Speaker: Devadatta Hegde
Affiliation: Brandeis University
Title: Poles of non-cuspidal Eisenstein series
Date and Time: December 9, 2024, 11:30:00 Hours
Venue: AG-77
Asbtract: 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 $G(\mathbb{Q})\backslash G(\mathbb{A})$ built from a square-integrable automorphic form $\varphi$ on $M(\mathbb{Q})\backslash M(\mathbb{A})^{1}$ 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 $1960$s, 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 $\widehat{M}$ on $\text{Lie}(\widehat{N})$, where $\ \widehat{\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 $1980$s, at least for several important examples, that there is an Arthur parameter $SL_{2} \mathbb{C})\to\widehat{M}$ attached to a non-cuspidal $\varphi$ on $M(\mathbb{Q})\backslash M(\mathbb{A})^{1}$. The simplest example is when $\varphi=1$, where the Arthur parameter is the principal homomorphism $SL_{2}(\mathbb{C})\to\widehat{M}$. 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 determined by the highest weights occurring in the decomposition of the $SL_{2}(\mathbb{C})$ representation on $\text{Lie}(\widehat{N})$ 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.
Affiliation: Brandeis University
Title: Poles of non-cuspidal Eisenstein series
Date and Time: December 9, 2024, 11:30:00 Hours
Venue: AG-77
Asbtract: 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 $G(\mathbb{Q})\backslash G(\mathbb{A})$ built from a square-integrable automorphic form $\varphi$ on $M(\mathbb{Q})\backslash M(\mathbb{A})^{1}$ 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 $1960$s, 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 $\widehat{M}$ on $\text{Lie}(\widehat{N})$, where $\ \widehat{\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 $1980$s, at least for several important examples, that there is an Arthur parameter $SL_{2} \mathbb{C})\to\widehat{M}$ attached to a non-cuspidal $\varphi$ on $M(\mathbb{Q})\backslash M(\mathbb{A})^{1}$. The simplest example is when $\varphi=1$, where the Arthur parameter is the principal homomorphism $SL_{2}(\mathbb{C})\to\widehat{M}$. 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 determined by the highest weights occurring in the decomposition of the $SL_{2}(\mathbb{C})$ representation on $\text{Lie}(\widehat{N})$ 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.