 Fix a Haar measure on "G"(A) and let "L"20("G"("K") \ "G"(A), ω) denote the Hilbert space of measurable complex-valued functions, "f", on "G"(A) satisfying
The vector space "L"20("G"("K") \ "G"(A), ω) is called the space of cusp forms with central character ω on "G"(A).A function appearing in such a space is called a cuspidal function.A cuspidal function generates a unitary representation of the group "G"(A) on the complex Hilbert space formula_5 generated by the right translates of "f".