Irreducible reps of maximal tori

[jdstokes]

Hi all,

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

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

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

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

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

 

[jdstokes]

This is actually really easy if you consider the derivative of $e^{2pi x(t)}$ and $(e^{2pi x(t)})^n$ 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 $x \mapsto nx$ as required.