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Dirac cohomology of some Harish-Chandra modules. (English) Zbl 1179.22013
Let \(G\) be a connected real reductive Lie group with Cartan involution \(\theta\), such that \(K=G^{\theta}\) is a maximal compact subgroup of \(G\). For example, \(G\) can be a connected semisimple Lie group with finite centre. In the paper under review the authors determine the Dirac cohomology of irreducible unitary Harish-Chandra modules \(A_{\mathfrak q}(\lambda)\). This family of modules includes all irreducible unitary representations with nonzero \(({\mathfrak g},K)\)-cohomology and all irreducible unitary Harish-Chandra modules with strongly regular infinitesimal character. As an illustration of the applications the authors relate their results with those of Vogan and Zuckerman on the \(({\mathfrak g},K)\)-cohomology of \(A_{\mathfrak q}(\lambda)\) modules.
The results of the present paper can be considered as a supplement to [J.-S. Huang and P. Pandžić, J. Am. Math. Soc. 15, No. 1, 185–202 (2002; Zbl 0980.22013)], where two of the authors obtained a necessary and sufficient condition for some Harish-Chandra modules to have nonzero Dirac cohomology.

MSC:
22E46 Semisimple Lie groups and their representations
18G60 Other (co)homology theories (MSC2010)
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[1] A. Borel, N. R.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
[2] T. Enright, Analogues of Kostant’s \(\mathfrak{u}\) -cohomology formulas for unitary highest weight modules, J. Reine. Angew. Math. 392 (1988), 27–36. · Zbl 0651.17003
[3] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. Russian transl.: Дж. Хамфрис, Введение в теорию алгебр Ли и их представлений, МЦНМО, М, 2003. · Zbl 0342.02023
[4] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978. · Zbl 0451.53038
[5] J.-S. Huang, P. Pandžić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), 185–202. · Zbl 0980.22013
[6] J.-S. Huang, P. Pandžić, Dirac Operators in Representation Theory, Mathematics: Theory and Applications, Birkhäuser, 2006.
[7] J.-S. Huang, P. Pandžić, D. Renard, Dirac operators and Lie algebra cohomology, Represent. Theory 10 (2006), 299–313. · Zbl 1134.22011
[8] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil Theorem, Ann. of Math. 74 (1961), 329–387. · Zbl 0134.03501
[9] B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J. 100 (1999), 447–501. · Zbl 0952.17005
[10] B. Kostant, Dirac cohomology for the cubic Dirac operator, in: Studies in Memory of Issai Schur, Progress in Mathematics, Vol. 210, Birkhäuser, Boston, 2003, pp. 69–93. · Zbl 1165.17301
[11] P. Pandžić, Coproducts for Clifford algebras, Glas. Mat. Ser. III 39 (2005), 207–211.
[12] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1–30. · Zbl 0249.22003
[13] S. A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the A \(\mathfrak{q}\) ({\(\lambda\)}) modules: The strongly regular case, Duke Math. J. 96 (1998), 521–546. · Zbl 0941.22014
[14] D. A. Vogan, Jr., Representations of Real Reductive Groups, Birkhäuser, Boston, 1981. · Zbl 0469.22012
[15] D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. 120 (1984), 141–187. · Zbl 0561.22010
[16] D. A. Vogan, Jr., Dirac operators and unitary representations, Three Talks at MIT Lie Groups Seminar, Fall 1997.
[17] D. A. Vogan, Jr., G. J. Zuckerman, Unitary representations with nonzero cohomology, Comp. Math. 53 (1984), 51–90. · Zbl 0692.22008
[18] N. R. Wallach, Real Reductive Groups, Vol. I, Academic Press, New York, 1988. · Zbl 0666.22002
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