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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generalized moment graphs and the equivariant intersection cohomology of BXB-orbit closures in the wonderful compactification of a group


Author: Stephen Oloo
Journal: Trans. Amer. Math. Soc. 373 (2020), 6451-6478
MSC (2010): Primary 14F43, 14M27; Secondary 32S60
DOI: https://doi.org/10.1090/tran/8109
Published electronically: July 8, 2020
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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.


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Additional Information

Stephen Oloo
Affiliation: Department of Mathematics, Kalamazoo College, Kalamazoo, Michigan 49048
Email: stephen.oloo@kzoo.edu

DOI: https://doi.org/10.1090/tran/8109
Received by editor(s): September 14, 2018
Received by editor(s) in revised form: January 9, 2020
Published electronically: July 8, 2020
Article copyright: © Copyright 2020 American Mathematical Society