{"id":2797,"date":"2011-09-06T17:42:43","date_gmt":"2011-09-06T15:42:43","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=2797"},"modified":"2024-02-20T13:40:36","modified_gmt":"2024-02-20T11:40:36","slug":"another-exercise-on-compact-groups","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2011\/09\/06\/another-exercise-on-compact-groups\/","title":{"rendered":"Another exercise on compact groups"},"content":{"rendered":"

While writing the chapters about compact groups in my notes<\/a>, I had a few times the impression that it would be useful to use the fact that there is a basis of conjugacy-invariant<\/i> 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.
\nThis happened again during the last week-end, with the difference that I thought of trying to prove this property using<\/i> 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<\/a>, 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. <\/p>\n

So I tried again to use representations, and indeed it works! Here’s the idea: the regular representation $latex \\rho$ of $latex G$ on $latex H=L^2(G)$ (with respect to Haar measure) is faithful, and gives a continuous injection
\n$latex G\\rightarrow U(H)$
\nwhere the unitary group $latex U(H)$ is given the strong operator topology<\/a> (if $latex G$ 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 $latex G$). What we gain from seeing $latex G$ 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.<\/p>\n

Indeed, from the definition of the strong topology on $latex U(H)$, any open neighborhood $latex U$ of 1 contains a finite intersection of sets of the type
\n$latex U_{f,\\epsilon}=\\{g\\in G\\,\\mid\\, \\|\\rho(g)f-f\\|\\text{\\textless} \\epsilon\\}$
\nwhere $latex f\\in H$ has norm 1 and $latex \\epsilon\\text{\\textgreater} 0$. It is then easy to see, using unitarity, that the conjugacy-invariant subset
\n$latex V_{f,\\epsilon}=\\bigcap_{x\\in G}{x^{-1}U_{f,\\epsilon}x}\\subset U_{f,\\epsilon}$
\nis equal to
\n$latex V_{f,\\epsilon}=\\{g\\in G\\,\\mid\\, \\|\\rho(g)\\varphi-\\varphi\\|\\text{\\textless} \\epsilon\\text{ for all }\\varphi\\in A_f\\}$
\nwhere
\n$latex A_f=\\{\\rho(x)f\\,\\mid\\, x\\in G\\}\\subset H$
\nis the orbit of $latex f$ under $latex G$. But the strong continuity of $latex \\rho$ implies that the orbit map $latex g\\mapsto \\rho(g)f$ is continuous for fixed $latex f$, so that $latex A_f$ is compact in $latex H$ (as image of a compact set under a continuous map; here $latex H$ has the usual norm topology).
\nIt is then a quick application of a standard compactness argument and splitting of epsilons to check that $latex V_{f,\\epsilon}$ is a neighborhood of 1 contained in $latex U_{f,\\epsilon}$, and intersecting finitely many of these, we see that $latex U$ contains indeed a conjugacy invariant neighborhood of 1…<\/p>\n

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 $latex G$ has a faithful finite-dimensional representation, this can be used instead of $latex \\rho$, but the purely topological argument becomes also much simpler.)<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading Another exercise on compact groups<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1086,2],"tags":[],"class_list":["post-2797","post","type-post","status-publish","format-standard","hentry","category-exercise","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2797","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=2797"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/2797\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=2797"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=2797"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=2797"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}