The purpose of this paper is to provide a detailed description of the spaces that can be specified as L2 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 Brüning 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.
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