Large Schubert varieties

Abstract

For a semisimple adjoint algebraic group G G and a Borel subgroup B B , consider the double classes B w B BwB in G G and their closures in the canonical compactification of G 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 B . We also construct a degeneration of the flag variety G / B G/B embedded diagonally in G / B × G / B G/B\times G/B , into a union of Schubert varieties. This yields formulae for the class of the diagonal of G / B × G / B G/B\times G/B in T T -equivariant K K -theory, where T T is a maximal torus of B B .