Quantum K-theory of flag varieties via non-abelian localization

In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big $\mathcal{J}$-function, for any partial flag variety. The reconstruction may start with any Weyl-group-invariant value of the well-understood big $\mathcal{J}$-function of an associated toric variety. We generalize the recursive method \cite{Givental:perm2}, based on torus fixed point localization, to deal with non-isolated one-dimensional toric orbits, through incorporating ``balanced broken orbits'' into consideration and subsequently proving a vanishing result of their contribution. Furthermore, we extend the study to twisted generating functions of the flag variety, and demonstrate properties including a non-abelian quantum Lefschetz theorem and a duality between level structures. In the end, we construct a K-theoretic mirror in terms of Jackson-type integrals.