A BGG type resolution of holomorphic Verma modules

Abstract

For a Hermitian symmetric space $X = G/K$ of non-compact type let $\theta$ denote the Cartan involution of the semisimple Lie group $G$ with respect to the maximal compact subgroup $K$ of $G$, and let $q$ denote a $\theta$-stable parabolic subalgebra of the complexified Lie algebra $g$ of $G$ with corresponding Levi subgroup $L$ of $G$. Given a finite-dimensional irreducible $L$ module $F_{L}$ we find Bernstein-Gelfand-Gelfand type resolutions of the induced $(g, L \cap K)$ module $U(g) \otimes_{U(q)}F_{L}$ and its Hermitian dual, the produced module $\mathrm{Hom}_{U(\bar{q})}(U(g),F_{L})_{L \cap K-\mathrm{finite}}$, where $U(\cdot)$ is the universal enveloping algebra functor and $\bar{q}$ is the complex conjugate of $q$. The results coupled with a Grothendick spectral sequence provide for application to certain $(g,K)$ modules obtained by cohomological parabolic induction, and they extend results obtained initially by Stanke.