Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition

TL;DR

This paper determines the unipotent orbits associated with degenerate Eisenstein series on general linear groups, confirming a conjecture and showing all unipotent orbits can be realized by specific automorphic representations.

Contribution

It introduces the descending decomposition, a new combinatorial tool for analyzing Weyl group elements, and confirms a conjecture about unipotent orbits in automorphic forms.

Findings

Confirmed Ginzburg's conjecture on unipotent orbits

Showed all unipotent orbits occur in automorphic representations

Introduced the descending decomposition method

Abstract

We determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups. This confirms a conjecture of David Ginzburg. This also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The key ingredient is a root-theoretic result. To prove it, we introduce the notion of the descending decomposition, which expresses every Weyl group element as a product of simple reflections in a certain way. It is suitable for induction and allows us to translate the question into a combinatorial statement.