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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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##### References:
 [1] ARABIA (A.) . - Thèse de doctorat , Université de Paris VII, 1985 . · Zbl 0558.18003 [2] ARABIA (A.) . - Cycles de Schubert et cohomologie équivariante de K/T , Invent. Math., t. 85, 1986 , p. 39-52. MR 87g:32036 | Zbl 0624.22005 · Zbl 0624.22005 [3] ARABIA (A.) . - Cohomologie T-équivariante de G/B pour un groupe G de Kač-Moody , C. R. Acad. Sc. Paris, t. 302, Série I, n^\circ 17, 1986 . MR 87h:32062 | Zbl 0596.22009 · Zbl 0596.22009 [4] ARABIA (A.) . - Cohomologie T-équivariante de la variété de drapeaux d’un groupe de Kač-Moody , Préprint, École Polytechnique, 1986 . [5] ATIYAH (M. F.) and BOTT (R.) . - The moment map and equivariant cohomology , Topology, t. 23, 1984 , p. 1-28. MR 85e:58041 | Zbl 0521.58025 · Zbl 0521.58025 [6] BERLINE (N.) and VERGNE (M.) . - Fourier transforms of orbits of the coadjoint representation , Representation theory of reductive groups [Park City, Utah, 1982 ], pp. 53-67. - Prog. Math. 40, Birkhaüser, Boston, Mass. 1983 . Zbl 0527.22010 · Zbl 0527.22010 [7] BERLINE (N.) et VERGNE (M.) . - Zéros d’un champ de vecteurs et classes caractéristiques équivariantes , Duke Math. J., t. 50, 1983 , p. 539-549. Article | MR 84i:58114 | Zbl 0515.58007 · Zbl 0515.58007 [8] BOREL (A.) . - Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts , Annals of Math., Vol. 57, n^\circ 1, 1953 . MR 14,490e | Zbl 0052.40001 · Zbl 0052.40001 [9] CARTAN (H.) . - Notions d’algèbre différentielle, applications aux groupes de Lie... et La transgression dans un groupe de Lie et dans un espace fibré principal , Colloque de topologie algébrique, Bruxelles 1950 , pp. 16-27 et pp. 57-71. Zbl 0045.30701 · Zbl 0045.30701 [10] KAČ (V.) . - Constructing groups associated to infinite dimensional Lie algebras, Infinite dimensional groups with applications , M. S. R. I. publications n^\circ 4, Springer Verlag, 1985 . Zbl 0614.22006 · Zbl 0614.22006 [11] KAČ (V.) and PETERSON (D.) . - Cohomology of infinite dimensional groups and their flag varieties , à paraître. [12] KOSTANT (B.) and KUMAR (S.) . - The Nil Hecke ring and cohomology of G/P for a Kač-Moody group G , Proc. Nat. Acad. Sci. USA, t. 83, 1986 , p. 1543. MR 88b:17025a | Zbl 0588.17012 · Zbl 0588.17012 [13] KOSTANT (B.) and KUMAR (S.) . - The Nil Hecke ring and cohomology of G/P for a Kač-Moody group G , Advances in Math., t. 68, 1986 , p. 187-237. MR 88b:17025b | Zbl 0641.17008 · Zbl 0641.17008 [14] SPANIER (E. H.) . - Algebraic topology . - 1966 , Springer-Verlag, New York. MR 35 #1007 | Zbl 0145.43303 · Zbl 0145.43303
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