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Through differentiable cohomology are induced Fréchet G - modules that construct the irreducible G - modules of infinite dimension, whose cohomology of infinite dimension representations is applicable to the Langlands classification obtaining the corresponding unitary representations. The obtained extensions are developed with u-cohomology and intertwining operators of induced representations applicable to certain Langlands data and through sheaves of holomorphic bundles. After are realized applications in the geometrical Langlands program to field theory and their ramifications in derived geometry. In this book are included several research exercises and research projects that can be developed in postgraduate courses and PhD dissertations.