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

This thesis studies the topology of a particularly nice compactification that exists for semisimple adjoint algebraic groups: the wonderful compactification. The compactifica- tion 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 inter- esting, 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 re- ferred to as GKM theory after Goresky, Kottowitz and MacPherson. Furthermore in the GKM setting we can recover ordinary intersection cohomology from the equivariant inter- section cohomology. Unfortunately the GKM theorems are not practical when computing intersection cohomology since for singular varieties we may not a priori know the local equivariant intersection cohomology at the torus fixed points. Braden and MacPherson address this problem, showing how to algorithmically apply GKM theory to compute the equivariant intersection cohomology for a large class of varieties that includes Schubert varieties. Our setting is more complicated than that of Braden and MacPherson in that we must use some larger torus orbits than just the 0 and 1-dimensional orbits. Nonetheless we are able to extend the moment graph approach of Braden and MacPherson. We define a more general notion of moment graph and identify canonical sheaves on the generalized moment graph whose sections are the equivariant intersection cohomology of the Borel orbit closures of the wonderful compactification.
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