# Peter–Weyl theory for vector fields

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).

• 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 ? – David Lehavi May 5 at 9:42
• @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. – Tobias Diez May 5 at 11:16