×

zbMATH — the first resource for mathematics

Dirac series for some real exceptional Lie groups. (English) Zbl 1453.22006
Summary: Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: \( \operatorname{EI} = E_{6 ( 6 )}\), \(\operatorname{EIV} = E_{6 ( - 26 )}\), \(\operatorname{FI} = F_{4 ( 4 )}\), \(\operatorname{FII} = F_{4 ( - 20 )} \). Along the way, we find an irreducible unitary representation of \(F_{4 ( 4 )}\) whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology.

MSC:
22E46 Semisimple Lie groups and their representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, J.; van Leeuwen, M.; Trapa, P.; Vogan, D., Unitary representations of real reductive groups, Astérisque (2020), in press, see also
[2] Barbasch, D.; Pandžić, P., Dirac Cohomology and Unipotent Representations of Complex Groups, Noncommutative Geometry and Global Analysis, Contemp. Math., vol. 546, 1-22 (2011), Amer. Math. Soc. · Zbl 1238.22008
[3] Barbasch, D.; Pandžić, P., Dirac cohomology of unipotent representations of \(S p(2 n, \mathbb{R})\) and \(U(p, q)\), J. Lie Theory, 25, 1, 185-213 (2015) · Zbl 1387.22016
[4] Ding, J.; Dong, C.-P., Spin norm, K-types, and tempered representations, J. Lie Theory, 26, 3, 651-658 (2016) · Zbl 1350.22012
[5] Ding, J.; Dong, C.-P., Unitary representations with Dirac cohomology: a finiteness result for complex Lie groups, Forum Math. (2020)
[6] Dong, C.-P., On the Dirac cohomology of complex Lie group representations, Transform. Groups. Transform. Groups, Transform. Groups, 18, 2, 595-597 (2013), Erratum: · Zbl 1267.22004
[7] Dong, C.-P., Unitary representations with non-zero Dirac cohomology for complex \(E_6\), Forum Math., 31, 1, 69-82 (2019) · Zbl 1429.22014
[8] Dong, C.-P., Unitary representations with Dirac cohomology: finiteness in the real case, Int. Math. Res. Not. IMRN
[9] Dong, C.-P.; Huang, J.-S., Dirac cohomology of cohomologically induced modules for reductive Lie groups, Am. J. Math., 137, 1, 37-60 (2015) · Zbl 1321.22014
[10] Dong, C.-P.; Wong, K. D., On the Dirac series of \(U(p, q) (2020)\), preprint
[11] Harish-Chandra, Harmonic analysis on real reductive groups. I. the theory of the constant term, J. Funct. Anal., 19, 104-204 (1975) · Zbl 0315.43002
[12] Huang, J.-S., Dirac Cohomology, Elliptic Representations and EndoscopyRepresentations of Reductive Groups, Progr. Math., vol. 312, 241-276 (2015), Birkhäuser/Springer: Birkhäuser/Springer Cham
[13] Huang, J.-S.; Pandžić, P., Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Am. Math. Soc., 15, 1, 185-202 (2002) · Zbl 0980.22013
[14] Huang, J.-S.; Pandžić, P., Dirac Operators in Representation Theory, Mathematics: Theory and Applications (2006), Birkhäuser
[15] Knapp, A., Lie Groups, Beyond an Introduction (2002), Birkhäuser · Zbl 1075.22501
[16] Knapp, A.; Vogan, D., Cohomological Induction and Unitary Representations (1995), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J. · Zbl 0863.22011
[17] Mehdi, S.; Pandžić, P.; Vogan, D., Translation principle for Dirac index, Am. J. Math., 139, 1465-1491 (2017) · Zbl 1384.17013
[18] Parthasarathy, R., Dirac operators and the discrete series, Ann. Math., 96, 1-30 (1972) · Zbl 0249.22003
[19] Parthasarathy, R., Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci., 89, 1, 1-24 (1980) · Zbl 0434.22011
[20] Paul, A., Cohomological induction in Atlas, slides of July 14 (2017), available from
[21] Parthasarathy, R.; Ranga Rao, R.; Varadarajan, S., Representations of complex semi-simple Lie groups and Lie algebras, Ann. Math., 85, 383-429 (1967) · Zbl 0177.18004
[22] Salamanca-Riba, S., On the unitary dual of real reductive Lie groups and the \(A_{\mathfrak{q}}(\lambda)\) modules: the strongly regular case, Duke Math. J., 96, 3, 521-546 (1999) · Zbl 0941.22014
[23] Salamanca-Riba, S.; Vogan, D., On the classification of unitary representations of reductive Lie groups, Ann. Math., 148, 3, 1067-1133 (1998) · Zbl 0918.22009
[24] Vogan, D., Representations of Real Reductive Lie Groups (1981), Birkhäuser · Zbl 0469.22012
[25] Vogan, D., Unitarizability of certain series of representations, Ann. Math., 120, 1, 141-187 (1984) · Zbl 0561.22010
[26] Vogan, D., The unitary dual of \(G_2\), Invent. Math., 116, 1-3, 677-791 (1994) · Zbl 0808.22003
[27] Vogan, D., Dirac operators and unitary representations, (3 Talks at MIT Lie Groups Seminar (Fall 1997))
[28] Atlas of Lie groups and representations, for more about the software
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.