Large Schubert varieties

Brion, Michel; Polo, Patrick

doi:10.1090/S1088-4165-00-00069-8

Large Schubert varieties
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by Michel Brion and Patrick Polo

Represent. Theory 4 (2000), 97-126

DOI: https://doi.org/10.1090/S1088-4165-00-00069-8

Published electronically: February 23, 2000

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Abstract:

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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