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 \(QK_{T}(G/B)\) $Q{K}_T(G/B)$ of a (finite-dimensional) flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of \(QK_{T}(G/B)\) $Q{K}_T(G/B)$ . In type \(A_{n-1}\) $A_{n-1}$ , we prove that the so-called quantum Grothendieck polynomials indeed represent (opposite) Schubert classes in the (non-equivariant) quantum K-theory \(QK(SL_{n}/B)\) $QK(S{L}_n/B)$ ; we also obtain very explicit information about the coefficients in the respective Chevalley formula.