Automorphic tensor products and cuspidal cohomology of the ${\rm GL}_4$

TL;DR

This paper investigates the contribution of certain automorphic representations of GL(4) to cuspidal cohomology, establishing lower bounds and exploring relationships between tensor products and symmetric powers of GL(2) representations.

Contribution

It provides an asymptotic lower bound for cuspidal automorphic representations of GL(4) from tensor products of GL(2) representations and proves non-equivalence of symmetric cube and tensor product representations.

Findings

Established lower bounds on cuspidal cohomology contributions.

Proved symmetric cube and tensor product representations are not equivalent.

Analyzed automorphic tensor products of GL(2) representations.

Abstract

In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\rm GL}_2$ and the automorphic tensor product of two representations of ${\rm GL}_2$ can not be equal (up to a twist by a character of ${\rm GL}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.