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.
— 1. Proof of the Peter-Weyl theorem —
Throughout these notes, is a fixed compact group.
Let and be unitary representations. An (linear) equivariant map is defined to be a continuous linear transformation such that for all .
A fundamental fact in representation theory, known as Schur’s lemma, asserts (roughly speaking) that equivariant maps cannot mix irreducible representations together unless they are isomorphic. More precisely:
Lemma 2 (Schur’s lemma for unitary representations) Suppose that and are irreducible unitary representations, and let be an equivariant map. Then is either the zero transformation, or a constant multiple of an isomorphism. In particular, if , then there are no non-trivial equivariant maps between and .
Proof: The adjoint map of the equivariant map is also equivariant, and thus so is . As is also a bounded self-adjoint operator, we can apply the spectral theorem to it. Observe that any closed invariant subspace of is -invariant, and is thus either or . By the spectral theorem, this forces to be a constant multiple of the identity. Similarly for . This forces to either be zero or a constant multiple of a unitary map, and the claim follows. (Thanks to Frederick Goodman for this proof.)
Schur’s lemma has many foundational applications in the subject. For instance, we have the following generalisation of the well-known fact that eigenvectors of a unitary operator with distinct eigenvalues are necessarily orthogonal:
Proof: Apply Schur’s lemma to the orthogonal projection from to .
Another application shows that finite-dimensional linear representations can be canonically identified (up to constants) with finite-dimensional unitary representations:
Corollary 4 Let be a linear representation on a finite-dimensional space . Then there exists a Hermitian inner product on that makes this representation unitary. Furthermore, if is irreducible, then this inner product is unique up to constants.
Proof: To show existence of the Hermitian inner product that unitarises , take an arbitrary Hermitian inner product and then form the average
(this is the “Weyl averaging trick”, which crucially exploits compactness of ). Then one easily checks (using the fact that is finite dimensional and thus locally compact) that is also Hermitian, and that is unitary with respect to this inner product, as desired. (This part of the argument does not use finite dimensionality.)
To show uniqueness up to constants, assume that one has two such inner products , on , and apply Schur’s lemma to the identity map between the two Hilbert spaces and . (Here, finite dimensionality is used to establish
A third application of Schur’s lemma allows us to express the trace of a linear operator as an average:
where is the identity operator.
Proof: The right-hand side is equivariant, and hence by Schur’s lemma is a multiple of the identity. Taking traces, we see that the right-hand side also has the same trace as . The claim follows.
Let us now consider the irreducible subrepresentations of the left-regular representation . From Corollary 3, we know that those subrepresentations coming from different isomorphism classes in are orthogonal, so we now focus attention on those subrepresentations coming from a single class . Define the -isotypic component of the regular representation to be the finite-dimensional subspace of spanned by the functions of the form
where are arbitrary vectors in . This is clearly a left-invariant subspace of (in fact, it is bi-invariant, a point which we will return to later), and thus induces a subrepresentation of the left-regular representation. In fact, it captures precisely all the subrepresentations of the left-regular representation that are isomorphic to :
Proposition 6 Let . Then every irreducible subrepresentation of the left-regular representation that is isomorphic to is a subrepresentation of . Conversely, is isomorphic to the direct sum of copies of . (In particular, has dimension ).
Proof: Let be a subrepresentation of the left-regular representation that is isomorphic to . Thus, we have an equivariant isometry whose image is ; it has an adjoint .
Let and . The convolution
can be re-arranged as
In particular, we see that for every . Letting be a sequence (or net) of approximations to the identity, we conclude that as well, and so , which is the first claim.
To prove the converse claim, write , and let be an orthonormal basis for . Observe that we may then decompose as the direct sum of the spaces
for . The claim follows.
From Corollary 3, the -isotypic components for are pairwise orthogonal, and so we can form the direct sum , which is an invariant subspace of that contains all the finite-dimensional irreducible subrepresentations (and hence also all the finite-dimensional representations, period). The essence of the Peter-Weyl theorem is then the assertion that this direct sum in fact occupies all of :
Proposition 7 We have .
Proof: Suppose this is not the case. Taking orthogonal complements, we conclude that there exists a non-trivial which is orthogonal to all , and is in particular orthogonal to all finite-dimensional subrepresentations of .
Now let be an arbitrary self-adjoint kernel, thus for all . The convolution operator is then a self-adjoint Hilbert-Schmidt operator and is thus compact. (Here, we have crucially used the compactness of .) By the spectral theorem, the cokernel of this operator then splits as the direct sum of finite-dimensional eigenspaces. As is equivariant, all these eigenspaces are invariant, and thus orthogonal to ; thus must lie in the kernel of , and thus vanishes for all self-adjoint . Using a sequence (or net) of approximations to the identity, we conclude that vanishes also, a contradiction.
— 2. Nonabelian Fourier analysis —
Given , the space of linear transformations from to is a finite-dimensional Hilbert space, with the Hilbert-Schmidt inner product ; it has a unitary action of as defined by . For any , the function can be easily seen to lie in , giving rise to a map . It is easy to see that this map is equivariant.
Proof: As and are finite-dimensional spaces with the same dimension , it suffices to show that this map is an isometry, thus we need to show that
for all . By bilinearity, we may reduce to the case when are rank one operators
for some , where is the dual vector to , and similarly for . Then we have
The latter expression can be rewritten as
Applying Fubini’s theorem, followed by Corollary 5, this simplifies to
which simplifies to , and the claim follows.
As a corollary of the above proposition, the orthogonal projection of a function to can be expressed as
the Fourier coefficient of at , thus the projection of to is the function
which has an norm of . From the Peter-Weyl theorem we thus obtain the Fourier inversion formula
and the Plancherel identity
where the dilation of a Hilbert space is formed by using the inner product . This is an isomorphism not only of Hilbert spaces, but of the left-action of . Indeed, it is an isomorphism of the bi-action of on both the left and right of both and , defined by
It is easy to see that each of the are irreducible with respect to the action. Indeed, first observe from Proposition 8 that is surjective, and thus must span all of . Thus, any bi-invariant subspace of must also be invariant with respect to left and right multiplication by arbitrary elements of , and in particular by rank one operators; from this one easily sees that there are no non-trivial bi-invariant subspaces. Thus we can view the Peter-Weyl theorem as also describing the irreducible decomposition of into -irreducible components.
Remark 1 In view of (1), it is natural to view as being the “spectrum” of , with each “frequency” occuring with “multiplicity” .
In the abelian case, any eigenspace of one unitary operator is automatically an invariant subspace of all other , which quickly implies (from the spectral theorem) that all irreducible finite-dimensional unitary representations must be one-dimensional, in which case we see that the above formulae collapse to the usual Fourier inversion and Plancherel theorems for compact abelian groups.
In the case of a finite group , we can take dimensions in (1) to obtain the identity
In the finite abelian case, we see in particular that and have the same cardinality.
Direct computation also shows other basic Fourier identities, such as the convolution identity
for , thus partially diagonalising convolution into multiplication of linear operators on finite-dimensional vector spaces . (Of course, one cannot expect complete diagonalisation in the non-abelian case, since convolution would then also be non-abelian, whereas diagonalised operators must always commute with each other.)
Call a function a class function if it is conjugation-invariant, thus for all . It is easy to see that this is equivalent to each of the Fourier coefficients also being conjugation-invariant: . By Lemma 5, this is in turn equivalent to being equal to a multiple of the identity:
where the character of the representation is given by the formula
The Plancherel identity then simplifies to
thus the form an orthonormal basis for the space of class functions. Analogously to (1), we have
(In particular, in the case of finite groups , has the same cardinality as the space of conjugacy classes of .)
Characters are a fundamentally important tool in analysing finite-dimensional representations of that are not necessarily irreducible; indeed, if decomposes into irreducibles as , then the character then similarly splits as
and so the multiplicities of each component in can be given by the formula
In particular, these multiplicities are unique: all decompositions of into irreducibles have the same multiplicities.
Remark 2 Representation theory becomes much more complicated once one leaves the compact case; convolution operators are no longer compact, and can now admit continuous spectrum in addition to pure point spectrum. Furthermore, even when one has pure point spectrum, the eigenspaces can now be infinite dimensional. Thus, one must now grapple with infinite-dimensional irreducible representations, as well as continuous combinations of representations that cannot be readily resolved into irreducible components. Nevertheless, in the important case of locally compact groups, it is still the case that there are “enough” irreducible unitary representations to recover a significant portion of the above theory. The fundamental theorem here is the Gelfand-Raikov theorem, which asserts that given any non-trivial group element in a locally compact group, there exists a irreducible unitary representation (possibly infinite-dimensional) on which acts non-trivially. Very roughly speaking, this theorem is first proven by observing that acts non-trivially on the regular representation, which (by the Gelfand-Naimark-Segal (GNS) construction) gives a state on the *-algebra of measures on that distinguishes the Dirac mass at from the Dirac mass from the origin. Applying the Krein-Milman theorem, one then finds an extreme state with this property; applying the GNS construction, one then obtains the desired irreducible representation.