Representations of a product of Lie groups

Question

Let $G=G_1\times G_2$ be a product of two compact Lie groups. Is every finite dimensional irreducible representation of $G$ a tensor product of irreducible representations of $G_1$ and $G_2$? This seems to be true (at least in many cases) but I have been unable to find a reference. Could someone provide a reference or a proof for this?

Answer 1

This is a consequence of the Peter-Weyl theorem, which says that the components $\phi_{ij}$ of the irreducible representations $\phi$ of $G_1$ form an orthogonal basis for $L^2(G_1)$, and similarly the components $\psi_{ij}$ of the irreducible representations $\psi$ of $G_2$ form an orthogonal basis for $L^2(G_2)$. The components of the tensor product $\theta=\phi \otimes \psi$ have the form $\theta_{ijkl}(g_1,g_2)=\phi_{ij}(g_1)\psi_{kl}(g_2)$. A straightforward computation shows that $\langle \chi_{\theta}, \chi_{\theta'} \rangle$ equals 1 if $\theta=\theta'$ and is zero otherwise. This shows that the tensor products $\theta=\phi \otimes \psi$ are distinct irreducible representations of $G$, as $\phi$ and $\psi$ range over the irreducible representations of $G_1$ and $G_2$. Thus the components $\theta_{ijkl}$ form an orthogonal set in $L^2(G)$.

By the completeness of $\phi_{ij}$ and $\psi_{ij}$ in $L^2(G_1)$ and $L^2(G_2)$ respectively, any indicator function (i.e., characteristic function) $1_{A}$ on a Borel-measurable subset $A \subseteq G_1$ may be well-approximated (in the $L^2$ norm) by a finite linear combination of the $\phi_{ij}$, and likewise an indicator function $1_{B}$ on a Borel-measurable subset $B \subseteq G_2$ may be well-approximated by a finite linear combination of the $\psi_{ij}$. It follows that $1_{A\times B}(g_1,g_2)=1_A(g_1)1_B(g_2)$ may be well approximated by a finite linear combination of the functions $\phi_{ij}(g_1)\psi_{kl}(g_2)$, i.e. by the $\theta_{ijkl}$. Since the linear combinations of $1_{A\times B}$ are dense in $L^2(G)$, this shows that the $\theta_{ijkl}$, when suitably normalized, form a complete orthonormal set in $L^2(G)$. Any other irreducible representation of $G_1 \times G_2$ would have to be orthogonal to all the $\theta_{ijkl}$ (by again applying Peter-Weyl), but the completeness of $\theta_{ijkl}$ rules this possibility out. This proves that all irreducible representations of $G= G_1 \times G_2$ are of the form $\phi \otimes \psi$ for irreducible representation $\phi$ of $G_1$ and $\psi$ of $G_2$.