Abstract :
I will review Braverman-Finkelberg’s geometric Satake correspondence conjecture for affine Lie algebras via instanton moduli spaces on C^2/(Z/\ell) and their refinement by the use of Coulomb branches of affine quiver gauge theories. I will briefly explain proofs of most of the statements in type A, which I believe work in general. Then I would like to spend most of my time explaining a new approach to one remaining statement. (This is a joint work with Dinakar Muthiah.) It is a description of the intersection cohomology as a graded vector space, which is given in terms of Brylinski-Kostant filtration in the usual geometric Satake. The new approach is regarded as an affine Lie algebra version of a work by Ginzburg-Riche. It gives a coset VOA module structure on the equivariant intersection cohomology groups, as conjectured by Belavin-Feigin, Nishioka-Tachikawa, as a higher level AGT. It is also regarded as a Coulomb branch type construction of the cotangent bundle of an affine flag variety and its quantization.