Let G be a connected simple linear algebraic group defined over a number field F. It is a fundamental problem in number theory and the theory of automorphic forms to describe the spectral decomposition of the unitary representation L~2(G(F)G(A)). By abstract results of functional analysis, such a unitary representation possesses an orthogonal decomposition L~2(G(F)G(A)) = L_d~2(G(F)G(A)) ? L_(cont)~2 (G(F)G(A)) into the direct sum of its discrete spectrum and its continuous spectrum. The theory of Eisenstein series reduces the description of L_(cont)~2 (G(F)G(A)) to that of the discrete spectrum of certain reductive subgroups of G, and thus the basic question is the understanding of the discrete spectrum L_d~2(G(F)G(A)). The discrete spectrum has a further orthogonal decomposition L_d~2(G(F)G(A)) = L_(cusp)~2 ? L_(res)~2 where L_(cusp)~2 is the subspace of cusp forms, and L_(res)~2 is the so-called residual spectrum. Let us write: L_(cusp)~2 = ?-(circumflex)_πm_(cusp)π·π and L_(res)~2 = ?-(circumflex)_πm_(res)π·π.
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