The Borel–Weil theorem for reductive Lie groups

José Araujo and Tim Bratten

Vol. 277 (2015), No. 2, 257–285

DOI: 10.2140/pjm.2015.277.257

Abstract

In this manuscript we consider the extent to which an irreducible representation for a reductive Lie group can be realized as the sheaf cohomology 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 nonnegative integer $q$ such that the compactly supported sheaf cohomology groups $H_c^p\left(S,{\mathcal{O}}_{\lambda }{|}_S\right)$ vanish except in degree $q$, in which case $H_c^q\left(S,{\mathcal{O}}_{\lambda }{|}_S\right)$ is the minimal globalization of an associated standard Beilinson–Bernstein module. In this study, we show that the $q$-th compactly supported cohomology group $H_c^q\left(U,{\mathcal{O}}_{\lambda }{|}_U\right)$ defines, in a natural way, a nonzero submodule of $H_c^q\left(S,{\mathcal{O}}_{\lambda }{|}_S\right)$, which is irreducible (i.e., realizes the unique irreducible submodule of $H_c^q\left(S,{\mathcal{O}}_{\lambda }{|}_S\right)$) when an associated algebraic variety is nonsingular. By a tensoring argument, we can show that the result holds, more generally (for nonsingular associated variety), when the representation $H_c^q\left(S,{\mathcal{O}}_{\lambda }{|}_S\right)$ is what we call a classifying module.

Keywords

reductive Lie group, representation theory, flag manifold

Mathematical Subject Classification 2010

Primary: 22E46

Milestones

Received: 27 June 2014

Revised: 14 March 2015

Accepted: 7 April 2015

Published: 15 September 2015