Functoriality of Coulomb branches

We prove that the affine closure of the cotangent bundle of the parabolic base affine space for $\mathrm{GL}_n$ or $\mathrm{SL}_n$ is a Coulomb branch, which confirms a conjecture of Bourget-Dancer-Grimminger-Hanany-Zhong. In particular, we show that the algebra of functions on the cotangent bundle of the parabolic base affine space of $\mathrm{GL}_n$ or $\mathrm{SL}_n$ is finitely generated. We prove this by showing that, if we are given a map $H \to G$ of complex reductive groups and a representation of $G$ satisfying an assumption we call gluable, then the Coulomb branch for the induced representation of $H$ is obtained from the corresponding Coulomb branch for $G$ by a certain Hamiltonian reduction procedure. In particular, we show that the Coulomb branch associated to any quiver with no loops can be obtained from Coulomb branches associated to quivers with exactly two vertices using this procedure.