On quantum $K$-groups of partial flag manifolds

Abstract: We show that the equivariant small quantum $K$-group of a partial flag manifold is a quotient of that of the full flag manifold in a way it respects the Schubert basis. This is a $K$-theoretic analogue of the parabolic version of Peterson's theorem [Lam-Shimozono, Acta Math. {\bf 204} (2010)] that exhibits different shape from the case of quantum cohomology. Our quotient maps send some of the Novikov variables to $1$, and its geometric meaning is unclear in quantum $K$-theory. This paper can be seen as a continuation of [K, Ann. of Math. (to appear), and Forum of Math., Pi {\bf 9} (2021)].