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Let be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then has a number of unitary representations, i.e. continuous homomorphisms to the group of unitary operators on a Hilbert space , equipped with the strong operator topology. In particular, one has the left-regular representation , where we equip with its normalised Haar measure (and the Borel -algebra) to form the Hilbert space , and is the translation operation
We call two unitary representations and isomorphic if one has for some unitary transformation , in which case we write .
Given two unitary representations and , one can form their direct sum in the obvious manner: . Conversely, if a unitary representation has a closed invariant subspace of (thus for all ), then the orthogonal complement is also invariant, leading to a decomposition of into the subrepresentations , . Accordingly, we will call a unitary representation irreducible if is nontrivial (i.e. ) and there are no nontrivial invariant subspaces (i.e. no invariant subspaces other than and ); the irreducible representations play a role in the subject analogous to those of prime numbers in multiplicative number theory. By the principle of infinite descent, every finite-dimensional unitary representation is then expressible (perhaps non-uniquely) as the direct sum of irreducible representations.
The Peter-Weyl theorem asserts, among other things, that the same claim is true for the regular representation:
Theorem 1 (Peter-Weyl theorem) Let be a compact group. Then the regular representation is isomorphic to the direct sum of irreducible representations. In fact, one has , where is an enumeration of the irreducible finite-dimensional unitary representations of (up to isomorphism). (It is not difficult to see that such an enumeration exists.)
In the case when is abelian, the Peter-Weyl theorem is a consequence of the Plancherel theorem; in that case, the irreducible representations are all one dimensional, and are thus indexed by the space of characters (i.e. continuous homomorphisms into the unit circle ), known as the Pontryagin dual of . (See for instance my lecture notes on the Fourier transform.) Conversely, the Peter-Weyl theorem can be used to deduce the Plancherel theorem for compact groups, as well as other basic results in Fourier analysis on these groups, such as the Fourier inversion formula.
Because the regular representation is faithful (i.e. injective), a corollary of the Peter-Weyl theorem (and a classical theorem of Cartan) is that every compact group can be expressed as the inverse limit of Lie groups, leading to a solution to Hilbert’s fifth problem in the compact case. Furthermore, the compact case is then an important building block in the more general theory surrounding Hilbert’s fifth problem, and in particular a result of Yamabe that any locally compact group contains an open subgroup that is the inverse limit of Lie groups.
I’ve recently become interested in the theory around Hilbert’s fifth problem, due to the existence of a correspondence principle between locally compact groups and approximate groups, which play a fundamental role in arithmetic combinatorics. I hope to elaborate upon this correspondence in a subsequent post, but I will mention that versions of this principle play a crucial role in Gromov’s proof of his theorem on groups of polynomial growth (discussed previously on this blog), and in a more recent paper of Hrushovski on approximate groups (also discussed previously). It is also analogous in many ways to the more well-known Furstenberg correspondence principle between ergodic theory and combinatorics (also discussed previously).
Because of the above motivation, I have decided to write some notes on how the Peter-Weyl theorem is proven. This is utterly standard stuff in abstract harmonic analysis; these notes are primarily for my own benefit, but perhaps they may be of interest to some readers also.