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K-theoretic Gromov-Witten invariants of lines in homogeneous spaces

Let X=G/P be a homogeneous space and e_k be the class of a simple coroot in H_2(X). A theorem of Strickland shows that for almost all X, the variety of pointed lines of degree e_k, denoted Z_k(X), is again a homogeneous space. For these X we show that the 3-point, genus 0, equivariant K-theoretic Gromov-Witten invariants of lines of degree e_k are equal to quantities obtained in the (ordinary) equivariant K-theory of Z_k(X). We apply this to compute the structure constants N_{u,v}^{w, e_k} for degree e_k from the multiplication of two Schubert classes 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,e_k}. This generalizes many results about K-theory and quantum cohomology of X, and also gives new identities among the structure constants in equivariant K-theory of X.