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Kostant homology formulas for oscillator modules of Lie superalgebras. (English) Zbl 1210.17026

In [“Lie algebra cohomology and the generalized Borel-Weil theorem,” Ann. Math. (2) 74, 329–387 (1961; Zbl 0134.03501)] B. Kostant computed the (co)homology groups for finite dimensional simple modules of a semisimple Lie algebra using by the Borel-Weil-Bott theorem. In this calculation the Weyl character formula has been recovered in a purely algebraic way. Kostant’s calculations has been generalized to integrable modules of Kac-Moody algebras, unitarizable highest weight modules of Hermitian symmetric pairs, finite dimensional modules of general linear superalgebras and recently to modules of infinite dimensional Lie superalgebras.
On the other hand Howe’s theory of reductive dual pairs has played important roles in the representation theory of real and \(p\)-adic Lie groups and there have been generalizations in different directions.
The main aim of this paper is to develop a conceptual approach to computing Kostant homology groups with coefficients in the oscillator \(\bar{\mathfrak{g}}\)-modules. To do this first they give a review and some notations for various Howe dualities involving the infinite dimensional Lie algebras \(\mathfrak{g}\) and finite dimensional Lie superalgebras \(\bar{\mathfrak{g}}\). Then they compute the character formulas of the oscillator \(\bar{\mathfrak{g}}\)-modules from Howe dualities. Finally, they calculate Casimir eigenvalues of representations of \(\bar{\mathfrak{g}}\) and \(\mathfrak{g}\) and obtain formulas for Kostant homology groups for oscillator \(\bar{\mathfrak{g}}\)-modules.
Reviewer: Cenap Özel (Bolu)

MSC:

17B55 Homological methods in Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B65 Infinite-dimensional Lie (super)algebras
17B56 Cohomology of Lie (super)algebras

Citations:

Zbl 0134.03501
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References:

[1] Aribaud, F., Une nouvelle démonstration d’un théorème de R. Bott et B. Kostant, Bull. Soc. Math. France, 95, 205-242 (1967) · Zbl 0155.06901
[2] Berele, A.; Regev, A., Hook Young diagrams with applications to combinatorics and representations of Lie superalgebras, Adv. Math., 64, 118-175 (1987) · Zbl 0617.17002
[3] Bröcker, T.; tom Dieck, T., Representations of Compact Lie Groups (1995), Springer-Verlag · Zbl 0874.22001
[4] Cheng, S.-J.; Kwon, J.-H., Howe duality and Kostant’s homology formula for infinite dimensional Lie superalgebras, Int. Math. Res. Not. (IMRN) (2008), Art. ID rnn 085, 52 pp
[5] Cheng, S.-J.; Lam, N.; Wang, W., Super duality and irreducible characters of ortho-symplectic Lie superalgebras, preprint · Zbl 1246.17007
[6] Cheng, S.-J.; Lam, N.; Zhang, R. B., Character formula for infinite dimensional unitarizable modules of the general linear superalgebra, J. Algebra, 273, 780-805 (2004) · Zbl 1041.17006
[7] Cheng, S.-J.; Wang, W., Howe duality for Lie superalgebras, Compos. Math., 128, 55-94 (2001) · Zbl 1023.17017
[8] Cheng, S.-J.; Wang, W., Lie subalgebras of differential operators on the super circle, Publ. Res. Inst. Math. Sci., 39, 545-600 (2003) · Zbl 1049.17021
[9] Cheng, S.-J.; Wang, W.; Zhang, R. B., Super duality and Kazhdan-Lusztig polynomials, Trans. Amer. Math. Soc., 360, 5883-5924 (2008) · Zbl 1234.17004
[10] Cheng, S.-J.; Zhang, R. B., Analogue of Kostant’s \(u\)-cohomology formula for the general linear superalgebra, Int. Math. Res. Not., 2004, 31-53 (2004) · Zbl 1096.17005
[11] Cheng, S.-J.; Zhang, R. B., Howe duality and combinatorial character formula for orthosymplectic Lie superalgebras, Adv. Math., 182, 124-172 (2004) · Zbl 1056.17004
[12] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type, Phys. D, 4, 343-365 (1981/1982) · Zbl 0571.35100
[13] Davidson, M.; Enright, T.; Stanke, R., Differential operators and highest weight representations, Mem. Amer. Math. Soc., 94, 455 (1991) · Zbl 0759.22015
[14] Enright, T., Analogues of Kostant’s \(u\)-cohomology formulas for unitary highest weight modules, J. Reine Angew. Math., 392, 27-36 (1988) · Zbl 0651.17003
[15] Frenkel, I., Representations of affine Lie algebras, Hecke modular forms and Kortweg-de Vries type equations, Lecture Notes in Math., 933, 71-110 (1982)
[16] Garland, H.; Lepowsky, J., Lie algebra homology and the Macdonald-Kac formulas, Invent. Math., 34, 37-76 (1976) · Zbl 0358.17015
[17] Howe, R., Remarks on classical invariant theory, Trans. Amer. Math. Soc., 313, 539-570 (1989) · Zbl 0674.15021
[18] Howe, R., Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, (The Schur Lectures (1992) (Tel Aviv). The Schur Lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8 (1995)), 1-182 · Zbl 0844.20027
[20] Jurisich, E., An exposition of generalized Kac-Moody algebras, (Lie Algebras and Their Representations. Lie Algebras and Their Representations, Seoul, 1995. Lie Algebras and Their Representations. Lie Algebras and Their Representations, Seoul, 1995, Contemp. Math., vol. 194 (1996), AMS: AMS Providence, RI), 121-159 · Zbl 0867.17016
[21] Kac, V., Lie superalgebras, Adv. Math., 16, 8-96 (1977) · Zbl 0366.17012
[22] Kac, V., Infinite Dimensional Lie Algebras (1990), Cambridge University Press · Zbl 0716.17022
[23] Kac, V.; Radul, A., Representation theory of the vertex algebra \(W_{1 + \infty}\), Transform. Groups, 1, 41-70 (1996) · Zbl 0862.17023
[24] Kang, S. J.; Kwon, J.-H., Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras, Proc. London Math. Soc., 81, 675-724 (2000) · Zbl 1040.17028
[25] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math., 74, 329-387 (1961) · Zbl 0134.03501
[26] Lam, N.; Zhang, R. B., Quasi-finite modules for Lie superalgebras of infinite rank, Trans. Amer. Math. Soc., 358, 403-439 (2006) · Zbl 1105.17013
[27] Liu, L., Kostant’s formula for Kac-Moody Lie algebras, J. Algebra, 149, 155-178 (1992) · Zbl 0779.17024
[28] Macdonald, I. G., Symmetric Functions and Hall Polynomials, Oxford Math. Monogr. (1995), Clarendon Press: Clarendon Press Oxford · Zbl 0487.20007
[29] Serganova, V., Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \(gl(m | n)\), Selecta Math. (N.S.), 2, 607-651 (1996) · Zbl 0881.17005
[30] Sergeev, A., An analogue of classical invariant theory for Lie superalgebras II, Michigan Math. J., 49, 147-168 (2001) · Zbl 1002.17002
[31] Wang, W., Duality in infinite dimensional Fock representations, Commun. Contemp. Math., 1, 155-199 (1999) · Zbl 0951.17011
[32] Vogan, D., Irreducible characters of semisimple Lie groups II: The Kazhdan-Lusztig conjectures, Duke Math. J., 46, 805-859 (1979) · Zbl 0421.22008
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