Eisenstein series and decomposition theory over function fields

Abstract

The theory of Eisenstein series for SL2(IR) has been well-explored (cf. [11]). In [12], Langlands developed the theory of Eisenstein series in great generality, namely, for arbitrary reductive Lie groups. Harder in [5] studied the corresponding theory for simply connected Chevalley groups over function fields of one variable. The present paper is concerned with a situation which is more restricted than I"5] in the sense that only GL z is considered, and more general in the sense that the Eisenstein functions are supposed to be invariant only by an open compact subgroup. We prove that these Eisenstein series are rational functions, satisfy functional equations, etc., as expected (Sects. 2, 3, and 5). One of the main themes of this paper is to study the intertwining operators which arise from the constant Fourier coefficients of Eisenstein series (Sects. 3 and 6). These are rational operators. Since they occur in the functional equations for Eisenstein series, the degrees of their numerators are essential in computing the dimensions of the spaces of cusp forms for congruence subgroups. This will be shown in a subsequent paper [7] by Harder et al. On the other hand, the computations of these intertwining operators are inevitable if one wants to get hold of the "bad" terms occurring in the trace formula. Another main theme is discussed in Sect. 7. There we develop a decomposition theory on the space of automorphic functions (of some central quasi-character and invariant under a congruence subgroup) which are eigenfunctions of the Hecke operator T o with eigenvalue 2~ at some "good" place v. Namely, any such function is a direct sum of an Eisenstein series and a cusp form. If we interpret T~ as the "Beltrami operator" and view eigenfunctions of T~ with eigenvalue 2~ as "holomorphic functions" as explained in 1-15, 16], then this decomposition theory is an analogue of the classical decomposition theory introduced by Hecke (cf. [8, No. 24]). Consequently, the study of automorphic functions is reduced to the study of cusp forms. There is also a similar result from the representation-theoretic point