  Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let
$\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the
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. 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
$\operatorname{Res}^G_L\colon\operatorname{Rep}(G)\rightarrow\operatorname{Rep}(L)$
in terms of the hyperbolic localization functors for the generalized
transversal slices.
