Peter-Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Peter, in the setting of a compact topological group G (Peter & Weyl 1927). The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur.

Contents

Matrix coefficients

A matrix coefficient of the group G is a complex-valued function φ on G given as the composition

\phi = L\circ \pi

where π : G → GL(V) is a finite-dimensional (continuous) group representation of G, and L is a linear functional on the vector space of endomorphisms of V, which contains GL(V) as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.

The first part of the Peter-Weyl theorem asserts (Bump 2004, §4.1; Knapp 1986, Theorem 1.12):

This first result resembles the Stone-Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization. In fact, if G is a matrix group, then the result follows easily from the Stone-Weierstrass theorem (Knapp 1986, p. 17). Conversely, it is a consequence of the subsequent conclusions of the theorem that any compact Lie group is locally isomorphic to a matrix group (Knapp 1986, Theorem 1.15).

A corollary of this result is that the matrix coefficients of G are dense in L2(G).

Decomposition of a unitary representation

The second part of the theorem gives the existence of a decomposition of a unitary representation of G into finite-dimensional representations. To state this part of the theorem we need first the idea of a unitary representation in a Hilbert space. A Hilbert space is a complete inner product space. A representation ρ of G on a Hilbert space H is a group homomorphism of G into the group GL(H) of bounded linear isomorphisms of H with itself with bounded inverses such that the map \scriptstyle{(g,v)\mapsto \rho(g)v} is a continuous function

G\times H\to H.

The representation ρ is unitary if ρ(g) is a unitary operator for all g ∈ G; to wit,

\langle \rho(g)v,\rho(g)w\rangle = \langle v,w\rangle

for all v, w ∈ H.

The second part of the Peter-Weyl theorem asserts (Knapp 1986, Theorem 1.14):

  • Let ρ be a unitary representation of a compact group G on a Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Decomposition of square-integrable functions

To state the third and final part of the theorem, there is a natural Hilbert space over G consisting of square-integrable functions, L2(G); this makes sense because Haar measure exists on G. Calling this Hilbert space H, the group G has a unitary representation ρ on H by acting on the left, via

ρ(h)f(g) = f(hg).

The final statement of the Peter-Weyl theorem (Knapp 1986, Theorem 1.14) gives an explicit orthonormal basis of L2(G). Roughly it asserts that the matrix coefficients for G, suitably renormalized, are an orthonormal basis of L2(G). In particular, L2(G) decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree (that is, the dimension of the underlying space of the representation).

More precisely, suppose that a representative π is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such π by Σ. Let \scriptstyle{u_{ij}^{(\pi)}} be the matrix coefficients of π in an orthonormal basis. Thus

u^{(\pi)}_{ij}(g) = \langle \pi(g)e_i, e_j\rangle.

for each g ∈ G. Finally, let d(π) be the degree of the representation π. The theorem now asserts that the functions

\left\{\sqrt{d^{(\pi)}}u^{(\pi)}_{ij}\mid\, \pi\in\Sigma,\,\, 1\le i,j\le d^{(\pi)}\right\}

is an orthonormal basis of L2(G).

Consequences

Structure of compact topological groups

From the theorem, one can deduce a significant general structure theorem. Let G be a compact topological group, which we assume Hausdorff. For any finite-dimensional G-invariant subspace V in L2(G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the Lie group GL(V). It follows by a basic theorem (of Élie Cartan) that the image of G is a Lie group also.

If we now take the limit (in the sense of category theory) over all such spaces V, we get a result about G - because G acts faithfully on L2(G). We can say that G is an inverse limit of Lie groups. It may of course not itself be a Lie group: it may for example be a profinite group.

References

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