Residue distributions, iterated residues, and the spherical automorphic spectrum

2022/07/14 by Marcelo De Martino, De Martino, Marcelo, Volker Heiermann +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2207.06773

openalex publication_date 2022/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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,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 Λ_F contribute to the spectrum, while the singularities caused by the zeroes of Λ_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 L²(G(F)\backslash G(\mathbbA_F),ξ)^K_[T,O(1)] by a comparison of the iterated residues with the residue distributions of [HO1].

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