An explicit determination of the K-theoretic structure constants of the affine Grassmannian associated to SL₂

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Let G:=SL₂ denote the affine Kac-Moody group associated to SL₂ and X the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of 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.