  We propose a construction of the Coulomb branch of a $3d\ {\mathcal N}=4$
gauge theory corresponding to a choice of a connected reductive group $G$ and a
symplectic finite-dimensional reprsentation $\mathbf M$ of $G$, satisfying
certain anomaly cancellation condition. This extends the construction of
arXiv:1601.03586 (where it was assumed that ${\mathbf M}={\mathbf
N}\oplus{\mathbf N}^*$ for some representation $\mathbf N$ of $G$). Our
construction goes through certain "universal" ring object in the twisted
derived Satake category of the symplectic group $Sp(2n)$. The construction of
this object uses a categorical version of the Weil representation; we also
compute the image of this object under the (twisted) derived Satake equivalence
and show that it can be obtained from the theta-sheaf introduced by S.Lysenko
on $\operatorname{Bun}_{Sp(2n)}({\mathbb P}^1)$ via certain Radon transform. We
also discuss applications of our construction to a potential mathematical
construction of $S$-duality for super-symmetric boundary conditions in
4-dimensional gauge theory and to (some extension of) the conjectures of
D.Ben-Zvi, Y.Sakellaridis and A.Venkatesh.
