. A G ] 2 6 A pr 1 99 9 LARGE SCHUBERT VARIETIES MICHEL

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 com-pactification 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 filtration of the algebra of regular functions on B. We also construct a degeneration of the flag variety G/B embedded diagonally in G/B × 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 × G/B, where T is a maximal torus of B.