Positivity in T-equivariant K-theory of Kac-Moody flag varieties

We prove a positivity result for the T-equivariant K-theory of flag varieties in the Kac-Moody case. Specifically, we show sign-alternation of the structure constants in the Schubert basis, which generalizes the work of Anderson-Griffeth-Miller from the finite case to the Kac-Moody case. Specializing this result to the affine Grassmannian implies sign-alternation of the structure constants of the affine stable Grothendieck polynomials as was conjectured by Lam-Schilling-Shimozono. Further, in the case of the affine Grassmannian associated to SL2, we determine an inductive formula for the T-equivariant structure constants and explicit closed forms for the ordinary structure constants in both the Schubert and dual bases.Doctor of Philosoph