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 ${\Lambda }_G^{+}$ 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 ${\mathbf{p}}_{{\lambda }_1,{\lambda }_2}:{\mathbf{B}}^G({\lambda }_1)\otimes {\mathbf{B}}^G({\lambda }_2)\to {\mathbf{B}}^G({\lambda }_1+{\lambda }_2)\cup \{0\}$ in terms of multiplication of generalized transversal slices. Let $L\subset G$ be a Levi subgroup of $G$. We describe the restriction to Levi ${\mathrm{Res}}_L^G:\mathrm{Rep}(G)\to \mathrm{Rep}(L)$ in terms of the hyperbolic localization functors for the generalized transversal slices.

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