Equivariant Intersection Cohomology of BXB Orbit Closures in the Wonderful Compactification of a Group

Abstract

EQUIVARIANT INTERSECTION COHOMOLOGY OF B ×B ORBIT CLOSURES IN THE WONDERFUL COMPACTIFICATION OF A GROUP FEBRUARY 2016 STEPHEN O. OLOO B.A., AMHERST COLLEGE M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Tom Braden This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactification is equivariant, extending the left and right action of the group on itself, and we focus on the local and global topology of the closures of Borel orbits. It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be interesting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out to be the coefficients of Kazhdan-Lusztig polynomials. We compute equivariant intersection cohomology with respect to a torus action because such actions often have convenient localization properties enabling us to use data from the moment graph (roughly speaking the collection of 0 and 1-dimensional orbits) to compute the equivariant (intersection) cohomology of the whole space, an approach commonly referred to as GKM theory after Goresky, Kottowitz and MacPherson. Furthermore in the GKM setting we can recover ordinary intersection cohomology from the