Large Schubert Varieties

. 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 compacti øcation 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 van der Kallen's øltration of the algebra of regular functions on B. We also construct a degeneration of the AEag variety G=B embedded diagonally in G=B \Theta G=B, into a union of Schubert varieties. This leads to formulae for the class of the diagonal in T -equivariant K-theory of G=B \Theta G=B, where T is a maximal torus of B. Introduction Consider an adjoint semisimple algebraic group G and a Borel subgroup B. The Schubert varieties are the images in G=B of the closures in G of double classes BwB. These varieties are generally singular, but all of them are normal and Cohen-Macaulay ..