Residue distributions, iterated residues, and the spherical automorphic spectrum

TL;DR

This paper analyzes the spectral contributions of residues in the context of automorphic forms on split reductive groups, showing that only certain singularities of intertwining operators influence the spectrum, with implications for understanding automorphic spectra.

Contribution

It demonstrates that only poles of the completed Dedekind zeta function contribute to the automorphic spectrum, clarifying the role of singularities in residue computations.

Findings

Only singularities from poles of  contribute to the spectrum.

Zeroes of  do not affect the iterated residues.

Residue distributions can be explicitly related to spectral measures.

Abstract

Let $G$ be a split reductive group over a number field $F$. We consider the computation of the inner product of two $K$-spherical pseudo Eisenstein series of $G$ supported in $[T,\mathcal{O}(1)]$ by means of residues, following a classical approach initiated by Langlands. We show that only the singularities of the intertwining operators due to the poles of the completed Dedekind zeta function $\Lambda_F$ contribute to the spectrum, while the singularities caused by the zeroes of $\Lambda_F$ do not contribute to any of the iterated residues which arise as a result of the necessary contour shifts. In the companion paper [DMHO] we use this result to explicitly determine the spectral measure of $L^2(G(F)\backslash G(\mathbb{A}_F),\xi)^K_{[T,\mathcal{O}(1)]}$ by a comparison of the iterated residues with the residue distributions of [HO1].