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

Understand

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.