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We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things, we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.

In this sub-chapter two basic technical themes are discussed. The first centres around the Riemann-Roch and Serre duality theorems for any compact curve on a surface and is treated in Sects. 1–6. The second theme, developed in Sects. 7–8 is embedded resolution of singularities of curves and the application to the so-called simple singularities of curves.

The duality for a locally compact group established by Pontrjagin, Tannaka, Krein, Steinspring, Eymard and Tatsuuma is an important theoretical basis for the harmonic analysis. At the formal level of pure algebras, the notion of Hopf algebras is used to deal with the algebraic groups, discrete groups, or their dual objects at the same time. In order to control the infinite dimensional unitary representations, functional analysis is necessarily combined with the algebraic framework of Hopf algebras. This consideration suggests us to introduce the notion of Kac-algebras in the language of the Neumann algebras, in with the above mentioned duality is generalized by Kac [5], Takesaki [9] and Enock and Schwartz [2].

Linear duality deals with the relationship between two complementary orthogonal subspaces L and L ⊥ of Kn. The main theorem of linear duality, FARKAS’ Lemma, will be presented in Chapter 4. In this chapter we will derive FARKAS’ Lemma only for a special class of complementary pairs (L, L⊥) arising from directed graphs.