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Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to$SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determinean inductive formula for the Schubert basis structure constants in thetorus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case ofordinary (non-equivariant) $K$-theory we find an explicit closed form for thestructure constants. We also determine an inductive formula for the structureconstants in the torus-equivariant cohomology ring, and use this formula tofind closed forms for some of the structure constants.

An explicit determination of the $K$-theoretic structure constants of the affine Grassmannian associated to $SL_2$