A general Chevalley formula for semi inf

We present a combinatorial Chevalley formula for arbitrary weights in the torus-equivariant K-group of semi-infinite flag manifolds, utilizing the quantum alcove model. This study proves the Chevalley formula for anti-dominant fundamental weights in the context of the torus-equivariant quantum K-theory QKT (G/B) of the flag manifold G/B, addressing a longstanding conjecture regarding the multiplicative structure of QKT (G/B). Furthermore, for type An−1, we establish that quantum Grothendieck polynomials represent Schubert classes in the non-equivariant quantum K-theory QK(SLn/B). Our findings also yield explicit information about the coefficients in the respective Chevalley formula, expanding the application of the quantum alcove model beyond previous results, which primarily focused on dominant and anti-dominant weights. This work completes the framework of employing the quantum alcove model in geometric settings and extends the Chevalley formula within K-theory to accommodate a broader class of weights.