Geometric Satake conjecture for Kac–Moody Coulomb branches

Let $\mathcal M(\lambda,\mu)$ be the Coulomb branch associated with a quiver gauge theory, let $\chi$ be the one-parameter subgroup defined by a character of the gauge group, and let

$$\mathfrak A_\chi(\lambda,\mu)=\{x\in\mathcal M(\lambda,\mu)\mid \lim_{t\to0}\chi(t)x\text{ exists}\}$$

be its attracting set. Let $\mathfrak g$ be the associated Kac–Moody Lie algebra and $\mathfrak g^\vee$ its Langlands dual. The geometric Satake conjecture. (1) The intersection of $\mathfrak A_\chi(\lambda,\mu)$ with each symplectic leaf is either empty or a Lagrangian subvariety. (2) The direct sum

$$\bigoplus_\mu H_{\operatorname{top}}(\mathfrak A_\chi(\lambda,\mu))$$

has the structure of the integrable highest-weight representation of $\mathfrak g^\vee$ with highest weight $\lambda$. The conjecture is proved in finite type and for affine quivers of type $A$; the general Kac–Moody case remains open.