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Dirac operators on Weil representations. II. (English) Zbl 1207.22009
[Part I by the author in ibid. 15, No. 2, 401–410 (2010; Zbl 1207.22008).]
In this paper the author describes Dirac cohomology with respect to a noncompact subalgebra \({\mathfrak l}\) of \({\mathfrak g}= {\mathfrak {sp}}(4,\mathbb C)\) for the Harish-Chandra modules associated with the even and odd Weil representations of \({\mathfrak g}\). The obtained results are in sharp contrast with previously known results in other situations. In particular, it turns out that the square of the Dirac operator does not act semisimply, that the Dirac cohomology is not the same as (co)homology for a certain nilpotent Lie subalgebra and, finally, that Dirac cohomology cannot be calculated in stages with respect to the compact Cartan subalgebra of \({\mathfrak l}\).
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)