Meromorphic continuation of higher-rank Eisenstein series

Expanded version of a talk given in Tel Aviv, Israel, 22 March 2001. To trivialize the proof of meromorphic continuation of Eisenstein series, we want to use only standard facts, together with Bernstein’s formulation of a continuation principle. In particular, one should prove meromorphic continuation as a vector-valued function, from which one infers that the continuation is of moderate growth, as are any residues. To avoid inessential complications, we only consider automorphic forms on groups G = GL(n,R), left invariant under the full group Γ = GL(n,Z), right invariant under the corresponding orthogonal group K = O(n,R), and with trivial central character. A slightly more serious assumption is that the parabolic from which we ‘induce’ will be maximal (proper). This assumption is mostly a matter of convenience and simplicity. An undesirable but essential assumption (for the moment) is that the ‘data’ on the Levi component is cuspidal. For now, it does not seem that this can be dropped, although one may hope.