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The representations [itex]S^1 = \mathbb{R}/\mathbb{Z} \to U(1)[/itex] are of the form [itex]\rho_n : [x] \mapsto e^{2\pi i n x}[/itex] for any integer n. I'm trying to understand why the push-forward [itex](\rho_n)_\ast : x \mapsto nx[/itex].

The push-forward of [itex]\rho_n[/itex] is a map from the tangent space of S^1 to the tangent space of U(1).

ie it is the map [itex](\rho_n)_\ast : \lambda \frac{\partial }{\partial x} (\cdot) \mapsto \lambda\partial (\cdot \circ \rho_n) [/itex]

where the dot indicates a the argument function and lambda is any real number.

Presumably one should have [itex]\lambda\partial/\partial x \mapsto n\lambda \partial/\partial x[/itex] or something but I can't see how this follows from the above.

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# Irreducible reps of maximal tori

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