On the Functional Equations Satisfied by Eisenstein Series

ly seen, the main problem of the theory of Eisenstein series is to analyze thespace L(ξ) or the spaces L({P}, ξ) in terms of the cusp forms on the various M . Thisanalysis is carried out—in principle—in the text. However, one can be satisfied with amore perspicuous statement if one is content to analyze L(ξ) in terms of the representationsoccurring discretely in the spaces of automorphic forms on the groups M .It is clear thatL({P}, ξ) =∫ ⊕ D0L({P}, ζ) |dζ|.Let L(G, {P}, ζ) be the closure of the sum of irreducible invariant subspaces of L({P}, ζ)and letL({G}, {P}, ξ) = L(G, {P}, ξ) =∫ ⊕ D0L(G, {P}, ζ) |dζ|.We write {P} {P1} if there is a P ∈ {P} and a P1 ∈ {P1} with P ⊇ P1. We shallconstruct a finer decomposition(5)L({P1}, ξ) =⊕ {P} {P1}L({P}, {P1}, ξ). If P ∈ {P} let p =p({P1}) be the set of classes of associate parabolic subgroups P1(M) ofM of the formP1(M) =M ∩ P1with P1 ∈ {P1} and P1 ⊆ P . The space L({P}, {P1}, ξ) will be isomorphic to a subspace of(6)⊕