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I'm having difficulty with this result (given as 2 lines in my book):

Let $\Phi$ be a root system as defined http://en.wikipedia.org/wiki/Root_system and let $W$ be a group generated by the reflections $s_\alpha$ for $\alpha \in \Phi$.

Suppose that we know each element of $W$ fixes pointwise the orthogonal complement, $U^\perp$ of the subspace spanned by $\Phi$, $U$.

Somehow, this implies that only $e$ can fix $\Phi$.

I can kinda see how we would want this to be true, but I'm unsure how to prove it.

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    Apologies, I've updated the question2011-10-17

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Clearly if a linear morphism fixes pointwise $\Phi$ and the orthogonal complement of $U$ then it fixes the whole space, being generated by $\Phi$ and the complement of $U$. So if one element in $W$ does that it is $e$. Perhaps I missed something but it seems quite easy ...

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    The fact after updating your question that $W$ appears to be a Weil group does not change my answer in my mind : it only ensures the fact that it actually fixes the orthogonal complement of the root space ...2011-10-17
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Here's a related (simpler) problem that if you solve it first, should help you with your problem.

Suppose $G$ is a group of linear transformations of a vector space $V$ and suppose that $v_1, \dots, v_n$ is a basis of $V$ such that $g \cdot v_i = v_i$ for some $g \in G$. Then $g$ is the identity transformation.