carries almost all information about simple $G$-modules. Their program is subsequently completed by Littelmann-Seshadri [@LiSe03] and Lenart-Shimozono [@LeSh14] (see also Lenart-Postnikov [@LeP07]).

Peterson [@Pet97] noticed that the quantum $K$-theory of $X$ should be intimately connected with the $K$-theory of the “affine version" of $X$ (see Lam-Shimozono [@LS10] and Lam-Li-Mihalcea-Shimozono [@LLMS]). In view of Givental-Lee [@GL03] and Braverman-Finkelberg [@BF14a; @BF14c], the quantum $K$-theory of $X$ can be defined through the space of quasi-maps, whose union forms a dense subset of the formal power series model ${\mathbf{Q}}_G$ of semi-infinite flag manifolds (cf. Finkelberg-Mirković [@FM99]).

Therefore, it is quite natural to make some rigorous sense of $K_H ( {\mathbf{Q}_{G}})$ and provide the Pieri-Chevalley formula using SMT, which is compatible with the pictures provided by Pittie-Ram and Peterson. This is what we perform in this paper by affording two new theories: [**1)**]{} the Borel-Weil-Bott theory of ${\mathbf{Q}_{G}}$ that enables us to define and calculate a version of $K_H ( {\mathbf{Q}_{G}})$, and [**2)**]{} the SMT of level-zero modules over quantum affine algebras. We remark that the level-zero modules over quantum affine algebras admit an interpretation through the geometry of affine Grassmannian (of Langlands dual type), which is the “affine version” of $X$ (see, e.g., Lenart-Naito-Sagaki-Schilling-Shimozono 