Homomorphisms Between Generalized Verma Modules

Abstract

Let $\mathfrak {g}$ be a finite-dimensional complex semisimple Lie algebra and $\mathfrak {p}$ a parabolic subalgebra. The first result is a necessary and sufficient condition, in the spirit of the Bernstein-Gelfand-Gelfand theorem on Verma modules, for Lepowsky’s "standard map" between two generalized Verma modules for $\mathfrak {g}$ to be zero. The main result gives a complete description of all homomorphisms between the generalized Verma modules induced from one-dimensional $\mathfrak {p}$-modules, in the "hermitian symmetric" situation.