The Borel-Weil theorem for reductive Lie groups

DOI10.2140/PJM.2015.277.257zbMATH Open1330.22019arXiv1312.4978OpenAlexW3100135435WikidataQ115230581 ScholiaQ115230581MaRDI QIDQ498695

Tim Bratten, José O. Araujo

Publication date: 29 September 2015

Published in: Pacific Journal of Mathematics (Search for Journal in Brave)

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 $mathcalD\mathrm{\%}$-module context) is antidominant and regular. Let $SsubseteqX$ be a $G_0\mathrm{\%}$-orbit and suppose $UsupseteqS$ 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 $H_{extc}^q(S,mathcal{O}_{lambda}mi{d}_S)$ vanish except in degree $q$, in which case $H_{extc}^q(S,mathcal{O}_{lambda}mi{d}_S)$ is the minimal globalization of an associated standard Beilinson-Bernstein module. In this study we show that the $q$-th compactly supported cohomolgy group $H_{extc}^q(U,mathcal{O}_{lambda}mi{d}_U)$ defines, in a natural way, a nonzero submodule of $H_{extc}^q(S,mathcal{O}_{lambda}mi{d}_S)$, which is irreducible (i.e. realizes the unique irreducible submodule of $H_{extc}^q(S,mathcal{O}_{lambda}mi{d}_S)$) 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 $H_{extc}^q(S,mathcal{O}_{lambda}mi{d}_S)$ is what we call a classifying module.

Full work available at URL: https://arxiv.org/abs/1312.4978

zbMATH Keywords

flag manifoldrepresentation theoryreductive Lie group

Mathematics Subject Classification ID

Semisimple Lie groups and their representations (22E46)

Related Items (5)

A version of the Weyl complete reducibility theorem for not necessarily continuous representations of connected Lie groups ⋮ Borel–Weil theory for groups over commutative Banach algebras ⋮ On \(\theta\)-stable Borel subalgebras of large type for real reductive groups ⋮ A Freudenthal-Weil theorem for pro-Lie groups ⋮ Bessel \(F\)-isocrystals for reductive groups

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