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

Abstract

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$. Let $Λ^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\mathbf{B}^{G}(λ),\ λ\inΛ^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\mathbf{p}_{λ_{1},λ_{2}}\colon \mathbf{B}^{G}(λ_{1}) \otimes \mathbf{B}^{G}(λ_{2}) \rightarrow \mathbf{B}^{G}(λ_{1}+λ_{2})\cup\{0\}$ 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.