Capelli type identities on certain scalar generalized Verma modules

Abstract

We give an analogue of the Capelli identity. The analogue is constructed on certain scalar generalized Verma modules of the complex simple Lie algebras of Hermitian symmetric type, and has stronger compatibility with group actions than the original Capelli identity. As in the original case, the analogue expresses certain invariant operators as operators from the center of the universal enveloping algebra. We also give examples for all the possible cases except for $E_{7}$.