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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
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