Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.



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> My Question:
>  What is the geometric analogue of the restriction functor $Res^G\_L:Rep(G)\to Rep(L)$?
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To be a little bit more precise:


Let $\check{G}$ be the dual group of $G$. As usual define $\mathcal{K}:=\mathbb{C}$ and $\mathcal{O}:=\mathbb{C}[[t]]$. Let further
$Gr\_{\check{G}}:=\check{G}(\mathcal{K})/\check{G}(\mathcal{O})$ denote the affine Grassmannian and $P\_{\check{G}(\mathcal{O})}(Gr\_{\check{G}})$ the category of $\check{G}(\mathcal{O})$-equivariant perverse sheaves on $Gr$.
The geometric Satake Isomorphism gives an equivalence of tensor categories categories
$$P\_{\check{G}(\mathcal{O})}(Gr\_{\check{G}}) \cong Rep(G)$$ 



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> Is there a nice (from the geometric viewpoint) functor $\check{Res}:P\_{\check{G}(\mathcal{O})}(Gr\_{\check{G}})\to P\_{\check{L}(\mathcal{O})}(Gr\_{\check{L}})$ which corresponds under the geometric Satake isomorphism to the restriction functor $Res^G\_L$?
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Restriction to Levi Subgroups and the Affine Grassmannian
On the level of sheaves, the construction is indeed a pull-push formula. I believe it was first worked out by Beilinson and Drinfeld in section 5.3 of their preprint "Quantization of Hitchin's Hamiltonians and Hecke eigen-sheaves". For published references, look at section 2.4 in Braverman and Gaitsgory's paper "Crystals via the affine Grassmannian" (Duke Math. J. 107 (2001), 561–575) or at section 1.3 in Vasserot's paper "On the action of the dual group on the cohomology of perverse sheaves on the affine Grassmannian" (Compositio Math. 131 (2002), 51–60).


