On Positivity in T -equivariant K-theory of Flag Varieties

We prove some general results on the T -equivariant K-theory KT (G/P ) of the flag variety G/P , where G is a semisimple complex algebraic group, P is a parabolic subgroup and T is a maximal torus contained in P . In particular, we make a conjecture about a positivity phenomenon in KT (G/P ) for the product of two basis elements written in terms of the basis of KT (G/P ) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T -equivariant Ktheory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.