论文标题
还原性阿替型组和von Neumann尺寸的平心
Plancherel Measures of Reductive Adelic Groups and Von Neumann Dimensions
论文作者
论文摘要
给定一个数字字段$ f $和$ f $上的还原组$ g $，统一的双$ \ hat {g（\ mathbb {a} _f）} $的adelic group $ g（\ mathbb {a} a} _f）的$ g（\ mathbb {a} _f）$和Placherel Matues $ n placherel Matues $ν_{g（can in can plan）其本地组$ g（f_v）$的度量。给定一个子集$ x \ subset \ hat {g（\ mathbb {a} _f）} $的有限plancherel度量，令$ h_x $是$ x $中不可减少表示的直接组成。除了A $ G（\ Mathbb {a} _f）$ - 模块和a $ g（f）$ - 模块，$ h_x $也是组的模块，von neumann algebra $ \ mathcal $ \ mathcal {l}（l}（g（f）） $ \ dim _ {\ Mathcal {l}（g（f））} h_x \ in [0，\ infty）$。事实证明，$ g（\ mathbb {a} _f）$的plancherel度量与代数$ \ mathcal {l}（l}（g（f））$的尺寸重合。 $ \ dim _ {\ Mathcal {l}（g（f））} h_x =ν_{g（\ Mathbb {a} _f）}（x）$，如果$ g $是半imple，则简单地连接，$ g（\ mathbb {a} _f）$是the tamagagaagagaagagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaagaa。
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 $ν_{G(\mathbb{A}_F)}$ 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 $\mathcal{L}(G(F))$: $\dim_{\mathcal{L}(G(F))}H_X=ν_{G(\mathbb{A}_F)}(X)$, if $G$ is semisimple, simply connected and $G(\mathbb{A}_F)$ is equipped with the Tamagawa measure.
