  The purpose of this paper is to provide a detailed description of the spaces
that can be specified as $L^2$ domains for the operators of a first order
elliptic complex on a compact manifold with conical singularities. This entails
an analysis of the nature of the minimal domain and of a complementary space in
the maximal domain of each of the operators. The key technical result is the
nondegeneracy of a certain pairing of cohomology classes associated with the
indicial complex. It is further proved that the set of choices of domains
leading to Hilbert complexes in the sense of Br\"uning and Lesch form a
variety, as well as a theorem establishing a necessary and sufficient condition
for the operator in a given degree to map its maximal domain into the minimal
domain of the next operator.
