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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Large Schubert varieties
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by Michel Brion and Patrick Polo PDF
Represent. Theory 4 (2000), 97-126 Request permission


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
  • MR Author ID: 41725
  • Email:
  • Patrick Polo
  • Affiliation: Université Paris Nord, Département de Mathématiques, L.A.G.A., UMR 7539 du CNRS, 93430 Villetaneuse, France
  • Email:
  • Received by editor(s): April 27, 1999
  • Received by editor(s) in revised form: October 9, 1999
  • Published electronically: February 23, 2000
  • © Copyright 2000 American Mathematical Society
  • Journal: Represent. Theory 4 (2000), 97-126
  • MSC (2000): Primary 14M15, 14L30, 20G05, 19E08
  • DOI:
  • MathSciNet review: 1789463