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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 $$F(\mathcal{E};R[T])$$.

##### MSC:
 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
##### Keywords:
$$K$$-theory; equivariant cohomology
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