Plancherel Measures of Reductive Adelic Groups and Von Neumann Dimensions

Abstract

Given a number field $F$ and a reductive group $G$ over $F$, the unitary dual $\widehat{G({\mathbb{A}}_F)}$ 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 $X\subset \widehat{G({\mathbb{A}}_F)}$ 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.

18 pages