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In order to further advance with Peter-Weyl theory, we need to talk about integration over [math]G[/math]. In the finite group case the situation is trivial, as follows:

Proposition

Any finite group [math]G[/math] has a unique probability measure which is invariant under left and right translations,

[[math]] \mu(E)=\mu(gE)=\mu(Eg) [[/math]]

and this is the normalized counting measure on [math]G[/math], given by [math]\mu(E)=|E|/|G|[/math].

Show Proof

This is indeed something trivial, which follows from definitions.

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In the general, continuous case, let us begin with the following key result:

Proposition

Given a unital positive linear form [math]\psi:C(G)\to\mathbb C[/math], the limit

[[math]] \int_\varphi f=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\psi^{*k}(f) [[/math]]

exists, and for a coefficient of a representation [math]f=(\tau\otimes id)w[/math] we have

[[math]] \int_\varphi f=\tau(P) [[/math]]

where [math]P[/math] is the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\psi)w[/math].

Show Proof

By linearity it is enough to prove the first assertion for functions of the following type, where [math]w[/math] is a Peter-Weyl representation, and [math]\tau[/math] is a linear form:

[[math]] f=(\tau\otimes id)w [[/math]]

Thus we are led into the second assertion, and more precisely we can have the whole result proved if we can establish the following formula, with [math]f=(\tau\otimes id)w[/math]:

[[math]] \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\psi^{*k}(f)=\tau(P) [[/math]]

In order to prove this latter formula, observe that we have:

[[math]] \psi^{*k}(f) =(\tau\otimes\psi^{*k})w =\tau((id\otimes\psi^{*k})w) [[/math]]

Let us set [math]M=(id\otimes\psi)w[/math]. In terms of this matrix, we have:

[[math]] ((id\otimes\psi^{*k})w)_{i_0i_{k+1}} =\sum_{i_1\ldots i_k}M_{i_0i_1}\ldots M_{i_ki_{k+1}} =(M^k)_{i_0i_{k+1}} [[/math]]

Thus we have the following formula, valid for any [math]k\in\mathbb N[/math]:

[[math]] (id\otimes\psi^{*k})w=M^k [[/math]]

It follows that our Cesàro limit is given by the following formula:

[[math]] \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\psi^{*k}(f) =\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\tau(M^k) =\tau\left(\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^nM^k\right) [[/math]]

Now since [math]w[/math] is unitary we have [math]||w||=1[/math], and so [math]||M||\leq1[/math]. Thus the last Cesàro limit converges, and equals the orthogonal projection onto the [math]1[/math]-eigenspace of [math]M[/math]:

[[math]] \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^nM^k=P [[/math]]

Thus our initial Cesàro limit converges as well, to [math]\tau(P)[/math], as desired.

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When the linear form [math]\psi\in C(G)^*[/math] is faithful, we have the following finer result:

Proposition

Given a faithful unital linear form [math]\psi\in C(G)^*[/math], the limit

[[math]] \int_\psi f=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\psi^{*k}(f) [[/math]]

exists, and is independent of [math]\psi[/math], given on coefficients of representations by

[[math]] \left(id\otimes\int_\psi\right)w=P [[/math]]

where [math]P[/math] is the orthogonal projection onto the space [math]Fix(w)=\left\{\xi\in\mathbb C^n\big|w\xi=\xi\right\}[/math].

Show Proof

In view of Proposition 4.13, it remains to prove that when [math]\psi[/math] is faithful, the [math]1[/math]-eigenspace of the matrix [math]M=(id\otimes\psi)w[/math] equals the space [math]Fix(w)[/math].

“[math]\supset[/math]” This is clear, and for any [math]\psi[/math], because we have the following implication:

[[math]] w\xi=\xi\implies M\xi=\xi [[/math]]

“[math]\subset[/math]” Here we must prove that, when [math]\psi[/math] is faithful, we have:

[[math]] M\xi=\xi\implies w\xi=\xi [[/math]]

For this purpose, assume that we have [math]M\xi=\xi[/math], and consider the following function:

[[math]] f=\sum_i\left(\sum_jw_{ij}\xi_j-\xi_i\right)\left(\sum_kw_{ik}\xi_k-\xi_i\right)^* [[/math]]

We must prove that we have [math]f=0[/math]. Since [math]v[/math] is unitary, we have:

[[math]] \begin{eqnarray*} f &=&\sum_{ijk}w_{ij}w_{ik}^*\xi_j\bar{\xi}_k-\frac{1}{N}w_{ij}\xi_j\bar{\xi}_i-\frac{1}{N}w_{ik}^*\xi_i\bar{\xi}_k+\frac{1}{N^2}\xi_i\bar{\xi}_i\\ &=&\sum_j|\xi_j|^2-\sum_{ij}w_{ij}\xi_j\bar{\xi}_i-\sum_{ik}w_{ik}^*\xi_i\bar{\xi}_k+\sum_i|\xi_i|^2\\ &=&||\xi||^2- \lt w\xi,\xi \gt -\overline{ \lt w\xi,\xi \gt }+||\xi||^2\\ &=&2(||\xi||^2-Re( \lt w\xi,\xi \gt )) \end{eqnarray*} [[/math]]

By using now our assumption [math]M\xi=\xi[/math], we obtain from this:

[[math]] \begin{eqnarray*} \psi(f) &=&2\psi(||\xi||^2-Re( \lt w\xi,\xi \gt ))\\ &=&2(||\xi||^2-Re( \lt M\xi,\xi \gt ))\\ &=&2(||\xi||^2-||\xi||^2)\\ &=&0 \end{eqnarray*} [[/math]]

Now since [math]\psi[/math] is faithful, this gives [math]f=0[/math], and so [math]w\xi=\xi[/math], as claimed.

■

We can now formulate a main result, as follows:

Theorem

Any compact group [math]G[/math] has a unique Haar integration, which can be constructed by starting with any faithful positive unital form [math]\psi\in C(G)^*[/math], and setting:

[[math]] \int_G=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\psi^{*k} [[/math]]

Moreover, for any representation [math]w[/math] we have the formula

[[math]] \left(id\otimes\int_G\right)w=P [[/math]]

where [math]P[/math] is the orthogonal projection onto [math]Fix(w)=\left\{\xi\in\mathbb C^n\big|w\xi=\xi\right\}[/math].

Show Proof

Let us first go back to the general context of Proposition 4.13. Since convolving one more time with [math]\psi[/math] will not change the Cesàro limit appearing there, the functional [math]\int_\psi\in C(G)^*[/math] constructed there has the following invariance property:

[[math]] \int_\psi*\,\psi=\psi*\int_\psi=\int_\psi [[/math]]

In the case where [math]\psi[/math] is assumed to be faithful, as in Proposition 4.14, our claim is that we have the following formula, valid this time for any [math]\varphi\in C(G)^*[/math]:

[[math]] \int_\psi*\,\varphi=\varphi*\int_\psi=\varphi(1)\int_\psi [[/math]]

Indeed, it is enough to prove this formula on a coefficient of a corepresentation:

[[math]] f=(\tau\otimes id)w [[/math]]

In order to do so, consider the following two matrices:

[[math]] P=\left(id\otimes\int_\psi\right)w\quad,\quad Q=(id\otimes\varphi)w [[/math]]

We have then the following formulae, which all follow from definitions:

[[math]] \left(\int_\psi*\,\varphi\right)f=\tau(PQ)\quad,\quad \left(\varphi*\int_\psi\right)f=\tau(QP)\quad,\quad \varphi(1)\int_\psi f=\varphi(1)\tau(P) [[/math]]

Thus, in order to prove our claim, it is enough to establish the following formula:

[[math]] PQ=QP=\psi(1)P [[/math]]

But this follows from the fact, that we know from Proposition 4.14, that [math]P=(id\otimes\int_\psi)w[/math] is the orthogonal projection onto [math]Fix(w)[/math]. Thus, we proved our claim. Now observe that, with [math]\Delta f(g\otimes h)=f(gh)[/math], this formula that we proved can be written as follows:

[[math]] \varphi\left(\int_\psi\otimes\,id\right)\Delta =\varphi\left(id\otimes\int_\psi\right)\Delta =\varphi\int_\psi(.)1 [[/math]]

This formula being true for any [math]\varphi\in C(G)^*[/math], we can simply delete [math]\varphi[/math], and we conclude that [math]\int_G=\int_\psi[/math] has the required left and right invariance property, namely:

[[math]] \left(\int_G\otimes\,id\right)\Delta =\left(id\otimes\int_G\right)\Delta =\int_G(.)1 [[/math]]

Finally, the uniqueness is clear as well, because if we have two invariant integrals [math]\int_G,\int_G'[/math], then their convolution equals on one hand [math]\int_G[/math], and on the other hand, [math]\int_G'[/math].

■

Summarizing, we know how to integrate over [math]G[/math]. Before getting into probabilistic applications, let us develop however more Peter-Weyl theory. We will need:

Proposition

We have a Frobenius type isomorphism

[[math]] Hom(v,w)\simeq Fix(v\otimes\bar{w}) [[/math]]

valid for any two representations [math]v,w[/math].

Show Proof

According to definitions, we have the following equivalences:

[[math]] \begin{eqnarray*} T\in Hom(v,w) &\iff&Tv=wT\\ &\iff&\sum_iT_{ai}v_{ij}=\sum_bw_{ab}T_{bj},\forall a,j \end{eqnarray*} [[/math]]

On the other hand, we have as well the following equivalences:

[[math]] \begin{eqnarray*} T\in Fix(v\otimes\bar{w}) &\iff&(v\otimes\bar{w})T=\xi\\ &\iff&\sum_{bi}v_{ji}\bar{w}_{ab}T_{bi}=T_{aj}\forall a,j \end{eqnarray*} [[/math]]

With these formulae in hand, both inclusions follow from the unitarity of [math]v,w[/math].

■

We can now formulate a third Peter-Weyl theorem, as follows:

Theorem (Peter-Weyl 3)

The dense subalgebra [math]\mathcal C(G)\subset C(G)[/math] generated by the coefficients of the fundamental representation decomposes as a direct sum

[[math]] \mathcal C(G)=\bigoplus_{w\in Irr(G)}M_{\dim(w)}(\mathbb C) [[/math]]

with the summands being pairwise orthogonal with respect to the scalar product

[[math]] \lt f,g \gt =\int_Gf\bar{g} [[/math]]

where [math]\int_G[/math] is the Haar integration over [math]G[/math].

Show Proof

By combining the previous two Peter-Weyl results, Theorems 4.10 and 4.11, we deduce that we have a linear space decomposition as follows:

[[math]] \mathcal C(G) =\sum_{w\in Irr(G)}C_w =\sum_{w\in Irr(G)}M_{\dim(w)}(\mathbb C) [[/math]]

Thus, in order to conclude, it is enough to prove that for any two irreducible representations [math]v,w\in Irr(G)[/math], the corresponding spaces of coefficients are orthogonal:

[[math]] v\not\sim w\implies C_v\perp C_w [[/math]]

But this follows from Theorem 4.15, via Proposition 4.16. Let us set indeed:

[[math]] P_{ia,jb}=\int_Gv_{ij}\bar{w}_{ab} [[/math]]

Then [math]P[/math] is the orthogonal projection onto the following vector space:

[[math]] Fix(v\otimes\bar{w}) \simeq Hom(v,w) =\{0\} [[/math]]

Thus we have [math]P=0[/math], and this gives the result.

■

Finally, we have the following result, completing the Peter-Weyl theory:

Theorem (Peter-Weyl 4)

The characters of irreducible representations belong to the algebra

[[math]] \mathcal C(G)_{central}=\left\{f\in\mathcal C(G)\Big|f(gh)=f(hg),\forall g,h\in G\right\} [[/math]]

called algebra of central functions on [math]G[/math], and form an orthonormal basis of it.

Show Proof

Observe first that [math]\mathcal C(G)_{central}[/math] is indeed an algebra, which contains all the characters. Conversely, consider a function [math]f\in\mathcal C(G)[/math], written as follows:

[[math]] f=\sum_{w\in Irr(G)}f_w [[/math]]

The condition [math]f\in\mathcal C(G)_{central}[/math] states then that for any [math]w\in Irr(G)[/math], we must have:

[[math]] f_w\in\mathcal C(G)_{central} [[/math]]

But this means that [math]f_w[/math] must be a scalar multiple of [math]\chi_w[/math], so the characters form a basis of [math]\mathcal C(G)_{central}[/math], as stated. Also, the fact that we have an orthogonal basis follows from Theorem 4.17. As for the fact that the characters have norm 1, this follows from:

[[math]] \int_G\chi_w\bar{\chi}_w =\sum_{ij}\int_Gw_{ii}\bar{w}_{jj} =\sum_i\frac{1}{M} =1 [[/math]]

Here we have used the fact, coming from Theorem 4.15 and Proposition 4.16, that the integrals [math]\int_Gw_{ij}\bar{w}_{kl}[/math] form the orthogonal projection onto the following vector space:

[[math]] Fix(w\otimes\bar{w})\simeq End(w)=\mathbb C1 [[/math]]

Thus, the proof of our theorem is now complete.

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General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].