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Equivariant cohomology and \(K\)-theory of Bott-Samelson varieties and flag varieties. (Cohomologie et \(K\)-théorie équivariantes des variétés de Bott-Samelson et des variétés de drapeaux.) (French) Zbl 1087.19004
The author computes the equivariant cohomology and \(K\)-theory of the Bott-Samelson varieties and deduces some results about flag varieties of Kac-Moody groups. For definitions and results about Kac-Moody groups and Kac-Moody algebras see V. G. Kac [“Infinite dimensional Lie algebras”, Cambridge Universsity Press (1985; Zbl 0574.17010)], V. G. Kac and D. H. Peterson, [“Regular functions on certain infinite-dimensional groups”, in: Arihtmetic and Geometry, Vol. II, Prog. Math. 36, 141–166 (1983; Zbl 0578.17014)] and S. Kumar [Invent. Math. 123, 471–506 (1996; Zbl 0863.14031)]. The Kac-Moody algebra \(\mathfrak{g=g}(A) \) for a generalized \( n\times n\;\)Cartan matrix \(A\) is the complex Lie algebra generated by a certain vector space \(\mathfrak{h}\) over \(\mathbb{C}\) of dimension \(2n-\)rank\( (A) \) and certain symbols satisfying some well defined relations. The algebra \( \mathfrak{b}\) is the Borel subalgebra of \(\mathfrak{g}\) and satisfies \( \mathfrak{h\subset b\subset g}\); we denote by \(H\subset B\subset G\) the corresponding Lie groups. One defines certain subgroups \(G_{k}\subset G\) for \(k=1,\ldots ,n\) and a Bott-Samelson variety \(\Gamma \) is defined to be the orbit space of the product \(G_{1}\times \ldots \times G_{n}\) under the right \(B^{n}\)-action given by \((g_{1},\ldots ,g_{n}) (b_{1},\ldots ,b_{n}) =(g_{1}b_{1},b_{1}^{-1}g_{2}b_{2},\ldots ,b_{n-1}^{-1}g_{n}b_{n}) \). Let \({\mathcal E}=\Gamma^H\) and denote by \(S\) the symmetric algebra of \(\mathfrak{h}^{\ast }\) identified inside \(H_{T}^{\ast }(\Gamma) \) where \(T\) is the maximal torus of the unitary form \(K\) of \(G\), by \(F(\mathcal{E};S) \) the \(S\)-algebra of the functions \(\mathcal{ E}\to S\) with pointwise addition and multiplication. Let \(X[T ] \) be the character group of T, and let us denote \(R[T] = \mathbb{Z}[X[T] ] \). The main result is that the \(T\)-equivariant cohomology of \(\Gamma ^{T}\) is isomorphic to \(F(\mathcal{E };S) \) and that its \(T\)-equivariant \(K\)-theory is isomorphic to \( F(\mathcal{E};R[T]) \).

19L47 Equivariant \(K\)-theory
55N91 Equivariant homology and cohomology in algebraic topology
14M15 Grassmannians, Schubert varieties, flag manifolds
17B45 Lie algebras of linear algebraic groups
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