  We revisit the factorisation of supersymmetric partition functions of 3d
$\mathcal{N}=4$ gauge theories. The building blocks are hemisphere partition
functions of a class of UV $\mathcal{N}=(2,2)$ boundary conditions that mimic
the presence of isolated vacua at infinity in the presence of real mass and FI
parameters. These building blocks can be unambiguously defined and computed
using supersymmetric localisation. We show that certain limits of these
hemisphere partition functions coincide with characters of lowest weight Verma
modules over the quantised Higgs and Coulomb branch chiral rings. This leads to
expressions for the superconformal index, twisted index and $S^3$ partition
function in terms of such characters. On the way we uncover new connections
between boundary 't Hooft anomalies, hemisphere partition functions and lowest
weights of Verma modules.
