Multiplicity of Eisenstein series in cohomology and applications to $GSp_4$ and $G_2$

TL;DR

This paper develops a framework to determine how often automorphic representations appear in cohomology, applying it to Eisenstein series on GSp_4 and G_2, revealing new insights into their structure and representations.

Contribution

It introduces a general method for computing automorphic representation multiplicities in cohomology, with specific applications to GSp_4 and G_2, including new results on CAP representations.

Findings

Computed exact multiplicities in cohomology for GSp_4 and G_2

Provided new information on archimedean components of CAP representations for G_2

Enhanced understanding of automorphic representations in cohomological contexts

Abstract

We set up a general framework to compute the exact multiplicity with which certain automorphic representations appear in both the cuspidal and Eisenstein cohomology of locally symmetric spaces. We apply this machinery to Eisenstein series on $GSp_4$ and split $G_2$. In the case of $G_2$, we will also obtain new information about the archimedean components of certain CAP representations using Arthur's conjectures.