  In this paper, we compute K-theoretic $I$-function with level structure
(defined by quasi-map theory) of GIT-quotient of a vector space via abelian and
non-abelian correspondence. As a consequence, we generalize Givental-Lee's
result to find the analogous "Toda operators" for the $I$-function with
nontrivial level structures in the case of complete flag variety.
