A general Chevalley formula for semi-infinite flag manifolds and quantum K-theory

We give a Chevalley formula for an arbitrary weight for the torus-equivariant K -group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for an anti-dominant fundamental weight for the (small) torus-equivariant quantum K -theory Q K T ( G / B ) of a (finite-dimensional) flag manifold G / B ; this has been a longstanding conjecture about the multiplicative structure of Q K T ( G / B ) . In type A n - 1 , we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K -theory Q K ( S L n / B ) ; we also obtain very explicit information about the coefficients in the respective Chevalley formula.