A new, configuration-space picture of a formalism of group quantization, the GAQ
formalism, is presented in the context of a previous, algebraic generalization. This presentation serves to make a comprehensive discussion in which other extensions of the
formalism, principally to incorporate gauge symmetries, are developed as well. Both
images are combined in order to analyse, in a systematic manner and with complete generality, the case of linear fields (abelian current groups). To ilustrate these developments we particularize them for several fields and, in particular, we carry out the quantization the abelian Chern-Simons models over an arbitrary closed surface in detail.M.N. is grateful to the Imperial College, where this paper has mainly been written, for its hospitality. M.N. is also grateful to the Spanish MEC, CSIC and IMAFF (Madrid) for a research contract. M.C. is grateful to the Spanish MEC for a FPI fellowship
