Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group
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Abstract:
We provide a functorial description of the torus equivariant intersection cohomology of the Borel orbit closures in the wonderful compactification of a semisimple adjoint complex algebraic group. Ours is an adaptation of the moment graph approach of Goresky, Kottwitz, and MacPherson. We first define a generalized moment graph, a combinatorial object determined by the structure of a finite collection of low-dimensional torus orbits. We then show how an algorithm of Braden and MacPherson computes the stalks of the equivariant intersection cohomology complex, producing “sheaves” on the generalized moment graph that functorially encode the equivariant intersection cohomology. Because the Braden-MacPherson algorithm uses only basic commutative algebra, this approach greatly simplifies the a priori very difficult task of computing intersection cohomology stalks. This paper extends previous work of Springer, who had computed the (ordinary) intersection cohomology Betti numbers of the Borel orbit closures in the wonderful compactification.References
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Additional Information
- Stephen Oloo
- Affiliation: Department of Mathematics, Kalamazoo College, Kalamazoo, Michigan 49048
- Email: stephen.oloo@kzoo.edu
- Received by editor(s): September 14, 2018
- Received by editor(s) in revised form: January 9, 2020
- Published electronically: July 8, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 6451-6478
- MSC (2010): Primary 14F43, 14M27; Secondary 32S60
- DOI: https://doi.org/10.1090/tran/8109
- MathSciNet review: 4155182