  We study the hemisphere partition function of a three-dimensional
$\mathcal{N}=4$ supersymmetric $U(N)$ gauge theory with one adjoint and one
fundamental hypermultiplet -- the ADHM quiver theory. In particular, we propose
a distinguished set of UV boundary conditions which yield Verma modules of the
quantised chiral rings of the Higgs and Coulomb branches. In line with a recent
proposal by two of the authors in collaboration with M. Bullimore, we show
explicitly that the hemisphere partition functions recover the characters of
these modules in two limits, and realise blocks gluing exactly to the partition
functions of the theory on closed three-manifolds. We study the geometry of the
vortex moduli space and investigate the interpretation of the vortex partition
functions as equivariant indices of quasimaps to the Hilbert scheme of points
in $\mathbb{C}^2$. We also investigate half indices of the ADHM quiver gauge
theory in the presence of a line operator and discuss their geometric
interpretation. Along the way we find interesting relations between our
hemisphere blocks and related quantities in topological string theory and
equivariant quantum K-theory.
