While writing the chapters about compact groups in my notes, I had a few times the impression that it would be useful to use the fact that there is a basis of conjugacy-invariant neighborhoods of 1 in such a group. This thought would then fork in two subthreads, one in which I noticed that I didn’t really see how to prove this, and the second in which I realized I didn’t need it anyway.
This happened again during the last week-end, with the difference that I thought of trying to prove this property using representation theory. I failed at first, and finally tried to look it up online to make sure the property was actually true. Indeed, it is, and I found this book, which has three proofs (of a slightly more general statement). But I have to admit to having an inferiority complex with respect to general topology: any argument that goes on for more than a few lines without being transparent tends to make me uneasy and confuse me, and this is what happened with these arguments.
So I tried again to use representations, and indeed it works! Here’s the idea: the regular representation of on (with respect to Haar measure) is faithful, and gives a continuous injection
where the unitary group is given the strong operator topology (if is infinite, it is not continuous for the operator norm topology). Hence this map is a homeomorphism onto its image (we use here the compactness of ). What we gain from seeing in this way as a subgroup of the unitary group of a (typically) infinite-dimensional space, and with a “strange” topology to boot, is a concrete description of a basis of open neighborhoods of 1, which is open to further manipulations.
Indeed, from the definition of the strong topology on , any open neighborhood of 1 contains a finite intersection of sets of the type
where has norm 1 and . It is then easy to see, using unitarity, that the conjugacy-invariant subset
is equal to
is the orbit of under . But the strong continuity of implies that the orbit map is continuous for fixed , so that is compact in (as image of a compact set under a continuous map; here has the usual norm topology).
It is then a quick application of a standard compactness argument and splitting of epsilons to check that is a neighborhood of 1 contained in , and intersecting finitely many of these, we see that contains indeed a conjugacy invariant neighborhood of 1…
I really can’t say that it is simpler than the purely topological argument (mostly because one needs to know about Haar measure!), but I find this rather nice as an exercise. It illustrates how representation theory can be useful to study a general compact group, at a very basic level, and also shows that it may be useful to embed (or project, with the orbit map) a compact group into complicated-looking infinite-dimensional beasts… (Of course, if has a faithful finite-dimensional representation, this can be used instead of , but the purely topological argument becomes also much simpler.)