Seidel and Pieri products in cominuscule quantum K-theory

We prove a collection of formulas for products of Schubert classes in the quantum K-theory ring QK(X) of a cominuscule flag variety X. This includes a K-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter q. We also prove new Pieri formulas for the quantum K-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum K-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements qd[OXu] of QK(X). Along the way we give a simple formula for K-theoretic Gromov–Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.