 Here the action of "g" ∈ "G"(A) on formula_6 is given by

The space of cusp forms with central character ω decomposes into a direct sum of Hilbert spaces
where the sum is over irreducible subrepresentations of "L"20("G"("K") \ "G"(A), ω) and the "m" are positive integers (i.e.each irreducible subrepresentation occurs with "finite" multiplicity).A cuspidal representation of "G"("A") is such a subrepresentation (, "V") for some "ω".