Tempered vs generic automorphic functions and the canonical filtration on automorphic functions

TL;DR

This paper introduces a filtration on automorphic functions derived from the Langlands conjecture's spectral side, linking cohomological support to Hecke operator spectra in the function field setting.

Contribution

It proposes a new filtration on automorphic functions based on spectral and cohomological support, connecting geometric and analytic perspectives.

Findings

Defines a filtration on automorphic functions via spectral support.

Proposes conjectures linking cohomological and Hecke spectra.

Establishes a framework for understanding automorphic functions in the function field case.

Abstract

We introduce and study the filtration on the space of automorphic functions (in the everywhere unramified situation for the function field case) obtained by transferring the filtration on the spectral side of the classical Langlands conjecture, induced by coherent singular support. We propose a number of conjectures that tie this filtration (which, by design, arises from the notion of cohomological support) to a filtration on the space of C-valued automorphic functions that arises by considering the analytic spectrum of Hecke operators.