# Mackey Analogy of Complex Semisimple Groups

## Meeting Details

Abstract: In 1975, George Mackey proposed that there should be an analogy (perhaps a bijection) between unitary representations of semisimple groups $G$ and unitary representations of its associated semidirect product $K\ltimes V$ where $K$ is a maximal compact subgroup of $G$ and $V$ is a vector space. A full understanding of unitary representations (even subcategories of unitary representations) of $G$ has proven to be very difficult (in fact, it is still an open question). However, unitary representations on $K\ltimes X$ is very easy and so can be used to provide insight into representations on $G$. It turns out that there is indeed an analogy for the case of general complex semisimple groups, and for some real semisimple groups. To keep things simple, I will emphasize the example of $SL(2,\mathbb{C})$, and provide necessary background information, including definitions of irreducible, unitary representations, Iwasawa decomposition of groups, induced representations, and principal series, so that I may present the Mackey Analogy for complex semisimple groups. If there is time, I may present the difficulty that arises when considering the real case or present how I am approaching this problem using $C^*$-algebras.