Real groups, symmetric varieties and Langlands duality

Let $G_\mathbb R$ be a connected real reductive group and let $X$ be the corresponding complex symmetric variety under the Cartan bijection. We construct a canonical equivalence between the relative Satake category of $G(\mathcal O)$-equivariant $\mathbb C$-constructible complexes on the loop space of $X$ and the real Satake category of $G_\mathbb R(\mathcal O_\mathbb R)$-equivariant $\mathbb C$-constructible complexes on the real affine Grassmannian. We show that the equivalence is $t$-exact with respect to the natural perverse $t$-structures and is compatible with the fusion products and Hecke actions. We further show that the relative Satake category is equivalent to the category of $\mathbb C$-constructible complexes on the moduli stack of $G_\mathbb R$-bundles on the real projective line $\mathbb P^1(\mathbb R)$ and hence provides a connection between the relative Langlands program and the geometric Langlands program for real groups. We provide numerous applications of the main theorems to real and relative Langlands duality including the formality and commutativity conjectures for the real and relative Satake categories and an identification of the dual groups for $G_\mathbb R$ and $X$.