Fourier Coefficients for Degenerate Eisenstein Series and the Descending Decomposition

We prove a Lie-theoretic result for type A root systems. This allows us to determine the unipotent orbits attached to degenerate Eisenstein series on general linear groups, confirming a conjecture of David Ginzburg. In particular, this also shows that any unipotent orbit of general linear groups does occur as the unipotent orbit attached to a specific automorphic representation. The proof of the Lie-theoretic result relies on 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.