Automorphic forms constructed from Whittaker vectors

Abstract

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.