Littlewood Richardson coefficients for Kac-Moody flag varieties

Let G be a complex simple Lie group or Kac-Moody group and P a parabolic subgroup. One of the goals Schubert calculus is to understand the product structure of the cohomology ring H^*(G/P) with respect to its basis of Schubert classes. If G/P is the Grassmannian, then the structure constants corresponding to the Schubert basis are the classical Littlewood-Richardson coefficients which appear in various topics such as enumerative geometry, algebraic combinatorics and representation theory.

In this talk, I will discuss joint work with A. Berenstein in which we give a combinatorial formula for these coefficients in terms of the Cartan matrix corresponding to G. In particular, our formula implies positivity of the “generalized” Littlewood-Richardson coefficients in the case where the corresponding Weyl group of G is a free Coxeter group (i.e. no braid relations). Moreover, this positivity result extends to the torus-equivariant coeffients of H^*_T(G/P) and does not rely on the geometry of the flag variety G/P.