In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.
A Riemannian manifold （M, g） is called Einstein manifold if its Ricci tensor satisfies r=c.g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.
