In the 1970's, George Mackey pointed out an analogy that exists betweentempered representations of semisimple Lie groups and unitary representationsof associated semidirect product Lie groups. More recently, Nigel Higsonrefined Mackey's analogy into a one-to-one correspondence for connected complexsemisimple groups, and in doing so obtained a novel verification of theBaum-Connes conjecture with trivial coefficients for such groups. Here weextend Higson's results to any Lie group with finitely many connectedcomponents whose connected component of the identity is complex semisimple. Ourmethods include Mackey's description of unitary representations of groupextensions involving projective unitary representations, as well as the notionof twisted crossed product $C^*$-algebra introduced independently by Green andDang Ngoc.
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机译：在1970年代，乔治·麦基（George Mackey）指出了一个类比，该类比存在于半简单李群的调和表示与相关的半直接产品李群的统一表示之间。最近，奈杰尔·希格森（Nigel Higson）将Mackey的类比提炼为连接的复杂半简单群的一对一对应关系，并以此方式获得了对鲍姆-康尼斯猜想的新颖验证，其中这些群具有微不足道的系数。在这里，我们将Higson的结果扩展到具有有限多个连接组件的李群，而这些组件的标识的连接组件是复半简单的。我们的方法包括Mackey对涉及投影unit表示的群扩展的unit表示的描述，以及Green和Dan Ngoc独立引入的扭曲交叉乘积$ C ^ * $-代数的概念。
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