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Dirac cohomology of highest weight modules. (English) Zbl 1257.22012
In this paper the authors explicitly describe the Dirac cohomology of simple highest weight modules over a finite dimensional complex semi-simple Lie algebra. The answer is given in terms of relative Kazhdan-Lusztig polynomials.

MSC:
22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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[1] Bernstein J., Gelfand I., Gelfand S.: Category of $${\(\backslash\)mathfrak{g}}$$ -modules. Funct. Anal. Appl. 10, 87–92 (1976) · Zbl 0353.18013
[2] Beilinson A., Ginzburg V., Soergel W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9, 473–527 (1996) · Zbl 0864.17006
[3] Boe B., Hunziker M.: Kostant modules in blocks of category $${\(\backslash\)mathcal{O}\^\(\backslash\)mathfrak{p}}$$ . Commun. Algebra 37, 323–356 (2009) · Zbl 1243.17003
[4] Casian L., Collingwood D.: The Kazhdan–Lusztig conjecture for parabolic Verma modules. Math. Z. 195, 581–600 (1987) · Zbl 0624.22010
[5] Casselman W., Osborne M.: The $${\(\backslash\)mathfrak{n}}$$ -cohomology of representations with an infinitesimal character. Comp. Math. 31, 219–227 (1975) · Zbl 0343.17006
[6] Deodhar V.: On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan–Lusztig polynomials. J. Algebra 111, 483–506 (1987) · Zbl 0656.22007
[7] Dong C.-P., Huang J.-S.: Jacquet modules and Dirac cohomology. Adv. Math. 226, 2911–2934 (2011) · Zbl 1221.22014
[8] Enright T.: Analogues of Kostant’s $${\(\backslash\)mathfrak{u}}$$ -cohomology formulas for unitary highest weight modules. J. Reine Angew. Math. 392, 27–36 (1988) · Zbl 0651.17003
[9] Enright, T., Sheldon, B.: Categories of highest weight modules: applications to classical Hermitian symmetric pairs. Mem. Am. Math. Soc. 67(367), iv+94 (1987) · Zbl 0621.17004
[10] Huang J.-S, Pandžić P.: Dirac cohomology, unitary representations and a proof of a conjecture of Vogan. J. Am. Math. Soc. 15, 185–202 (2002) · Zbl 0980.22013
[11] Huang, J.-S, Pandžić, P.: Dirac Operator in Representation Theory. Mathematics Theory and Applications. Birkhäuser, Boston (2006)
[12] Huang J.-S., Kang Y.-F., Pandžić P.: Dirac cohomology of some Harish-Chandra modules. Transform. Groups 14, 163–173 (2009) · Zbl 1179.22013
[13] Huang J.-S, Pandžić P., Renard D.: Dirac operators and Lie algebra cohomology. Represent. Theory 10, 299–313 (2006) · Zbl 1134.22011
[14] Humphreys, J.: Representations of Semisimple Lie Algebras in the BGG Category $${\(\backslash\)mathcal{O}}$$ . GSM, vol. 94. American Mathematical Soceity, Providence (2008) · Zbl 1177.17001
[15] Irving R.: Singular blocks of the category $${\(\backslash\)mathcal{O}}$$ . Math. Z. 204, 209–224 (1990) · Zbl 0715.17010
[16] Kostant B.: A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups. Duke Math. J. 100, 447–501 (1999) · Zbl 0952.17005
[17] Kostant B.: A generalization of the Bott–Borel–Weil theorem and Euler number multiplets of representations. Lett. Math. Phys. 52, 61–78 (2000) · Zbl 0960.22011
[18] Kostant B.: Dirac cohomology for the cubic Dirac operator, studies in memory of Issai Schur. Prog. Math. 210, 69–93 (2003) · Zbl 1165.17301
[19] Schmid W.: Vanishing theorems for Lie algebra cohomology and the cohomology of discrete subgroups of semisimple Lie groups. Adv. Math. 41, 78–113 (1981) · Zbl 0472.22003
[20] Soergel W.: $${\(\backslash\)mathfrak{n}}$$ -cohomology of simple highest weight modules on walls and purity. Invent. Math. 98, 565–580 (1989) · Zbl 0781.22011
[21] Soergel W., Kategorie O.: Perverse Garben Und Moduln Uber Den Koinvariantez Zur Weylgruppe. J. Am. Math. Soc. 3, 421–445 (1990) · Zbl 0747.17008
[22] Vogan, D. Jr.: Dirac operators and unitary representationss, 3 talks at MIT Lie groups seminar, Fall (1997)
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