This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group $G$ identifies the tensor category $\mathrm{Rep}(G^{\vee })$ of finite-dimensional representations of the Langlands dual group $G^{\vee }$ with the tensor category ${\mathrm{Perv}}_{G(O)}({\mathrm{Gr}}_G)$ of $G(O)$-equivariant perverse sheaves on the affine Grassmannian ${\mathrm{Gr}}_G=G(K)/G(O)$ of $G$. (Here $K=\mathbb{C}((t))$ and $O=\mathbb{C}[[t]]$.) As a by-product one gets a description of the irreducible $G(O)$-equivariant intersection cohomology (IC) sheaves of the closures of $G(O)$-orbits in ${\mathrm{Gr}}_G$ in terms of $q$-analogs of the weight multiplicity for finite-dimensional representations of $G^{\vee }$.

The purpose of this article is to try to generalize the above results to the case when $G$ is replaced by the corresponding affine Kac-Moody group $G_{\mathrm{aff}}$. (We refer to the (not yet constructed) affine Grassmannian of $G_{\mathrm{aff}}$ as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various $G_{\mathrm{aff}}(O)$-orbits inside the closure of another $G_{\mathrm{aff}}(O)$-orbit in ${\mathrm{Gr}}_{G_{\mathrm{aff}}}$. We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding $q$-analog of the weight multiplicity for the Langlands dual affine group $G_{\mathrm{aff}}^{\vee }$, and we check this conjecture in a number of cases.

Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian, and so forth) will be addressed in another publication

We prove a relative trace identity between ${\mathrm{GL}}_{2n}$ and ${\widetilde{\mathrm{S}\mathrm{p}}}_n$, using Ginzburg, Rallis, and Soudry's work on automorphic descent. This should serve as a model on using automorphic descent to establish a relative trace identity

We prove the existence of minimizers for Hartree-Fock-Bogoliubov (HFB) energy functionals with attractive two-body interactions given by Newtonian gravity. This class of HFB functionals serves as a model problem for self-gravitating relativistic Fermi systems, which are found in neutron stars and white dwarfs. Furthermore, we derive some fundamental properties of HFB minimizers such as a decay estimate for the minimizing density. A decisive feature of the HFB model in gravitational physics is its failure of weak lower semicontinuity. This fact essentially complicates the analysis compared to the well-studied Hartree-Fock theories in atomic physics

We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy