Let $G$
be a split semisimple group over $\mathbb{Q}$
and let $P = N \rtimes M$
be a maximal parabolic subgroup of $G$
defined over $\mathbb{Q}$
. The Eisenstein series $E_P(s, \varphi)$
is an automorphic form on $G(\mathbb{Q})\G(\mathbb{A})$
built from a square-integrable automorphic form $\varphi$
on $M(\mathbb{Q})\M(\mathbb{A})^1$
and depends meromorphically on a complex spectral parameter $s \in \mathbb{C}$
. The poles of these Eisenstein series in the region $\text{Re}(s) > 0$
play a central role in the spectral decomposition of automorphic forms.
In the 1960s, Langlands showed that when $\varphi$
is a cuspform, the poles of $E_P(s, \varphi)$
are determined via $L$
-functions attached to $\varphi$
using
the adjoint representation of $\hat{M}$
on $\text{Lie}(\hat{N})$
, where $\hat{\cdot}$
stands for the Langlands dual group. No such structural result was known about the poles of $E_P(s, \varphi)$
when $\varphi$
is not a cuspform.
It has been known since the 1980s, at least for several important examples, that there is an Arthur parameter $\mathrm{SL}_2(\mathbb{C}) \to \hat{M}$
attached to a non-cuspidal $\varphi$
on $M(\mathbb{Q})\M(\mathbb{A})^1$
. The simplest example is when $\varphi = 1$
, where the Arthur parameter is the principal homomorphism
$\mathrm{SL}_2(\mathbb{C}) \to \hat{M}$
.
In this talk, we provide evidence that the poles of non-cuspidal Eisenstein series $E_P(s, \varphi)$
in the region $\text{Re}(s) > 0$
are related to the highest weights occurring in the decomposition of the $\mathrm{SL}_2(\mathbb{C})$
representation on $\text{Lie}(\hat{N})$
induced by the corresponding Arthur parameter, by determining the poles of the unramified degenerate Eisenstein series $E_P(s, \varphi = 1)$
using a straightforward global argument.