Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

Griffiths-harris Rigidity of Compact Hermitian Symmetric Spaces

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

Dirac Operators and Lie Algebra Cohomology

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the correspondin...

Lie Algebra Cohomology and the Representations Ofsemisimple Lie Groups

The Lie algebra cohomology of jets

Let g be a finite-dimensional complex reductive semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra zg[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible repr...

Lie Algebra Cohomology and the Borel-Weil-Bott Theorem

We have seen that irreducible finite dimensional representations of a complex simple Lie algebra g or corresponding compact Lie group are classified and can be constructed starting from an integral dominant weight. The dominance condition depends upon a choice of positive roots (or equivalently, a choice of invariant complex structure on the flag manifold.) An obvious question is that of what h...