Plancherel Measures of Reductive Adelic Groups and Von Neumann Dimensions

TL;DR

This paper establishes a deep connection between the Plancherel measure of adelic groups and von Neumann algebra dimensions, providing a new perspective on harmonic analysis of reductive groups over number fields.

Contribution

It proves that for semisimple, simply connected groups, the Plancherel measure equals the von Neumann dimension over the group algebra, linking harmonic analysis and operator algebras.

Findings

Plancherel measure coincides with von Neumann dimension for certain groups.

The result applies to groups with Tamagawa measure.

Provides a new interpretation of harmonic analysis in terms of operator algebras.

Abstract

Given a number field $F$ and a reductive group $G$ over $F$, the unitary dual $\hat{G(\mathbb{A}_F)}$ of the adelic group $G(\mathbb{A}_F)$ and the Placherel measure $\nu_{G(\mathbb{A}_F)}$ on it can be determined by the Plancherel measure of its local groups $G(F_v)$. Given a subset $X\subset \hat{G(\mathbb{A}_F)}$ of finite Plancherel measure, let $H_X$ be the direct integral of the irreducible representations in $X$. Besides a $G(\mathbb{A}_F)$-module and a $G(F)$-module, $H_X$ is also a module over the group von Neumann algebra $\mathcal{L}(G(F))$, hence there is a canonical dimension $\dim_{\mathcal{L}(G(F))}H_X\in [0,\infty)$. It is proved that the Plancherel measure of $G(\mathbb{A}_F)$ coincides with the dimension over the algebra $\mathcal{L}(G(F))$: $\dim_{\mathcal{L}(G(F))}H_X=\nu_{G(\mathbb{A}_F)}(X)$, if $G$ is semisimple, simply connected and $G(\mathbb{A}_F)$ is equipped with the Tamagawa measure.