5 n - dimensional global correspondences of Langlands

The program of Langlands is studied here on the basis of: • new concepts of global class field theory related to the explicit construction of global class fields and of reciprocity laws; • the representations of the reductive algebraic groups GL(n) constituting the n-dimensional representations of the associated global Weil-Deligne groups; • a toroidal compactification of the conjugacy classes of these reductive algebraic groups whose analytic representations constitute the cuspidal representations of these groups GL(n) in the context of harmonic analysis. This leads us to build two types of n-dimensional global bilinear correspondences of Langlands by taking into account the irreducibility or the reducibility of the representations of the considered algebraic groups. The major outcome of this global approach is the generation of general algebraic symmetric structures, consisting of double symmetric towers of conjugacy class representatives of algebraic groups, so that the analytic toroidal representations of these conjugacy class representatives are the equivalence classes of the cuspidal representations of these algebraic groups.