# zbMATH — the first resource for mathematics

Dirac operators and Lie algebra cohomology. (English) Zbl 1134.22011
The paper under review studies the Dirac cohomology of unitary modules for the Kostant cubic Dirac operator and its relation to nilpotent Lie algebra cohomology, and shows that in some cases these two cohomologies coincide.

##### MSC:
 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
Full Text:
##### References:
 [1] Anton Alekseev and Eckhard Meinrenken, Lie theory and the Chern-Weil homomorphism, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 303 – 338 (English, with English and French summaries). · Zbl 1105.17015 [2] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. · Zbl 0980.22015 [3] Jing-Song Huang and Pavle Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185 – 202. · Zbl 0980.22013 [4] Jing-Song Huang and Pavle Pandžić, Dirac operators in representation theory, Representations of real and \?-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 2, Singapore Univ. Press, Singapore, 2004, pp. 163 – 219. · Zbl 1057.22016 [5] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329 – 387. · Zbl 0134.03501 [6] Bertram Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), no. 3, 447 – 501. · Zbl 0952.17005 [7] Bertram Kostant, A generalization of the Bott-Borel-Weil theorem and Euler number multiplets of representations, Lett. Math. Phys. 52 (2000), no. 1, 61 – 78. Conference Moshé Flato 1999 (Dijon). · Zbl 0960.22011 [8] Bertram Kostant, Dirac cohomology for the cubic Dirac operator, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 69 – 93. · Zbl 1165.17301 [9] Shrawan Kumar, Induction functor in noncommutative equivariant cohomology and Dirac cohomology, J. Algebra 291 (2005), no. 1, 187 – 207. · Zbl 1090.22007 [10] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1 – 30. · Zbl 0249.22003 [11] David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. · Zbl 0469.22012 [12] D.A. Vogan, Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997. [13] D.A. Vogan, $$\mathfrak{n}$$-cohomology in representation theory, a talk at “Functional Analysis VII”, Dubrovnik, Croatia, September 2001. [14] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. · Zbl 0666.22002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.