Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum $K$-group of partial flag manifolds

In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $-{\varpi }_k$ in the torus-equivariant quantum $K$-group of the partial flag manifold $G\mathrm{/}{P}_J$ (where $J=I\setminus \{k\}$) corresponding to the maximal (standard) parabolic subgroup $P_J$ of minuscule type in type $A$, $D$, $E$, or $B$. This result is obtained by proving a similar formula in a torus-equivariant $K$-group of the semi-infinite partial flag manifold ${\mathbf{Q}}_J$ of minuscule type, and then by making use of the isomorphism between the torus-equivariant quantum $K$-group of $G\mathrm{/}{P}_J$ and the torus-equivariant $K$-group of ${\mathbf{Q}}_J$, recently established by Kato.