Shimura invariant differential operators and their eigenvalues

Abstract

Let $\bar D$ be the covariant Cauchy-Riemann operator and $\mathcal D$ the covariant holomorphic differential operator on a line bundle over a Hermitian symmetric space $G/K$ . We study the Shimura invariant differential operators defined via $\bar D$ and $\mathcal D$ . We find the eigenvalues of a family of the Shimura operators and of the generators.