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Harish-Chandra modules for quantum symmetric pairs

Author: Gail Letzter
Journal: Represent. Theory 4 (2000), 64-96
MSC (2000): Primary 17B37
Published electronically: February 18, 2000
MathSciNet review: 1742961
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Abstract: Let $U$ denote the quantized enveloping algebra associated to a semisimple Lie algebra. This paper studies Harish-Chandra modules for the recently constructed quantum symmetric pairs $U$,$B$ in the maximally split case. Finite-dimensional $U$-modules are shown to be Harish-Chandra as well as the $B$-unitary socle of an arbitrary module. A classification of finite-dimensional spherical modules analogous to the classical case is obtained. A one-to-one correspondence between a large class of natural finite-dimensional simple $B$-modules and their classical counterparts is established up to the action of almost $B$-invariant elements.

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  • [CP] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1995. MR 96h:17014
  • [DK] C. DeConcini and V.G. Kac, Representations of quantum groups at roots of 1, In: Operator Algebras, Unitary Representations, Envelping Algebras, and Invariant Theory, Prog. Math. 92 (1990), 471-506. MR 92g:17012
  • [Di] M.S. Dijkhuizen, Some remarks on the construction of quantum symmetric spaces, In: Representations of Lie groups, Lie algebras and their quantum analogues, Acta Appl. Math. 44 (1996), no. 1-2, 59-80. MR 98c:33020
  • [D] J. Dixmier, Algebres Enveloppantes, Cahiers Scientifiques, XXXVII, Gauthier-Villars, Paris (1974). MR 58:16803a
  • [GI] A.M. Gavrilik and N.Z. Iorgov, $q$-deformed inhomogeneous algebras $U_q(so_n)^{\phantom{x}}$ and their representations, In: Symmetry in nonlinear mathematical physics, Vol. 1, 2, Natl. Acad. Sci. Ukraine, Inst. Math., Kiev (1997), 384-392. MR 99c:81087
  • [Ja] N. Jacobson, Basic Algebra. II, W. H. Freeman and Co., San Francisco, CA, 1980. MR 81g:00001
  • [J] J.C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), 53-65. MR 55:12783
  • [JL1] A. Joseph, and G. Letzter, Local finiteness of the adjoint action for quantized enveloping algebras, Journal of Algebra, 153 (1992), 289-318. MR 94b:17023
  • [JL2] A. Joseph, and G. Letzter, Separation of variables for quantized enveloping algebras, American Journal of Mathematics, 116, (1994), 127-177. MR 95e:17017
  • [Jo] A. Joseph, Quantum Groups and their Primitive Ideals, Springer-Verlag, New York (1995). MR 96d:17015
  • [Jo2] A. Joseph, On a Harish-Chandra homomorphism, C. R. Acad. Sci. Paris Ser. I Math. 324 (1997), no. 7, 759-764. MR 98d:17017
  • [Ke] M.S. Kebe, ${\mathcal O}$-algebres quantiques, C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), no. 1, 1-4. MR 97c:17022
  • [Kn] A.W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, Birkhauser, Boston, MA 140 (1996). MR 98b:22002
  • [K1] B. Kostant, On the existence and Irreducibility of Certain Series of Representations. In: Lie groups an their representations, Summer School Conference, Budapest, 1971, Halsted press, New York (1975), 231-331. MR 53:3206
  • [K2] B. Kostant, Lie group representations on polynomial rings, American Journal of Mathematics 85 (1963), 327-404. MR 28:1252
  • [L1] G. Letzter, Subalgebras which appear in quantum Iwasawa decompositions, Canadian Journal of Mathematics 49 (1997), no. 6, 1206-1223. MR 99g:17022
  • [L2] G. Letzter, Symmetric Pairs for Quantized Enveloping algebras Journal of Algebra, 220, (1999), 729-767. CMP 2000:03
  • [N] M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Advances in Mathematics 123 (1996) no. 1, 16-77. MR 98a:33004
  • [NS] M. Noumi and T. Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, Group Theoretical Methods in Physics (ICGTMP), (Toyonaka, Japan, 1994) World Sci. Publishing, River Edge, N.J. (1995) 28-40. MR 97h:33033
  • [R1] M. Rosso, Groupes Quantiques, Representations Lineaires et Applications, Thesis Paris 7, (1990).
  • [R2] M. Rosso, Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988) 581-593. MR 90c:17019

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Additional Information

Gail Letzter
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Received by editor(s): October 22, 1999
Received by editor(s) in revised form: November 19, 1999
Published electronically: February 18, 2000
Additional Notes: The author was supported by NSF grant no. DMS-9753211
Article copyright: © Copyright 2000 American Mathematical Society