Book Review: Spectral decomposition and Eisenstein series

Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition

We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of th...

Eisenstein Series*

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...

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-duality groups Sl(d, Z), SO(d, d, Z) or E d+1(d+1) (Z) respectively. Using G(Z)-invariant mass formulae, we construct invariant modular functions on the symmetric space K\G(...

Eisenstein Series and Their Applications

The simplest Eisenstein series

We explain some essential aspects of the simplest Eisenstein series for SL2(Z) on the upper half-plane H. There are many different proofs of meromorphic continuation and functional equation of the simplest Eisenstein series for Γ = SL2(Z). We will follow [Godement 1966a] rewriting of a Poisson summation argument that appeared in [Rankin 1939], if not earlier. This argument is the most elementar...