## Dirac cohomology for the cubic Dirac operator.(English)Zbl 1165.17301

Joseph, Anthony (ed.) et al., Studies in memory of Issai Schur. Basel: Birkhäuser (ISBN 0-8176-4208-0/hbk). Prog. Math. 210, 69-93 (2003).
Summary: Let $${\mathfrak g}$$ be a complex semisimple Lie algebra and let $${\mathfrak r}\subset{\mathfrak g}$$ be any reductive Lie subalgebra such that $$B|{\mathfrak r}$$ is nonsingular where $$B$$ is the Killing form of $${\mathfrak g}$$. Let $$Z({\mathfrak r})$$ and $$Z({\mathfrak g})$$ be, respectively, the centers of the enveloping algebras of $${\mathfrak r}$$ and $${\mathfrak g}$$. Using a Harish-Chandra isomorphism one has a homomorphism $$\eta:Z({\mathfrak g})\to Z({\mathfrak r})$$ which, by a well-known result of H. Cartan, yields the the relative Lie algebra cohomology $$H({\mathfrak g},{\mathfrak r})$$.
Let $$V$$ be any $${\mathfrak g}$$-module. For the case where $${\mathfrak r}$$ is a symmetric subalgebra, Vogan has defined the Dirac cohomology $$\text{Dir}(V)$$ of $$V$$. Using the cubic Dirac operator we extend his definition to the case where $${\mathfrak r}$$ is arbitrary subject to the condition stated above. We then generalize results of J.-S. Huang and P. Pandžić [J. Am. Math. Soc. 15, No. 1, 185–202 (2002; Zbl 0980.22013)] on a proof of a conjecture of Vogan. In particular $$\text{Dir}(V)$$ has a structure of a $$Z({\mathfrak r})$$-module relative to a “diagonal” homomorphism $$\gamma:Z({\mathfrak r})\to \text{End\,Dir}(V)$$. In case $$V$$ admits an infinitesimal character $$\chi$$ and $$I$$ is the identity operator on $$\text{Dir}(V)$$ we prove
$\gamma\circ\eta= \chi I.\tag{A}$
In addition we also prove that $$V$$ always exists (in fact $$V$$ can taken to be an object in category $$O$$) such that $$\text{Dir}(V)\neq 0$$. If $${\mathfrak r}$$ has the same rank as $${\mathfrak g}$$ and $$V$$ is irreducible and finite dimensional, then (A) generalizes a result of Gross-Kostant-Ramond-Sternberg.
For the entire collection see [Zbl 1005.00049].

### MSC:

 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras 17B35 Universal enveloping (super)algebras

Zbl 0980.22013
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