Cycles de Schubert et cohomologie équivariante de K/T. (Schubert cycles and equivariant cohomology of K/T). (French) Zbl 0624.22005

Let K be a compact connected Lie group, T be a maximal torus in K and T’ be its normalizer in K. The flag variety \(X=K/T\) admits a cellular decomposition \(X=\cup X_ w\) with cells indexed by all elements w of the Weyl group \(W=T'/T\). The closures \(\bar X{}_ w\) of these cells determine elements in the dual space to the complex cohomology space \(H^*(X)\) that are called Schubert cycles.
The aim of the present paper is to define and to study the analogues of Schubert cycles for the equivariant cohomology \(H^*_ K(X)\). Identifying (by the Chern-Weil isomorphism) \(H^*_ K(X)\) with the space of polynomial functions on the Lie algebra \({\mathfrak t}\) of T the author gives an explicit formula for the equivariant Schubert cycles in terms of the reduced decompositions of elements of W.
Reviewer: S.I.Gel’fand


22E40 Discrete subgroups of Lie groups
55N25 Homology with local coefficients, equivariant cohomology
14M15 Grassmannians, Schubert varieties, flag manifolds
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