# Classifying representation through extensions

Let $G$ be a group (you preferred type: finite, compact, ...):

Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism: $$\sigma : G \rightarrow N$$ by some irreducible representations of subgroups of $N$ and some projective representations $G/ kern(\sigma)$, where $kern(\sigma)$ is a so called type-1 subgroup (i.e. the von Neumann group algebra is a direct integral type $1$ factors), e.g. take a finite, abelian, compact or amenable group.

Q1: Is there a nice reference for a condensed treatment of these results?

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Re: Q1, until real answers appear.

There is a survey article: "Projective representations and the Mackey obstruction - a survey", Contemporary Mathematics, v. 449 (2008), pp. 345-378.

Besides this article, there are a lot of interesting tidbits in the above-mentioned volume. (Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey) One of the papers in here (the one by Kirillov) contains a simplified proof of the imprimitivity theorem in the Lie group case.

There is also an article by V.S. Varadarajan (posted on his website) about the work of Mackey, but the UCLA website seems to be misbehaving right now, so you'll have to check this later.

BTW: There is another closely related question this morning. Do you guys know each other?

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No, I guess this question was unrelated. –  Marc Palm Apr 2 '11 at 11:26