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In this paper, we prove a class of nontrivial q-Pochhammer symbol identitieswith extra parameters by iterated residue method. Then we use these identitiesto find relations of the quasi-map $K$-theoretical $I$-functions with levelstructure between Grassmannian and its dual Grassmannian. Here we find aninterval of levels within which two $I$-functions are the same, and on theboundary of that interval, two $I$-functions are intertwining with each other.We call this phenomenon level correspondence in Grassmann duality.

Level correspondence of $K$-theoretic $I$-function in Grassmann duality