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.
Citation
Floyd Williams. "A BGG type resolution of holomorphic Verma modules." Illinois J. Math. 43 (4) 633 - 653, Winter 1999. https://doi.org/10.1215/ijm/1256060683
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