K-theoretic Gromov-Witten Invariants of Lines in Homogeneous Spaces

Let X = G/P be a homogeneous space and epsilon(k) the homology class of a simple coroot. For almost all X, the variety Z(k)(X) of degree epsilon(k) pointed lines in X is known to be homogeneous. For these X, we show that the 3-point, genus 0, equivariant K-theoretic Gromov-Witten invariants of lines of degree epsilon(k) equal quantities obtained in the (ordinary) equivariant K-theory of Z(k)(X). We apply this to compute the Schubert structure constants N-u,N-vw,epsilon(k) in the equivariant quantum K-theory ring of X. Using geometry of spaces of lines through Schubert or Richardson varieties we prove vanishing and positivity properties of N-u,v(w,epsilon k). This generalizes many results from K-theory and quantum cohomology of X and gives new identities among the structure constants in the equivariant K-theory of X.