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:

 2.2e+47 Semisimple Lie groups and their representations 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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