The T-equivariant Cohomology of Bott-Samelson varieties

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We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of the associated Weyl group such that its global sections give the T-equivariant cohomology of our Bott-Samelson resolution.