This talk is based on the paper https://arxiv.org/abs/1805.07721 (joint with M. Finkelberg and I. Mirković).
Let $G$ be a reductive complex algebraic group. Recall that a geometric Satake isomorphism is an equivalence between the category of $G(\mathcal{O})$-equivariant perverse sheaves on the affine Grassmannian for $G$ and the category of finite dimensional representations of the Langlands dual group $\check{G}$.
It follows that for any perverse sheaf $\mathcal{P}$ there exists an action of $\check{\mathfrak{g}}$ on the global cohomology of $\mathcal{P}$.
We will explain how to construct this action explicitly. To do so, we will describe a new geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $U(\check{\mathfrak{g}})$ based on certain one-parametric deformation of zastava spaces. We will start from the case $G=SL_2$, where everything can be described explicitly.