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Positivity in equivariant Schubert calculus. (English) Zbl 1069.14055

Summary: We prove a positivity property for the cup product in the \(T\)-equivariant cohomology of the flag variety. This was conjectured by D. Peterson and has as a consequence a conjecture of S. Billey [Duke Math. J. 96, No.1, 205–224 (1999; Zbl 0980.22018)]. The result for the flag variety follows from a more general result about algebraic varieties with an action of a solvable linear algebraic group such that the unipotent radical acts with finitely many orbits. The methods are those used by S. Kumar and M. Nori [Int. Math. Res. Not. 1998, No.14, 757–763 (1998; Zbl 1014.17023)].

MSC:

14M17 Homogeneous spaces and generalizations
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14M15 Grassmannians, Schubert varieties, flag manifolds
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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