  In this paper we construct certain quantum spin systems on moduli spaces of
$G$-connections on a connected oriented finite graph, with $G$ a simply
connected compact Lie group. We construct joint eigenfunctions of the commuting
quantum Hamiltonians in terms of local invariant tensors. We determine
sufficient conditions ensuring superintegrability of the quantum spin system
using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra
and Lepowsky & McCollum.
  The resulting class of quantum superintegrable spin systems includes the
quantum periodic and open spin Calogero-Moser spin chains as special cases. In
the periodic case the description of the joint eigenfunctions in terms of local
invariant tensors are multipoint generalised trace functions, in the open case
multipoint spherical functions on compact symmetric spaces.
