Intersection cohomology of B×B-orbit closures in group compactifications

Abstract

An adjoint semi-simple group G has a “wonderful” compactification X, which is a smooth projective variety, containing G as an open subvariety. X is acted upon by G×G and, B denoting a Borel subgroup of G, the group B × B has finitely many orbits in X. The main results of this paper concern the intersection cohomology of the closures of the B × B-orbits. Examples of such closures are the “large Schubert varieties,” the closures in X of the double cosets BwB in G. After recalling some basic results about the wonderful compactification, we discuss in Section 1 the description of the B ×B-orbits, and establish some basic results. In Section 2 the “Bruhat order” of the set V of orbits is introduced and described explicitly. As an application we obtain cellular decompositions of the large Schubert varieties. Let H be the Hecke algebra associated to G, it is a free module over an algebra of Laurent polynomials Z[u,u−1]. As a particular case of results of [MS], the spherical G × G-variety X defines a representation of the Hecke algebra associated to G×G, i.e. H⊗Z[u,u−1] H, in a free module M over an extension of Z[u,u−1], with a basis (mv) indexed by V . The definition of M is sheaftheoretical, working over the algebraic closure of a finite field. This is discussed in Section 3. On the model of [LV] a duality map ∆ is introduced on M, coming from Verdier duality in sheaf theory. The matrix coefficients of ∆ relative to the