Is there a Peter-Weyl-Theorem over the real numbers?

Question

I'd like to know: Is there a Peter-Weyl-like Theorem over the real numbers? If so, can you provide a reference?

With "over the real numbers" I mean that the representations considered act on vector spaces over the real numbers.

Best, Leon

Answer 1

Here is what Knapp calls the Peter-Weyl theorem. Often people mean just statement (1), but the other follow easily from it.

Let $G$ be a compact real Lie group. Then

1. The matrix coefficients of the finite-dimensional complex irreducible representations are dense in $L^2(G)$;
2. If $\{\Phi^\alpha\}$ is a maximal set of mutually inequivalent finite-dimensional irreducible complex unitary representations of $G$, and $(d^\alpha)^{1/2}\Phi_{ij}^\alpha\}$ is the corresonding orthonormal set of matrix coefficients, then this second set is an orthonormal basis for $L^2(G)$.
3. Every irreducible complex unitary representation of $G$ is finite-dimensional.
4. Every unitary complex representation of $G$ is an othogonal direct sum of finite-dimensional irreducible representations.
5. Let $\Phi$ be a unitary complex representation on a Hilbert space $V$. For every irreducible representation $\tau$,let $E_\tau$ be the orthogonal projection onto the closure of the sum of invariant subspaces of $V$ isomorphic to $\tau$.Then $E_\tau=(\dim\tau)\Phi(\overline{\chi_\tau})$, where $\chi_\tau$ is the character of $\tau$. The $E_\tau$ are orthogonal idempotents, and every $v\in V$ satisfies $v=\sum_\tau E_\tau v$.

To prove (1) one can either assume $G$ is a matrix group or not. If not, one uses the Hilbert-Schmidt theorem, and I think this argument actually goes through for real representations. If one does assume that $G$ is a matrix group, then the span of the matrix coefficients of finite-dimensional irreducible real representations

• Separates points, by looking at the given representation of the matrix group
• Vanishes nowhere, by looking at the same;
• Is an algebra, as if $f$ and $g$ are matrix-coefficients of representations $V$ and $W$, then $fg$ is a matrix coefficient of (a suitable irreducible subrepresentation of) $V\otimes W$.

Therefore by compactness of $G$, the matrix coefficients are uniformly dense in the continuous functions, hence dense in the norm topology in $L^2(G)$.

Point (3) follows from (4) and (4) follows from (1). The argument over $\mathbb{C}$ goes through also for $\mathbb{R}$: it is essentially Zorn's lemma and then using (1) to prove that the orthogonal complement to a maximal set of orthogonal finite-dimensional irreps itself contains a finite-dimensional irrep, which of course is a contradiction.

As you identified, the problem comes with (2) and (5). For (2), as you corrected an earlier version of this answer saying, while matrix coefficients of nonequivalent representations will be orthogonal, different matrix coefficients of same representation no longer need to be. For example, the circle group acting on the plane via rotations.

The proof of (5) requires in the end the existence of eigenvalues as it involves Schur's lemma. In your example the exact problem is that abelian groups can have two-dimensional real irreps, so in this case $V_m$ appears once in $L^2_\mathbb{R}$ so that its complexification gives two one-dimensional irreps each occuring once. In general I think that unless you can say that every irreducible representation occuring in $L^2_\mathbb{C}(G)$ arising as a subrepresentation of a complexified representation over $\mathbb{R}$ is contributed only by a single isomorphism class of representation over $\mathbb{R}$, there won't be an easy formula. I can't say I've thought about this (the only interesting part, as it turns out) at all, though.

Answer 2

I think that the Peter-Weyl theorem can be formulated for complex and real representations simultaneously using the ideas from the book by Bröcker and tom Dieck "Representations of compact Lie groups", Chapter III, Section 6, Exercise 4.

Let $\mathbb{K}$ be either $\mathbb{C}$ or $\mathbb{R}$ and let $\lambda$ be an irreducible representation of a compact topological group $G$ in a finite-dimensional $\mathbb{K}$-linear space $E$ with $G$-invariant inner product. Let $D(E)$ be the set of linear operators in $E$ that intertwine $\lambda$ with itself. If $\mathbb{K}=\mathbb{C}$, then $D(E)=\mathbb{C}$, while if $\mathbb{K}=\mathbb{R}$, then $D(E)$ may take three values, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$, the skew field of quaternions.

Let $E^*$ be the linear space, dual to $E$, that is, the set of $\mathbb{K}$-linear functionals over $E$. It is a right $D(U)$-linear space via composition. On the other hand, $E$ is a left $D(U)$-linear space via evaluation. The tensor product $E^*\otimes_{D(U)}E$ is defined. A simple bookkeeping shows that the dimension of this $\mathbb{K}$-linear space is equal to $\frac{(\dim E)^2}{\dim D(E)}$. If $\mathbb{K}=\mathbb{C}$, then the denominator is equal to $1$, if $\mathbb{K}=\mathbb{R}$, then it may take three different values: $1$, $2$, and $4$.

The map that maps the tensor product $x^*\otimes y$ to the continuous function $f\colon G\to\mathbb{K}$, $f(g)=x^*(\lambda(g^{-1})x)$, can be extended by linearity to all of $E^*\otimes_{D(U)}E$. The image of this map is a finite-dimensional subspace of $C(G)$, call it $E_{\lambda}$. It is exactly the subspace grnerated by the matrix entries of the representation $\lambda$.

The Peter--Weyl theorem becomes:

1. If $\lambda_1$ and $\lambda_2$ are not equivalent, then the intersection of the spaces $E_{\lambda_1}$ and $E_{\lambda_2}$ is equal to $\{0\}$.

2. The set of finite $\mathbb{K}$-linear combinations of the elements of the spaces $E_{\lambda}$ is dense in both $C(G)$ and $L^2(G)$.

3. The space $E_{\lambda}$ carries the direct sum of $\frac{\dim E}{\dim D(E)}$ copies of the representation $\lambda$, an the Hilbert direct sum of all these representations is equivalent to the left regular representation of $G$ in $L^2(G)$.