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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).

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    $\begingroup$ A vector field is a map from the group to the Lie algebra, which is a finite dimensional vector space, I.e. X(G) maps to $(L^2(G))^{\mathrm{dim}(G}}$, or am I missing something ? $\endgroup$ – David Lehavi May 5 at 9:42
  • $\begingroup$ @DavidLehavi You are right, and this observation lead me to believe that there is a similar theory for vector fields. I was just hoping for a more invariant formulation (i.e. without choosing a basis in the Lie algebra) that behaves well with respect to derivatives (gradient map, commutator of vector fields)... Sorry for being a bit vague here. I have a problem involving vector fields on a Lie group, and I was hoping that one can do better then just applying Peter-Weyl to the coefficient functions with respect to a global frame; but I'm not sure what I want exactly in the end. $\endgroup$ – Tobias Diez May 5 at 11:16

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