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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 thequantum alcove model. As an application, we prove the Chevalley formula for ananti-dominant fundamental weight for the (small) torus-equivariant quantum$K$-theory $QK_{T}(G/B)$ of an (ordinary) flag manifold $G/B$; this has been alongstanding conjecture about the multiplicative structure of $QK_{T}(G/B)$. Intype $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomialsindeed represent (opposite) Schubert classes in the (non-equivariant) quantum$K$-theory $QK(SL_{n}/B)$; we also obtain very explicit information about thecoefficients in the respective Chevalley formula.

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