The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2023-03-01 DOI:10.1016/j.exmath.2022.09.001

L.C. Ruth

引用次数: 0

Abstract

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $P G L (2, F)$ and the formal dimension of a discrete series representation of $G L (2, F)$ . The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

查看原文本刊更多论文

格协体积与离散级数形式维数的乘积:p进GL(2)

设F为特征为0的非阿基米德局部域和不能被2整除的阶剩余域。我们展示了如何计算PGL(2,F)中无扭转晶格的协体积与GL(2,F)的离散级数表示的形式维数的乘积。协体积来自Ihara的一个定理，形式维数包含在Corwin、Moy和Sally的结果中。通过一个可以追溯到Atiyah的定理，以及一个自由群的第二个上同调群的平凡性，得到的乘积是一个离散级数表示的冯·诺依曼维，被认为是一个自由群因子的表示。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

期刊介绍： Our aim is to publish papers of interest to a wide mathematical audience. Our main interest is in expository articles that make high-level research results more widely accessible. In general, material submitted should be at least at the graduate level.Main articles must be written in such a way that a graduate-level research student interested in the topic of the paper can read them profitably. When the topic is quite specialized, or the main focus is a narrow research result, the paper is probably not appropriate for this journal. Most original research articles are not suitable for this journal, unless they have particularly broad appeal.Mathematical notes can be more focused than main articles. These should not simply be short research articles, but should address a mathematical question with reasonably broad appeal. Elementary solutions of elementary problems are typically not appropriate. Neither are overly technical papers, which should best be submitted to a specialized research journal.Clarity of exposition, accuracy of details and the relevance and interest of the subject matter will be the decisive factors in our acceptance of an article for publication. Submitted papers are subject to a quick overview before entering into a more detailed review process. All published papers have been refereed.