Inner cohomology of $G{L}_n$

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 $n\ge 5$ it is trivial in all but finitely many degrees where it has a `simple' description in terms of algebraic Hecke characters.

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