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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 identifies the tensor category of finite-dimensional representations of the Langlands dual group with the tensor category of -equivariant perverse sheaves on the affine Grassmannian of . (Here and .) As a by-product one gets a description of the irreducible -equivariant intersection cohomology (IC) sheaves of the closures of -orbits in in terms of -analogs of the weight multiplicity for finite-dimensional representations of .
The purpose of this article is to try to generalize the above results to the case when is replaced by the corresponding affine Kac-Moody group . (We refer to the (not yet constructed) affine Grassmannian of as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various -orbits inside the closure of another -orbit in . We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding -analog of the weight multiplicity for the Langlands dual affine group , 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 and , 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
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