( \lambda )]$ for each $\lambda \in P$ and relevant $y \in W_{{\mathrm{af}}}$. We also prove that our $K_{{\widetilde{{\mathbf{I}}}}}({\mathbf{Q}_{G}}^{\mathrm{rat}})$ is natural enough so that it admits a nil-DAHA action as an analog of Kostant-Kumar [@KK90] for ${\mathbf{Q}_{G}^{\mathrm{rat}}}$ (see Section \[sec:nDAHA\] for details).

Here we recall that in Ishii-Naito-Sagaki [@INS] and Naito-Sagaki [@NS16], the semi-infinite path model of the crystal basis of $V_{x}^{-} ( \lambda )$ is constructed for every $\lambda \in P^+$ and $x \in W_{{\mathrm{af}}}$; it is a specific subset of the set of “semi-infinite” LS paths ${\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda )$ of shape $\lambda$ parametrizing the global crystal basis of $V ( \lambda )$. Note that it is endowed with three functions $$\iota,\,\kappa : {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \longrightarrow W_{\mathrm{af}}\quad \text{and} \quad
{\mathop{\rm wt}\nolimits}: {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \longrightarrow P \oplus {\mathbb{Z}}\delta,$$ which are called the initial/final directions and the weight, respectively. We set $${\mathbb{B}^{{\frac{\infty}{2}}}}_{{\succeq}x} ( \lambda ) := 
 \bigl\{ \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}( \lambda ) \mid \kappa ( \eta ) {\succeq}x \bigr\}.$$

In order to make use of the path model above to derive the Pieri-Chevalley formula for ${\mathbf{Q}_{G}^{\mathrm{rat}}}$, we additionally need a combinatorial version of the semi-infinite SMT. This consists of the definition of the initial direction ${\iota(\eta,\,x)} \in W_{{\mathrm{af}}}$ of 