Let G be a connected reductive algebraic group over ℂ, and let Λ + be the monoid of dominant weights of G. We construct integrable crystals BG(λ), λ ∈ Λ + , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps $$P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} $$ in terms of multiplication in generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the functor Res : Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.