# The Borel-Weil theorem for reductive Lie groups

Research paper by **José Araujo, Tim Bratten**

Indexed on: **27 Apr '15**Published on: **27 Apr '15**Published in: **Mathematics - Representation Theory**

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#### 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.