Lifting $I$-functions from the Grassmannians to their cotangent bundles

We relate two fundamental enumerative functions, namely the $I$-functions in the quantum $K$-ring of $G(r,n)$ and of its cotangent bundle, by defining a $K$-theoretic operator on classes, called balancing. This operator lifts the $I$-function of $G(r,n)$ to that of $T^{\ast }G(r,n)$, providing an explicit geometric interpretation. We also define an operator acting on difference operators and show that, for certain $K$-theoretic functions and the corresponding difference operators that annihilate them, including the $I$-functions of projective spaces ${\mathbb{P}}^n$, the balancing operation on difference operators and on classes is compatible. Moreover, for general $G(r,n)$, we recover the Bethe-Ansatz equations for $T^{\ast }G(r,n)$ via a procedure inspired by both balancing and the abelian/non-abelian correspondence.