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Can an analytic function defined on a maximal torus be extended analytically to all the Lie ...

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 $T$?
Max Reinhold Jahnke's user avatar