Do Ngoc Diep

Abstract

          For a fixed percuspidal subgroup P = MAU and a fixed finite spectrum G-module V, the associated spectral sequence for the fibration U/{G ∩ U} ---> {^\circ}P --> M/{G_M} converges and the cohomology group H^*(K_M \{^\circ}P/{G ∩ P};V) is isomorphic to the direct sum of E₂ -terms. Every cohomology class of this type can be represented by an V-valued automorphic form. The restriction map send the cohomology classes at infinity of H^*(G;V), represented by singular values  of the associated Eisenstein series to the cohomology classes of the boundary ∂ ({\bar X}_cusp/G), compatible with its weight decomposition. All these together give us a decomposition of the cohomology of Langlands type discrete groups into the cuspidal and Eisenstein parts.