To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli space of vacua). It shoud be a Poisson variety, symplectic in the smooth locus, and, in some sense, hyper-Kähler.
When we have a connected complex reductive group $G$ and a representation $M$ of cotangent type, i.e., $M=N \oplus N^*$, where $N$ is a finite dimensional complex $G$-representation, Bravermann, Finkelberg and Nakajima succedded (Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II) in defining in a mathematically rigorous way the Coulomb branch $\mathcal{M}_C(G,M)$ of the associated quantum field theory $\mathcal{T}(G,M)$
In the previous paper by Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, I, he considered representations $M$ not necessarily of cotangent type, and described, in a heuristic way, the coordinate ring $\mathbb{C}[\mathcal{M}_C(G,M)]$ as a graded vector space, among other things. However, the product at the time could not be defined.
What progress has been made since the two above mentioned papers came out in trying to define rigorously Coulomb branches whose representation $M$ is not necessarily of cotangent type?
