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Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?

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This seems to be consequence of the paper

Cartan, Henri: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85 1957 77–99

Cartan shows more generally (see sections 6 and 7), that a real analytic function on a real analytic submanifold of $\mathbb R^n$ can be extended to a real analytic function on all of $\mathbb R^n$.

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