Chromatic aberrations of geometric Satake over the regular locus

Let $G$ be a connected, simply-laced, almost simple algebraic group over $\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, and let $T_c$ be a maximal torus therein. Let $\mathrm{Gr}_G$ denote the affine Grassmannian of $G$, and let $\check{G}$ denote the Langlands dual group to $G$ with Lie algebra $\check{\mathfrak{g}}$. The derived geometric Satake equivalence of Bezrukavnikov-Finkelberg gives an equivalence between the $\infty$-category $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ of $G_c$-equivariant local systems of $\mathbf{C}$-vector spaces on $\mathrm{Gr}_G$ and the $\infty$-category of quasicoherent sheaves on a large open substack of $\check{\mathfrak{g}}^\ast[2]/\check{G}$. In this article, we study the analogous story when $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ is replaced by the $\infty$-category of $T_c$-equivariant local systems of $k$-modules over $\mathrm{Gr}_G(\mathbf{C})$, where $k$ is ($2$-periodic) rational cohomology, (complex) K-theory, or elliptic cohomology. Crucial to our work is the genuine equivariant refinement of these cohomology theories. We show that, although there may not be an equivalence as in derived geometric Satake, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ admits a 1-parameter degeneration to an $\infty$-category of quasicoherent sheaves built out of the geometry of various Langlands-dual stacks associated to $k$ and the $1$-dimensional group scheme computing $S^1$-equivariant $k$-cohomology. For example, when $k$ is an elliptic cohomology theory with elliptic curve $E$, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ degenerates to the $\infty$-category of quasicoherent sheaves on a large open locus in the moduli stack of $\check{B}$-bundles of degree $0$ on $E$. We also study several applications of these equivalences.