The geometric Satake equivalence identifies for a suitable algebraic group $G$, and suitable local field $K$ with ring of integers $\mathcal{O}_K$, suitable functions on the double coset/Grassmannian $G(\mathcal{O}_K)\backslash G(K)/G(\mathcal{O}_K)$ with the representation ring of the Langlands dual group ${}^L G$.
Notice that the double coset appearing here is akin to that which controls the Langlands correspondence, whose geometric meaning is discussed for instance at Weil uniformization and at function field analogy.
Wikipedia, Satake isomorphism
Benedict Gross, On the Satake isomorphism (pdf)
Dennis Gaitsgory, Graduate Seminar. Geometric Representation theory. Fall 2009–Spring 2010, Feb. 23: Affine Grassmannian-II: the Satake equivalence (Ryan Reich) (pdf)
example 1.7 in these rough notes, apparently taken in some talk by Jacob Lurie: Talk by Jacob Lurie