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Irreducible reps of maximal tori

  1. Jan 29, 2008 #1
    Hi all,

    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.
    Last edited: Jan 29, 2008
  2. jcsd
  3. Feb 2, 2008 #2
    This is actually really easy if you consider the derivative of [itex]e^{2pi x(t)}[/itex] and [itex](e^{2pi x(t)})^n[/itex] with respect to t. Since both maps are complex analytic you will find that the derivative of the latter is n times the derivative of the first function, multiplied by some phase difference. From the point of view of the tangent spaces, this means that [itex] x \mapsto nx[/itex] as required.
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