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