Dirac cohomology, unitary representations and a proof of a conjecture of Vogan. (English) Zbl 0980.22013

In the present paper a proof is given of Vogan’s conjecture on Dirac cohomology.
Let \(G\) be a connected semisimple Lie group with finite centre, \(K\) the maximal subgroup of \(G\) corresponding to the Cartan involution \(\theta\), with Cartan decomposition \(g= k \oplus p\) and let \(g^C\) be the complexification of \(g\). If \(X\) is an irreducible unitarizable \((g^C, K)\)-module, the Dirac operator \(D\) acts on \(X \otimes S\), where \(S\) is the space of spinors for \(p\). The Vogan conjecture states that if \(D\) has nonzero kernel on \(X \otimes S\), then the infinitesimal character of \(X\) can be described in terms of the heighest weight of a \(\widetilde K\)-type in \(\ker D\), where \(\widetilde K\) is a double cover of \(K\) corresponding to the group \(Spin(p)\).
The main new idea for proving Vogan’s conjecture is introducing a differential \(d\) on the \(K\)-invariants in \(U(g^C)\otimes C(p^C)\) related to \(D\), where \(U(g^C)\) is the universal enveloping algebra of \(g^C\) and \(C(p^C)\) is the Clifford algebra of the complexification of \(p\). The conjecture follows by determining the cohomology of \(d\).


22E46 Semisimple Lie groups and their representations
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
Full Text: DOI


[1] Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1 – 62. , https://doi.org/10.1007/BF01389783 Michael Atiyah and Wilfried Schmid, Erratum: ”A geometric construction of the discrete series for semisimple Lie groups” [Invent. Math. 42 (1977), 1 – 62; MR 57 #3310], Invent. Math. 54 (1979), no. 2, 189 – 192. · Zbl 0413.22009 · doi:10.1007/BF01408936
[2] H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de Topologie algébrique, C.B.R.M. Bruxelles (1950), 57-71.
[3] William Casselman and M. Scott Osborne, The \?-cohomology of representations with an infinitesimal character, Compositio Math. 31 (1975), no. 2, 219 – 227. · Zbl 0343.17006
[4] Ryoshi Hotta, On a realization of the discrete series for semisimple Lie groups, J. Math. Soc. Japan 23 (1971), 384 – 407. · Zbl 0213.13701 · doi:10.2969/jmsj/02320384
[5] Ryoshi Hotta and R. Parthasarathy, A geometric meaning of the multiplicity of integrable discrete classes in \?²(\Gamma \?), Osaka J. Math. 10 (1973), 211 – 234. · Zbl 0337.22016
[6] B. Kostant, A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. Jour. 100 (1999), 447-501. CMP 2000:05
[7] B. Kostant, Dirac cohomology for the cubic Dirac operator, in preparation. · Zbl 1165.17301
[8] S. Kumaresan, On the canonical \?-types in the irreducible unitary \?-modules with nonzero relative cohomology, Invent. Math. 59 (1980), no. 1, 1 – 11. · Zbl 0442.22010 · doi:10.1007/BF01390311
[9] J.-S. Li, On the first eigenvalue of Laplacian on locally symmetric manifolds, First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001, 271-278. CMP 2001:12
[10] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1 – 30. · Zbl 0249.22003 · doi:10.2307/1970892
[11] Susana A. Salamanca-Riba, On the unitary dual of real reductive Lie groups and the \?_{\?}(\?) modules: the strongly regular case, Duke Math. J. 96 (1999), no. 3, 521 – 546. · Zbl 0941.22014 · doi:10.1215/S0012-7094-99-09616-3
[12] Wilfried Schmid, On the characters of the discrete series. The Hermitian symmetric case, Invent. Math. 30 (1975), no. 1, 47 – 144. · Zbl 0324.22007 · doi:10.1007/BF01389847
[13] David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. · Zbl 0469.22012
[14] David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141 – 187. · Zbl 0561.22010 · doi:10.2307/2007074
[15] D. A. Vogan, Jr., Dirac operator and unitary representations, 3 talks at MIT Lie groups seminar, Fall of 1997.
[16] D. A. Vogan, Jr., On the smallest eigenvalue of the Laplacian on a locally symmetric space, Lecture at the Midwest Conference on Representation Theory and Automorphic Forms, Chicago, June, 2000.
[17] David A. Vogan Jr. and Gregg J. Zuckerman, Unitary representations with nonzero cohomology, Compositio Math. 53 (1984), no. 1, 51 – 90. · Zbl 0692.22008
[18] 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.