Entirety of cuspidal Eisenstein series on loop groups

Abstract

In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the finite-dimensional setting, where such series only converge absolutely in a right half plane (and have poles elsewhere coming from $L$-functions in their constant terms). Our result is the ${\Bbb Q}$-analog of a theorem of A.~Braverman and D.~Kazhdan from the function field setting, who previously showed the analogous Eisenstein series there are finite sums.