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Representation Theory

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Large Schubert varieties

Authors: Michel Brion and Patrick Polo
Journal: Represent. Theory 4 (2000), 97-126
MSC (2000): Primary 14M15, 14L30, 20G05, 19E08
Published electronically: February 23, 2000
MathSciNet review: 1789463
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For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$; we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically a filtration à la van der Kallen of the algebra of regular functions on $B$. We also construct a degeneration of the flag variety $G/B$ embedded diagonally in $G/B\times G/B$, into a union of Schubert varieties. This yields formulae for the class of the diagonal of $G/B\times G/B$ in $T$-equivariant $K$-theory, where $T$ is a maximal torus of $B$.

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Additional Information

Michel Brion
Affiliation: Université de Grenoble I, Département de Mathématiques, Institut Fourier, UMR 5582 du CNRS, 38402 Saint-Martin d’Hères Cedex, France

Patrick Polo
Affiliation: Université Paris Nord, Département de Mathématiques, L.A.G.A., UMR 7539 du CNRS, 93430 Villetaneuse, France

Keywords: Schubert varieties, canonical compactifications of semisimple groups, filtrations of rational representations, equivariant $K$-theory
Received by editor(s): April 27, 1999
Received by editor(s) in revised form: October 9, 1999
Published electronically: February 23, 2000
Article copyright: © Copyright 2000 American Mathematical Society