Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Plancherel Measures of Reductive Adelic Groups and Von Neumann
  Dimensions   We first point out a connection between the Plancherel measure of a general
unimodular group of type I and the von Neumann dimensions of its lattices.
Given a reductive algebraic group $G$ over a number field $F$, we describe the
Plancherel measure $\nu_{G(\mathbb{A}_F)}$ on the unitary dual
$\widehat{G(\mathbb{A}_F)}$ of the adelic group $G(\mathbb{A}_F)$. For any
subset $X$ of $\widehat{G(\mathbb{A}_F)}$ with a finite measure, there is
naturally a representation $H_X$ of $G(\mathbb{A}_F)$ given by the direct
integral of irreducible ones in $X$. Here we consider the dimension of this
module over the von Neumann algebra $\mathcal{L}(G(F))$ of the discrete group
$G(F)$. If $G$ is semi-simple and simply-connected, we prove
$\dim_{\mathcal{L}(G(F))}(H_X)=\nu_{G(\mathbb{A}_F)}(X)$, once we take the
canonical Tamagawa measure on $G(\mathbb{A}_F)$.

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The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
3
The categories of this paper are: physics