K-theoretic Gromov-Witten invariants of line degrees on flag varieties

A homology class d is an element of H-2(X, Z) of a complex flag variety X = G/P is called a line degree if the moduli space M-0,M-0(X, d) of 0-pointed stable maps to X of degree d is also a flag variety G/P '. We prove a quantum equals classical formula stating that any n-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety G/P '. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags G/B. Our formulas make it straightforward to compute the big quantum K-theory ring QK(big)(X) modulo the ideal < Q(d)> generated by degrees d larger than line degrees.