Affine Kac–Moody geometric Satake conjecture for Coulomb branches

Let $Q$ be an affine quiver, let $\mathcal M_C(\lambda,\mu)$ be its Coulomb branch, and let $\mathfrak A_\chi(\lambda,\mu)$ be the attracting set for the chosen cocharacter $\chi$. Let $V(\lambda)_\mu$ be the $\mu$-weight space of the integrable affine Kac–Moody representation $V(\lambda)$; let $\Phi$ denote hyperbolic restriction, and let $\mathfrak g_{\mathrm{KM}}$ be the associated affine Kac–Moody algebra. Affine geometric Satake conjecture. (1) $\mathfrak A_\chi(\lambda,\mu)$ is empty if and only if $V(\lambda)_\mu=0$, and $\mathcal M_C(\lambda,\mu)^{\chi(\mathbb C^\times)}$ is a single point when nonempty. (2) The intersections of $\mathfrak A_\chi(\lambda,\mu)$ with symplectic leaves are Lagrangian; consequently hyperbolic restriction is hyperbolic semismall, remains perverse on the intersection-cohomology complex, and is identified with $H_{\mathrm{top}}(\mathfrak A_\chi(\lambda,\mu))$. (3) The direct sum of these top homology groups over $\mu$ has a $\mathfrak g_{\mathrm{KM}}$-module structure isomorphic to $V(\lambda)$, with the $\mu$-summand isomorphic to $V(\lambda)_\mu$. This is the proposed affine Kac–Moody analogue of geometric Satake; the paper records related results in finite type and affine type $A$, but leaves the general assertion open.