Richardson varieties and equivariant K-theory

We generalize Standard Monomial Theory (SMT) to intersections of Schubert varieties and opposite Schubert varieties; such varieties are called Richardson varieties. The aim of this article is to get closer to a geometric interpretation of the standard monomial theory as constructed in (P. Littelmann, J. Amer. Math. Soc. 11 (1998) 551–567). In fact, the construction given here is very close to the ideas in (P. Lakshmibai, C.S. Seshadri, J. Algebra 100 (1986) 462–557). Our methods show that in order to develop a SMT for a certain class of subvarieties in G/B (which includes G/B), it suffices to have the following three ingredients, a basis for , compatibility of such a basis with the varieties in the class, certain quadratic relations in the monomials in the basis elements. An important tool (as in (P. Lakshmibai, C.S. Seshadri, J. Algebra 100 (1986) 462–557)) will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above. This provides a direct connection to the equivariant K-theory (products of classes of structure sheaves with classes of line bundles), where the combinatorially defined notion of standardness gets a geometric interpretation.