A simple construction of the automorphic residual spectrum

Abstract

We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is nonzero and square-integrable. Our proof follows the philosophy of Kazhdan and Okounkov. We give a geometric interpretation of Langlands' square-integrability criterion in this setting and, using the equivariant integration formula, prove that the regularization satisfies this criterion. As an immediate consequence, we obtain a simple and uniform proof of Arthur's unitarity conjecture, without case-by-case analysis or machine computation.