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

Resumo

Let G be a connected reductive algebraic group over ℂ, and let Λ_G⁺ be the monoid of dominant weights of G. We construct integrable crystals B^G(λ), λ ∈ Λ_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 \(P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} \) in terms of multiplication in generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the functor Res_L^G: Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.