Automorphic forms constructed from Whittaker vectors

Let G be a semi-simple Lie group of split rank 1 and Γ a discrete subgroup of G of cofinite volume. If P is a percuspidal parabolic of G with unipotent radical N and if χ is a non-trivial unitary character of N such that χ ( Γ ∩ N ) = 1 then a meromorphic family of functions M( v ) on gG / G that satisfy all of the conditions in the definition of automorphic form except for the condition of moderate growth is constructed. It is shown that the principal part of M( v ) at a pole v 0 with Re v 0 ⩾ 0 is square integrable and that “essentially” all square integrable automorphic forms with non-zero χ-Fourier coefficient can be constructed using the principal parts of the M -series. For square integrable automorphic forms that are fixed under a maximal compact subgroup the proviso “essentially” can be dropped. The Fourier coefficients of the M -series are computed. A specific term in the χ-Fourier coefficient is shown to determine the structure of the singularities of the M -series. This term is related to Selberg's “Kloosterman-Zeta function.” A functional equation for the M -series is derived. For the case of SL (2, R) the results are made more explicit and a complete family of square integrable automorphic forms is constructed. Also the paper introduces the conjecture that for semi-simple Lie groups of split rank > 1 and irreducible Γ the condition of moderate growth in the definition of automorphic form is redundant. Evidence for this conjecture is given for SO ( n , 1) over a number field.