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Number Theory Seminar

Title: Poles of non-cuspidal Eisenstein series
Speaker: Devadatta Hegde (Brandeis University, USA)
Date: 31 December 2024
Time: 11 AM
Venue: LH-1

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.


Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 31 Jan 2025