Peter-Weyl Theorem

From Fourier to Compact Groups

Fourier analysis on the circle rests on a single structural fact. The exponentials \(e^{in\theta}\) form an orthonormal basis of the square-integrable functions on \(S^1\). Every periodic signal decomposes into these elementary frequencies, and the decomposition is orthogonal, complete, and norm-preserving. The exponentials are not an arbitrary choice. The circle is a group under addition of angles, and each \(e^{in\theta}\) is a one-dimensional representation of that group, since the map \(\theta \mapsto e^{in\theta}\) sends the group operation to multiplication of complex numbers. Classical Fourier analysis is, in this light, the harmonic analysis of the abelian group \(S^1\), and the basis functions are exactly its irreducible representations.

The aim of this page is to identify the analogue of this basis for a general compact group, where the group need not be abelian and its irreducible representations need not be one-dimensional. The replacement for the exponentials will be the matrix entries of the irreducible representations, and the replacement for the orthonormal basis of \(L^2(S^1)\) will be a basis of \(L^2(K)\) assembled from those entries. The central results are the orthogonality of characters and the completeness theorem of Peter and Weyl, which together promote the circle's Fourier expansion to a noncommutative harmonic analysis valid on every compact matrix Lie group.

Throughout, \(K\) denotes a compact matrix Lie group, that is, a compact closed subgroup of \(GL(n, \mathbb{C})\). Such a group is by Cartan's closed subgroup theorem a smooth embedded submanifold of \(GL(n, \mathbb{C})\) whose group operations are smooth, hence a Lie group in the sense of manifold theory, and every construction of that theory applies to it. Being a subset of \(M_n(\mathbb{C})\) it is metrizable, so it is also compact Hausdorff.

Integration over \(K\) is taken against the normalized Haar integral, written \(\int_K f(x)\, dx\). It is the integral against the Haar volume form \(\omega_K\), the unique left-invariant top-degree form on \(K\) of total integral \(1\). The hypothesis of that result, a left-invariant orientation, is met here. The group admits a left-invariant global frame, and the top-degree form taking the value \(1\) on that frame at every point is left-invariant and nowhere vanishing, so it determines an orientation, left-invariant because the form is. Normalization means \(\int_K 1\, dx = 1\), and left invariance means that \(\int_K f(yx)\, dx = \int_K f(x)\, dx\) for every fixed \(y \in K\).

The integral is right invariant as well. Write \(R_y\) for right translation by \(y \in K\), a diffeomorphism of \(K\) with inverse \(R_{y^{-1}}\). Since \(R_y \circ L_g = L_g \circ R_y\) and \(L_g^{*}\omega_K = \omega_K\), the pullback \(R_y^{*}\omega_K\) is again left-invariant. Top-degree alternating forms on a \(k\)-dimensional space form a one-dimensional space, here with \(k = \dim K\), so \(R_y^{*}\omega_K = c\,\omega_K\) for some function \(c\) on \(K\), and pulling back by \(L_g\) turns the last identity into \((c \circ L_g)\,\omega_K = c\,\omega_K\), forcing \(c\) to be constant. That constant is nonzero because \(\omega_K\) vanishes nowhere, so \(R_y\) is orientation-preserving when \(c \gt 0\) and orientation-reversing when \(c \lt 0\), and part (d) of the properties of the integral gives \[ c = \int_K R_y^{*}\omega_K = \operatorname{sgn}(c) \int_K \omega_K = \operatorname{sgn}(c), \] so \(c = \pm 1\). Applying the same part to \(f\omega_K\), whose pullback is \(c\,(f \circ R_y)\,\omega_K\), \[ c \int_K f(xy)\, dx = \int_K R_y^{*}(f\omega_K) = \operatorname{sgn}(c) \int_K f(x)\, dx = c \int_K f(x)\, dx, \] and dividing by \(c\) leaves \(\int_K f(xy)\, dx = \int_K f(x)\, dx\). Conjugation \(x \mapsto yxy^{-1}\) is a left translation followed by a right translation, so the integral is invariant under it too. We use these invariances repeatedly and without further comment.

The Haar integral also determines a measure, which is what gives meaning to \(L^2(K)\). The integral is positive, since for continuous \(f \geq 0\) not identically zero the form \(f\omega_K\) is everywhere non-negative and strictly positive somewhere, so part (c) of the properties of the integral makes \(\int_K f\, dx \gt 0\). Boundedness follows by applying positivity to \(\sup_K |f| - \operatorname{Re}(\alpha f)\), with \(\alpha\) a unimodular scalar chosen so that \(\alpha \int_K f\, dx = \left| \int_K f\, dx \right|\), which yields \(\left| \int_K f\, dx \right| \leq \sup_K |f|\). Because \(K\) is compact, \(C_0(K) = C(K)\), so the representation of positive functionals by measures presents the Haar integral as integration against a unique regular Borel measure \(\mu_K\) on \(K\), with \(\mu_K(K) = \int_K 1\, dx = 1\). We write \(L^2(K)\) for the \(L^2\) space of this measure and keep the notation \(\int_K f(x)\, dx\) for integration against it. That \(C(K)\) is dense in \(L^2(K)\) is a standard fact about regular Borel measures on compact spaces, which we take as given.

Characters and Class Functions

The objects that play the role of frequencies are built from traces of representations. We work throughout with finite-dimensional representations of \(K\) on complex vector spaces. Recall that every such representation of a compact group may be taken to be unitary with respect to an inner product obtained by averaging an arbitrary one over the group.

The character discards all information about \(\Pi\) except what survives the trace, yet it retains enough to detect the representation completely. Two features make it the right object. First, it is unchanged under isomorphism of representations, because the trace is invariant under conjugation. If \(\Pi' = S\Pi S^{-1}\) then \(\operatorname{tr}(\Pi'(x)) = \operatorname{tr}(S\Pi(x)S^{-1}) = \operatorname{tr}(\Pi(x))\). Second, and for the same reason, the character is constant on conjugacy classes of \(K\). This second property is fundamental enough to name.

Every character is a class function. The verification is immediate from the multiplicativity of \(\Pi\) and the cyclic invariance of the trace: for any \(x, y \in K\), \[ \chi_\Pi(yxy^{-1}) = \operatorname{tr}\big(\Pi(y)\Pi(x)\Pi(y)^{-1}\big) = \operatorname{tr}\big(\Pi(x)\big) = \chi_\Pi(x), \] where the middle equality uses \(\operatorname{tr}(ABA^{-1}) = \operatorname{tr}(B)\). The continuous class functions form a subspace of \(L^2(K)\), and the orthogonality theorem will show that the characters of the irreducible representations sit inside this subspace as an orthonormal system. The Peter-Weyl theorem will then show that this system is complete. It is to the space of class functions what the exponentials \(e^{in\theta}\) are to \(L^2(S^1)\). On the circle the two pictures coincide, because every element is its own conjugacy class and so every function is a class function. That degeneration is one we return to once the general theorem is in hand.

Orthonormality of Characters

We equip \(L^2(K)\) with the inner product \[ \langle f, g \rangle = \int_K \overline{f(x)}\, g(x)\, dx, \] linear in the second argument. The orthogonality theorem states that the characters of the irreducible representations form an orthonormal set with respect to this product. Distinct irreducibles have orthogonal characters, and each irreducible character has unit norm. This is the exact analogue of the relation \(\langle e^{im\theta}, e^{in\theta} \rangle = \delta_{mn}\) underlying ordinary Fourier series, now without any assumption of commutativity.

The proof rests on two preliminary facts. The first turns the Haar integral of a representation into a concrete geometric object, an orthogonal projection. The second computes the trace of a tensor product. After establishing these we assemble the orthogonality relation by recognizing the integral above as the dimension of a space of intertwining maps, which Schur's lemma then evaluates.

Averaging as a Projection

For any finite-dimensional representation \((\Pi, V)\) of \(K\), write \[ V^K = \{\, v \in V : \Pi(x)v = v \text{ for all } x \in K \,\} \] for the subspace of vectors fixed by every group element. Averaging the operators \(\Pi(x)\) over the group produces a projection onto this subspace.

When \(\Pi\) is irreducible and is not the trivial representation, the only vector fixed by all of \(K\) is the origin. The subspace \(V^K\) is invariant, it is not all of \(V\) because \(\Pi\) is not trivial, and irreducibility leaves only \(V^K = \{0\}\). The proposition then records that \(\int_K \Pi(x)\, dx = 0\).

The Trace of a Tensor Product

The orthogonality integral pairs two representations, and the product of their traces will be read as the trace of a single representation on the tensor product space. The underlying linear-algebra identity is the following, an immediate consequence of the characteristic property of the tensor product, which guarantees that \(A \otimes B\) is the well-defined operator acting by \((A \otimes B)(v \otimes w) = Av \otimes Bw\). That property is stated there for real vector spaces, and both its statement and its proof use only the vector space operations, so it transfers verbatim to complex ones.

Assembling the Orthogonality Relation

Two further ingredients connect the conjugate character to a dual representation. Fix an orthonormal basis for the invariant inner product recalled above, so that \(\Pi(x)\) is unitary and \(\Pi(x)^{-1}\) is its conjugate transpose. Then \([\Pi(x^{-1})]^\top = \overline{\Pi(x)}\), the entrywise conjugate, so the matrix of the dual representation, which acts by \(\Pi^*(x) = [\Pi(x^{-1})]^\top\), is the entrywise conjugate of the matrix of \(\Pi(x)\). Taking traces, \[ \overline{\chi_\Pi(x)} = \overline{\operatorname{tr}(\Pi(x))} = \operatorname{tr}\big(\overline{\Pi(x)}\big) = \operatorname{tr}\big(\Pi^*(x)\big) = \chi_{\Pi^*}(x). \] The conjugate of a character is therefore the character of the dual representation. We can now prove the theorem.

The orthonormality relation already separates representations. Two irreducibles with the same character are isomorphic, because \(\langle \chi_\Pi, \chi_\Sigma \rangle = 1\) forces \(\Pi \cong \Sigma\). What remains is the deeper question of completeness. Do the characters exhaust the class functions, leaving no direction in \(L^2(K)\) orthogonal to all of them? That is the content of the Peter-Weyl theorem, to which we now turn.

Matrix Entries and the Peter-Weyl Theorem

Completeness cannot be proved at the level of characters alone. A character is a single function extracted from a representation by the trace, and the trace collapses too much. The route to completeness passes instead through the full collection of matrix coefficients of a representation, of which the character is the special case formed by summing the diagonal. We isolate these coefficients, prove that they are dense in \(L^2(K)\), and only then descend to characters by averaging over conjugacy.

The basis-free description \(f(x) = \operatorname{tr}(\Pi(x)A)\) is the one we use, because it transforms cleanly under the group operations. Of these, the single fact the completeness proof relies on is how an integral of conjugated operators behaves on an irreducible representation.

The Algebra of Matrix Entries

Let \(\mathcal{A} \subseteq C(K)\) be the set of all functions expressible as a finite linear combination of matrix entries, ranging over all finite-dimensional representations of \(K\). The Peter-Weyl theorem asserts that \(\mathcal{A}\) is uniformly dense in \(C(K)\), and therefore dense in \(L^2(K)\). We prove this by verifying that \(\mathcal{A}\) satisfies the hypotheses of the Stone-Weierstrass theorem on the compact Hausdorff space \(K\): that it is a self-adjoint subalgebra containing the constants and separating points. The uniform closure of such an algebra is then all of \(C(K)\).

From Matrix Entries to Characters

Density of matrix entries is the Peter-Weyl theorem in its primary form. Completeness of characters is the statement that a continuous class function orthogonal to every irreducible character must vanish, and it follows by projecting matrix entries onto the class functions. The bridge is an averaging over conjugacy that sends each matrix entry to a linear combination of characters.

The Peter-Weyl Decomposition and the Window to GDL

The density of matrix entries upgrades at once to an orthonormal basis. The space \(L^2(K)\) is a Hilbert space, carrying the inner product \(\langle f, g \rangle = \int_K \overline{f}\, g\, dx\) and complete by the Riesz-Fischer theorem. We call a family in a Hilbert space an orthonormal basis when its members are orthonormal and their finite linear combinations are dense, which is the condition of the definition for sequences with the countability of the family dropped. The orthogonality relations for the matrix entries supply the first half of that condition and the density theorem supplies the second, and together they yield the decomposition that is the harmonic-analytic core of the theory.

Every \(f \in L^2(K)\) thus expands in matrix entries, with coefficients obtained by integrating \(f\) against their conjugates. These are the noncommutative Fourier coefficients of \(f\). Only countably many of them are nonzero for a fixed \(f\), since for any finite subfamily \(\{e_i\}\) of the rescaled entries the expansion of \(0 \leq \|f - \sum_i \langle e_i, f \rangle e_i\|_2^2 = \|f\|_2^2 - \sum_i |\langle e_i, f \rangle|^2\) leaves fewer than \(n^2\|f\|_2^2\) coefficients of modulus above \(1/n\). Enumerate the nonzero ones as a sequence and let \(M\) be the closed span of the corresponding entries, a Hilbert space because a closed subset of a complete space is complete, and separable because the rational combinations of that sequence are dense in it. Then \(f \in M\). Writing \(f = m + m^{\perp}\) as in the projection theorem, every entry of the enumerated sequence lies in \(M\) and is therefore orthogonal to \(m^{\perp}\), while every remaining entry is orthogonal to \(M\) and has coefficient zero against \(f\), hence is orthogonal to \(m^{\perp}\) as well. Being orthogonal to a basis, \(m^{\perp} = 0\), and Parseval's identity applied inside \(M\) gives both the \(L^2\)-convergent expansion of \(f\) and the energy identity \(\|f\|_2^2 = \sum_i |\langle e_i, f \rangle|^2\). This is the precise sense in which the Peter-Weyl theorem is the harmonic analysis of a compact group. It furnishes the basis, the transform, and the energy identity, with the irreducible representations playing the role of frequencies.

The Circle Recovered

Take \(K = U(1)\), the circle realized as a compact matrix Lie group inside \(GL(1, \mathbb{C})\). It is abelian, so its irreducible complex representations are one-dimensional, and they are exactly \(\Pi_n(e^{i\theta}) = e^{in\theta}\) for \(n \in \mathbb{Z}\), a classification we take for granted here. Each is its own single matrix entry, and the factor \(d_\Pi = 1\) makes the rescaling trivial, so the Peter-Weyl decomposition states that \(\{e^{in\theta}\}_{n \in \mathbb{Z}}\) is an orthonormal basis of \(L^2(U(1))\). Its normalized Haar measure is \(d\theta/2\pi\), and taking \(L = \pi\) in the completeness of the trigonometric system gives the family \((2\pi)^{-1/2}e^{-in\theta}\) in \(L^2\) of the unnormalized measure, which is the same family after the relabelling \(n \mapsto -n\) and the rescaling that normalizes the measure. The general theorem is the noncommutative extension of that one fact. Where the circle offered scalar exponentials indexed by integers, a nonabelian compact group offers matrix-valued entries indexed by its irreducible representations.