Towards geometric Satake correspondence for Kac-Moody algebras -- Cherkis bow varieties and affine Lie algebras of type $A$

We give a provisional construction of the Kac-Moody Lie algebra module structure on the hyperbolic restriction of the intersection cohomology complex of the Coulomb branch of a framed quiver gauge theory, as a refinement of the conjectural geometric Satake correspondence for Kac-Moody algebras proposed in arXiv:1604.03625. This construction assumes several geometric properties of the Coulomb branch under the torus action. These properties are checked in affine type A, via the identification of the Coulomb branch with a Cherkis bow variety established in arXiv:1606.02002.