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On the cohomology ring of the infinite flag manifold \(LG/T\). (English) Zbl 0957.17025

Author’s introduction: In J. Differ. Geom. 20, 389–431 (1984; Zbl 0565.17007), S. Kumar described the Schubert classes which are the dual to the closures of the Bruhat cells in the flag varieties of the Kac-Moody groups associated to the infinite dimensional Kac-Moody algebras. These classes are indexed by affine Weyl groups and can be chosen as elements of integral cohomologies of the homogeneous space \(\widehat{L}_{\text{pol}} G_{\mathbb C}/ \widehat{B}\) for any compact simply connected semi-simple Lie group \(G\). Later, S. Kumar and B. Kostant [Adv. Math. 62, 187–237 (1986; Zbl 0641.17008)] gave explicit cup product formulas of these classes in the cohomology algebras by using the relation between the invariant-theoretic relative Lie algebra cohomology theory (using the representation module of the nilpotent part) with the purely nil-Hecke rings. These explicit product formulas involve some BGG-type operators \(A^i\) and reflections. Using some homotopy equivalences, we determine cohomology ring structures of \(LG/T\) where \(LG\) is the smooth loop space on \(G\). Here, as an example we calculate the products and explicit ring structure of \(LSU_2/T\) using these ideas.
Note that these results grew out of a chapter of the author’s thesis [On the complex cobordism of flag varieties associated to loop groups, PhD thesis, University of Glasgow (1998)].

MSC:

17B56 Cohomology of Lie (super)algebras
57T15 Homology and cohomology of homogeneous spaces of Lie groups
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
22E67 Loop groups and related constructions, group-theoretic treatment
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