Eisenstein series and string thresholds

We investigate the relevance of Eisenstein series for representing certain G(Z)-invariant string theory amplitudes which receive corrections from BPS states only. G(Z) may stand for any of the mapping class, T-duality and U-d uality groups Sl(d, Z), SO(d, d, Z) or Ed+1(d+1)(Z) respectively. Using G(Z )-invariant mass formulae, we construct invariant modular functions on the symmetric space K\G(R) of non-compact type, with K the maximal compact subg roup of G (R), that generalize the standard nonholomorphic Eisenstein serie s arising in harmonic analysis on the fundamental domain of the Poincare up per half-plane. Comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising i n one- and g-loop string amplitudes, we obtain a manifestly T-duality invar iant representation of the latter, conjecture their non-perturbative U-dual ity invariant extension, and analyze the resulting non-perturbative effects . This includes the R-4 and (RN4g-4)-N-4 couplings in toroidal compactifica tions of M-theory to any dimension D greater than or equal to 4 and D great er than or equal to 6 respectively.