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Cohomologie T-équivariante de la variété de drapeaux d’un groupe de Kač-Moody. (T-equivariant cohomology of the flag manifold of a Kač-Moody group). (French) Zbl 0706.57024
Summary: Bernstein-Gel’fand-Gel’fand operators \({\mathcal A}_ i\) are defined over the integral T-equivariant cohomology \(H^*_ T({\mathcal F})\) of the flag manifold \({\mathcal F}=G/B\) of a Kač-Moody group G. By integration over the Schubert manifolds of \({\mathcal F}\), we characterize a family \(\{\) \({\mathcal L}_ w\}_{w\in W}\) of \(H^*_ T(\cdot)\)-linear forms over \(H^*_ T({\mathcal F})\), base of the dual of \(H^*_ T({\mathcal F})\). These canonical forms are related to the operators \({\mathcal A}_ i\) by the equality \({\mathcal L}_{wr_ i}={\mathcal L}_ w{\mathcal A}_ i\) whenever \(wr_ i>w\), implying the intrinsic character of the compositions \({\mathcal A}_ w\) of the \({\mathcal A}_ i's\). We show that each \({\mathcal A}_ w\) can be obtained by integration over fibers of certain fibrations above \({\mathcal F}.\)
By restriction to the subspace W of T-fixed points of \({\mathcal F}\), we given an injective homomorphism \(\Theta\) from \(H^*_ T({\mathcal F})\) into the algebra F(W;Q) of all maps defined on W with values in the fraction field Q of the polynomial algebra \(S={\mathbb{Z}}[\alpha_ 1,...,\alpha_ n]\), where \(\{\alpha_ 1,...,\alpha_ n\}\) denotes the simple root system of the Lie algebra of G. Explicit formulas for the localizations of the \({\mathcal L}_ w's\) over F(W;Q) are given. We determine also the localizations \(A_ i's\) of the \({\mathcal A}_ i's\) over F(W;Q), which allows us to characterize algebraically the image of \(\Theta\) as the greatest subset of F(W;S) of maps of bounded degrees stable under the action of the \(A_ i's\). We then easily identify this image to the Kostant-Kumar algebra \(\Lambda\), explaining the principal results of B. Kostant and S. Kumar [Proc. Natl. Acad. Sci. USA 83, 1543-1545 (1986; Zbl 0588.17012); Adv. Math. 62, 187-237 (1986; Zbl 0641.17008)].

MSC:
57T15 Homology and cohomology of homogeneous spaces of Lie groups
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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