On the analytic properties of intertwining operators I: global normalizing factors

TL;DR

This paper establishes a uniform estimate for the logarithmic derivatives of global normalizing factors related to intertwining operators across various reductive groups, aiding spectral analysis in automorphic forms and trace formulas.

Contribution

It provides a novel uniform estimate for the $L^1$-norm of logarithmic derivatives of normalizing factors for intertwining operators on multiple reductive groups, crucial for spectral analysis.

Findings

Uniform estimate for $L^1$-norm of logarithmic derivatives

Applicable to inner forms of $GL(n)$, classical groups, and $G_2$

Facilitates analysis of Arthur's trace formula and limit multiplicity

Abstract

We provide a uniform estimate for the $L^1$-norm (over any interval of bounded length) of the logarithmic derivatives of global normalizing factors associated to intertwining operators for the following reductive groups over number fields: inner forms of $GL(n)$; quasi-split classical groups and their similitude groups; the exceptional group $G_2$. This estimate is a key ingredient in the analysis of the spectral side of Arthur's trace formula. In particular, it is applicable to the limit multiplicity problem studied by the authors in earlier papers.