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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 \(d\).

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