Glasnik Matematicki, Vol. 50, No. 2 (2015), 397-414.

K-INVARIANTS IN THE ALGEBRA U(𝔤) ⊗ C(𝔭) FOR THE GROUP SU(2,1)

Ana Prlić

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: anaprlic@math.hr


Abstract.   Let 𝔤 = 𝔨 ⊕ 𝔭 be the Cartan decomposition of the complexified Lie algebra 𝔤 =𝔰𝔩 (3,C) of the group G=SU(2,1). Let K=S(U(2)× U(1)), so K is a maximal compact subgroup of G. Let U(𝔤) be the universal enveloping algebra of 𝔤, and let C(𝔭) be the Clifford algebra with respect to the trace form B(X,Y)=tr(XY) on 𝔭. We are going to prove that the algebra of K-invariants in U(𝔤) ⊗ C(𝔭) is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for (𝔤,K)-modules. Along the way we will also recover the (well known) structure of the algebra U(𝔤)K.

2010 Mathematics Subject Classification.   22E47, 22E46.

Key words and phrases.   Lie group, Lie algebra, representation, special unipotent representation, Dirac operator, Dirac cohomology.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.50.2.09


References:

  1. A. Alekseev and E. Meinrenken, Lie theory and the Chern-Weil homomorphism, Ann. Sci. Ecole. Norm. Sup. 38 (2005), 303-338.
    MathSciNet     CrossRef

  2. D. Barbasch, D. Ciubotaru and P.E. Trapa, Dirac cohomology for graded affine Hecke algebras, Acta Math. 209 (2012), 197-227.
    MathSciNet     CrossRef

  3. D. Barbasch and P. Pandžić, Dirac cohomology and unipotent representations of complex groups, in: Noncommutative Geometry and Global Analysis, eds. A. Connes, A. Gorokhovsky, M. Lesch, M. Pflaum, B. Rangipour, Contemporary Mathematics, vol. 546, American Mathematical Society, 2011, 1-22.
    MathSciNet    

  4. D. Barbasch and P. Pandžić, Dirac cohomology of unipotent representations of Sp(2n,R) and U(p,q), J. Lie Theory 25 (2015), 185-213.
    MathSciNet    

  5. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
    MathSciNet     CrossRef

  6. Harish-Chandra, Representations of semisimple Lie groups. II, Trans. Amer. Math. Soc. 76 (1954), 26-65.
    MathSciNet     CrossRef

  7. J.-S. Huang, Y.-F. Kang and P. Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009), 163-173.
    MathSciNet     CrossRef

  8. J.-S. Huang and P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185-202.
    MathSciNet     CrossRef

  9. J.-S. Huang and P. Pandžić, Dirac Operators in Representation Theory, Mathematics: Theory and Applications, Birkhauser, 2006.
    MathSciNet    

  10. J.-S. Huang, P. Pandžić, Dirac cohomology for Lie superalgebras, Transform. Groups 10 (2005), 201-209.
    MathSciNet     CrossRef

  11. J.-S. Huang, P. Pandžić and V. Protsak, Dirac cohomology of Wallach representations, Pacific J. Math. 250 (2011), 163-190.
    MathSciNet     CrossRef

  12. J.-S. Huang, P. Pandžić and D. Renard, Dirac operators and Lie algebra cohomology, Represent. Theory 10 (2006), 299-313.
    MathSciNet     CrossRef

  13. J.-S. Huang, P. Pandžić and F. Zhu, Dirac cohomology, K-characters and branching laws, Amer. J. Math. 135 (2013), no.5, 1253-1269.
    MathSciNet     CrossRef

  14. K. D. Johnson, The centralizer of a Lie algebra in an enveloping algebra, J. reine angew. Math. 395 (1989) 196-201.
    MathSciNet     CrossRef

  15. V. Kac, P. Möseneder Frajria and P. Papi, Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting, Adv. Math. 217 (2008), 2485-2562.
    MathSciNet     CrossRef

  16. F. Knop, A Harish-Chandra homomorphism for reductive group actions, Ann. of Math. (2) 140 (1994), 253-288.
    MathSciNet     CrossRef

  17. B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404.
    MathSciNet     CrossRef

  18. B. Kostant, Dirac cohomology for the cubic Dirac operator, Studies in Memory of Issai Schur, Progress in Mathematics, Vol. 210 (2003), 69-93.
    MathSciNet    

  19. B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809.
    MathSciNet     CrossRef

  20. S. Kumar, Induction functor in noncommutative equivariant cohomology and Dirac cohomology, J. Algebra 291 (2005), 187-207.
    MathSciNet     CrossRef

  21. J. Lepowsky and G. W. McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45-57.
    MathSciNet     CrossRef

  22. P. Pandžić and D. Renard, Dirac induction for Harish-Chandra modules, J. Lie Theory 20 (2010), 617-641.
    MathSciNet    

  23. R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.
    MathSciNet     CrossRef

  24. R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. 89 (1980), 1-24.
    MathSciNet    

  25. A. Prlić, Algebraic Dirac induction for nonholomorphic discrete series of SU(2,1), Ph.D. thesis, University of Zagreb, 2014.

  26. D. A. Vogan, Jr., Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997.

Glasnik Matematicki Home Page