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Journal of Lie Theory 21 (2011), No. 4, 861--884
Copyright Heldermann Verlag 2011

Cubic Dirac Cohomology for Generalized Enright-Varadarajan Modules

Salah Mehdi
Dép. de Mathématiques, UMR 7122 - CNRS, Université Paul Verlaine, 57045 Metz, France

Rajagopalan Parthasarathy
Dept. of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamil Nadu, India


\def\g{{\frak g}} \def\h{{\frak h}} \def\v{{\frak v}} For a complex semisimple Lie algebra $\g=\h\oplus\v$ where $\h$ is a quadratic subalgebra and $\h$ and $\v$ are orthogonal with respect to the Killing form, we construct a large family of $(\g,\h)$-modules with non-zero cubic Dirac cohomology. Our method uses analogue of the construction of generalized Enright-Varadarajan modules for what we call $(\h,\v)$-split parabolic subalgebras. This family of modules includes discrete series representations and ${\cal A}_{\q}(\lambda)$-modules.

Keywords: Quadratic subalgebra, generalized Enright-Varadrajan module, (g,h)-module, Verma modules, Kostant's cubic Dirac operator, Dirac cohomology.

MSC: 22E46, 22E47; 17B10

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