Computation of generalized equivariant cohomologies of Kac–Moody flag varieties

By Megumi Harada, A. Henriques, T. Holm. 2004

94 citations recorded by the source corpus.

Abstract In 1998, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring H T * ( X ) can be described by combinatorial data obtained from its orbit decomposition. In this paper, we generalize their theorem in three different ways. First, our group G need not be a torus. Second, our space X is an equivariant stratified space, along with some additional hypotheses on the attaching maps. Third, and most important, we allow for generalized equivariant cohomology theories E G * instead of H T * . For these spaces, we give a combinatorial description of E G * ( X ) as a subring of ∏ E G * ( F i ) , where the F i are certain invariant subspaces of X. Our main examples are the flag varieties G / P of Kac–Moody groups G , with the action of the torus of G . In this context, the F i are the T-fixed points and E G * is a T-equivariant complex oriented cohomology theory, such as H T * , K T * or MU T * . We detail several explicit examples.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ae45191b5e03dd277c7c9c073d7c1e47704de4b5

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Poset pinball, GKM-compatible subspaces, and Hessenberg varieties (2010, methodology): “Schubert classes in the equivariant K-theory and equivariant cohomology of the Grassmannian, http://arxiv.org/abs/math/0512204.”
A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, background): “This influential theory and its many consequences have been extensively generalized and used since [4, 13, 15, 17, 19, 20, 21, 23, 25].”
Equivariant oriented cohomology of flag varieties (2014, background): “In [HHH], Harada, Henriques and Holm prove the injectivity of the pull-back to fixed points map and the characterization of its image in the topological context of generalized cohomology theories, under an assumption that certain characteristic classes are prime to each other.”
K-theory Schubert calculus of the affine Grassmannian (2009, background): “See also [ HHH ] for more general cohomology theories and spaces.”