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Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let$\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct theintegrable crystals $\mathbf{B}^{G}(\lambda),\ \lambda\in\Lambda^{+}_{G}$,using the geometry of generalized transversal slices in the affine Grassmannianof 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 ofgeneralized transversal slices. Let $L \subset G$ be a Levi subgroup of $G$. Wedescribe the restriction to Levi$\operatorname{Res}^G_L\colon\operatorname{Rep}(G)\rightarrow\operatorname{Rep}(L)$in terms of the hyperbolic localization functors for the generalizedtransversal slices.

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