 1040894 (92e:17022)

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Additional Information

Martina Balagović
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): April 2, 2010
Received by editor(s) in revised form: June 10, 2011
Published electronically: September 8, 2011
Additional Notes: This work was partially supported by the NSF grant DMS-0504847.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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