Some Combinatorics in the Cancellation of Poles of Eisenstein Series for $GL(n,\mathbb{A}_\mathbb{Q})$

TL;DR

This paper extends previous work on pole cancellations of Eisenstein series to more complex cases involving Speh representations on GL(n), using combinatorial analysis of Weyl group cosets.

Contribution

It generalizes the method for pole cancellation of Eisenstein series to include those induced from Speh representations on GL(n).

Findings

Describes combinatorics of Weyl group cosets for these Eisenstein series.

Provides a generalized framework for understanding pole cancellations.

Extends prior results to a broader class of automorphic forms.

Abstract

The cancellations of poles of degenerate Eisenstein series were studied by Hanzer and Mui\'{c}. This paper generalizes the method and the result to Eisenstein series constructed from inducing two Speh representations $\Delta(\tau,m)|\cdot|^{s_1}\otimes\Delta(\tau,n)|\cdot|^{s_2}$ for the group $GL(m+n,\mathbb{A}_\mathbb{Q})$ for self-dual cuspidal automorphic representation $\tau$ by describing the combinatorics of the relevant Weyl group coset.