Geometric Satake correspondence for affine Lie groups

• Let $G$ be a complex reductive group, which is a complexification of a compact Lie group $G_c$. Let $\mathrm{Gr}_G$ be the affine Grassmannian associated with $G$. It is a partial flag variety associated with the affine Lie group $G_{\mathrm{aff}}$ for $G$, and related to moduli spaces of singular $G_c$-monopoles on $\R^3$. Geometric Satake correspondence gives a topological construction of representations of the Langlands dual group $G^\vee$via $\mathrm{Gr}_G$. If we replace $G$ by $G_{\mathrm{aff}}$, we cannot naively consider $\mathrm{Gr}_{G_{\mathrm{aff}}}$. Nevertheless we can still consider moduli spaces of $G_c$-instantons on the Taub-NUT space divided by a finite cyclic group （multi-Taub-NUT space）, and construct integrable representations of the Langlands dual group of $G_{\mathrm{aff}}$. （By technical reasons, a rigorous proof is given only in type A.）
• I will give a pedagogic presentation of materials, sacrificing mathematically rigorous introduction of basic tools, e.g., perverse sheaves, the definition of affine Grassmannian, and a definition of Coulomb branches via equivariant Borel-Moore homology. For mathematically oriented participants, I recommend two articles of Mark Andrea de Cataldo and Zinwen Zhu in Geometry of Moduli Spaces and Representation Theory, IAS Park City 24, AMS 2017 for the first two topics, and my expository article for Coulomb branches. An exposition of geometric Satake for affine Lie algebras is available at here.