Inner cohomology of $GL_n$

TL;DR

This paper explicitly describes the inner cohomology of adelic locally symmetric spaces associated with $GL_n$, revealing vanishing results and a simple structure for noncuspidal parts in certain cases.

Contribution

It provides an explicit description of the inner cohomology for $GL_n$, including vanishing results and a characterization of noncuspidal components for prime $n$.

Findings

Inner cohomology vanishes for nonconstant sheaves at all degrees for prime $n$.

Noncuspidal quotient modules are trivial for $n=2,3$ in all degrees.

For $n \\geq 5$, noncuspidal parts are trivial except in finitely many degrees, where they relate to algebraic Hecke characters.

Abstract

We give an explicit description of the inner cohomology of an adelic locally symmetric space of a given level structure attached to the general linear group of prime rank $n$, with coefficients in a locally constant sheaf of complex vector spaces. We show that for all prime $n$ the inner cohomology vanishes in all degrees for nonconstant sheaves, otherwise the quotient module of the inner cohomology classes that are not cuspidal is trivial in all degrees for primes $n = 2,3$, and for all primes $n \geq 5$ it is trivial in all but finitely many degrees where it has a `simple' description in terms of algebraic Hecke characters.