The Dirac cohomology of a finite dimensional representation

Mehdi, S.; Zierau, R.

doi:10.1090/S0002-9939-2014-11952-6

The Dirac cohomology of a finite dimensional representation
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by S. Mehdi and R. Zierau PDF

Proc. Amer. Math. Soc. 142 (2014), 1507-1512 Request permission

Abstract:

The Dirac cohomology of a finite dimensional representation of a complex semisimple Lie algebra $\mathfrak {g}$, with respect to any quadratic subalgebra $\mathfrak {h}$, is computed. This generalizes a formula obtained by Kostant in the case where $\mathfrak {g}$ and $\mathfrak {h}$ have equal rank, and by Huang, Kang and Pandžić in the case where $\mathfrak {h}$ is the fixed point of an involution.

References

Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
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. MR 1862801, DOI 10.1090/S0894-0347-01-00383-6
Jing-Song Huang, Yi-Fang Kang, and Pavle Pandžić, Dirac cohomology of some Harish-Chandra modules, Transform. Groups 14 (2009), no. 1, 163–173. MR 2480856, DOI 10.1007/s00031-008-9036-7
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. MR 1719734, DOI 10.1215/S0012-7094-99-10016-0
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. MR 1985723
R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30. MR 318398, DOI 10.2307/1970892