Integrable crystals and restriction to Levi via generalized slices in the affine Grassmannian

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let ${\mathrm{\Lambda }}_G^{+}$ be the monoid of dominant weights of $G$. We construct the integrable crystals ${\mathbf{B}}^G(\lambda ),\lambda \in {\mathrm{\Lambda }}_G^{+}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $$\mathbf{p}${\lambda{1},\lambda_{2}}\colon $\mathbf{B}$^{G}(\lambda_{1}) \otimes $\mathbf{B}$^{G}(\lambda_{2}) \rightarrow $\mathbf{B}$^{G}(\lambda_{1}+\lambda_{2})\cup{0}$ in terms of multiplication of generalized transversal slices. Let $L\subset G$ be a Levi subgroup of $G$. We describe the restriction to Levi ${\mathrm{Res}}_L^G\text{\hspace{0.17em}⁣}:\mathrm{Rep}(G)\to \mathrm{Rep}(L)$ in terms of the hyperbolic localization functors for the generalized transversal slices.