Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866914225345003520 |
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| author | Ali, Abid Carbone, Lisa Garrett, Paul |
| author_facet | Ali, Abid Carbone, Lisa Garrett, Paul |
| contents | Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $Γ$ be a non-uniform lattice in $G$. When $Γ$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $Γ\backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_02919 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields Ali, Abid Carbone, Lisa Garrett, Paul Number Theory Group Theory Let $G$ be an affine or hyperbolic rank 2 Kac--Moody group over a finite field $\mathbb F_q$. Let $X=X_{q+1}$ be the Tits building of $G$, the $(q+1)$--homogeneous tree, and let $Γ$ be a non-uniform lattice in $G$. When $Γ$ is a standard parabolic subgroup for the negative $BN$--pair, we define Eisenstein series on $Γ\backslash X$ and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on $G$. A crucial tool is a description of the vertices of $X$ in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building $X$ and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein's Continuation Principle. |
| title | Eisenstein series on arithmetic quotients of rank 2 Kac--Moody groups over finite fields |
| topic | Number Theory Group Theory |
| url | https://arxiv.org/abs/2108.02919 |