Peter–Weyl theory for vector fields

Question

Let $G$ be a compact Lie group. The classical Peter-Weyl theorem shows that $L^2(G)$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$. This is a powerful statement as it allows to answer questions about functions on $G$ in terms of matrix coefficients of irreducible representations.

I was wondering if there exist a decomposition of the space $\mathfrak{X}(G)$ of vector fields on $G$ that has a similar spirit than the Peter-Weyl theorem. In particular, I was hoping that the gradient map $C^\infty(G) \to \mathfrak{X}(G)$ (with respect to the Riemannian metric induced by the Killing form of $G$) has a nice behavior with respect to the decompositions of the spaces on both sides. I'm a bit vague here because I don't know what I can hope for. In the best case, the gradient map is diagonalized (similar to how the Laplacian is diagonalized by the classical Peter-Weyl theorem).

Answer 1

Let $\mathfrak g \to \mathfrak{X}(G)$ be the inclusion of right-invariant vector fields on $G$ into vector fields on $G$. Then we have an isomorphism $C^\infty(G) \otimes_{\mathbb R} \mathfrak g \to \mathfrak{X}(G)$ of left $C^\infty(G)$-modules defined by $f \otimes_{\mathbb R} x \mapsto fx$. It is $G \times G$-equivariant. Applying Peter-Weyl to $C^\infty(G)$ gives the desired decomposition. The gradient $C^\infty(G) \to \mathfrak{X}(G)$ is $G\times G$-equivariant if the metric is biinvariant.