Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$, and let $\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct integrable crystals $\mathbf{B}^{G}(\lambda)$, $\lambda\in\Lambda^+_G$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of $G$. We also construct tensor product maps $\mathbf{p}_{\lambda_{1},\lambda_{2}}\colon\mathbf{B}^{G}(\lambda_1)\otimes\mathbf{B}^{G}(\lambda_2) \to\mathbf{B}^{G}(\lambda_{1}+\lambda_{2})\cup\{0\}$ in terms of multiplication in generalized transversal slices. Let $L \subset G$> be a Levi subgroup of $G$. We describe the functor $\operatorname{Res}^G_L\colon\operatorname{Rep}(G)\to\operatorname{Rep}(L)$ of restriction to $L$ in terms of the hyperbolic localization functors for generalized transversal slices.