The Borel-Weil theorem for reductive Lie groups

Abstract

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomolgy of an equivariant holomorphic line bundle defined on an open invariant submanifold of a complex flag space. Our main result is the following: suppose $G_{0}$ is a real reductive group of Harish-Chandra class and let $X$ be the associated full complex flag space. Suppose $\mathcal{O}_{\lambda}$ is the sheaf of sections of a $G_{0}$-equivariant holomorphic line bundle on $X$ whose parameter $\lambda$ (in the usual twisted $\mathcal{D}% $-module context) is antidominant and regular. Let $S\subseteq X$ be a $G_{0}% $-orbit and suppose $U\supseteq S$ is the smallest $G_{0}$-invariant open submanifold of $X$ that contains $S$. From the analytic localization theory of Hecht and Taylor one knows that there is a nonegative integer $q$ such that the compactly supported sheaf cohomology groups $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ vanish except in degree $q$, in which case $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is the minimal globalization of an associated standard Beilinson-Bernstein module. In this study we show that the $q$-th compactly supported cohomolgy group $H_{\text{c}}^{q}(U,\mathcal{O}_{\lambda}\mid_{U})$ defines, in a natural way, a nonzero submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$, which is irreducible (i.e. realizes the unique irreducible submodule of $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$) when an associated algebraic variety is nonsingular. By a tensoring argument, we can show that the result holds, more generally (for nonsingular Schubert variety), when the representation $H_{\text{c}}^{q}(S,\mathcal{O}_{\lambda}\mid_{S})$ is what we call a classifying module.