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Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group.

Every finite-dimensional representation of G has a character, which is a function on G, T and T/W.

I want to prove that for two different points a and b in T/W, we can find such a character $\chi$ that $\chi(a)\neq \chi(b)$.

Sorry, Piotr Achinger, you are probably right. I have to reformulate my thoughts and repost it.

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Presumably $G$ is connected. Points $t, t' \in T$ have distinct images in $T/W$ iff they are not conjugate in $G$. Indeed, if $t' = gtg^{-1}$ then $gTg^{-1}$ and $T$ contain $t'$ and so lie in the connected compact Lie group $Z_G(t)^0$. By conjugacy of maximal tori in such Lie groups, there exists $h \in Z_G(t)^0$ such that $h(gTg^{-1}) = T$, so $t' = ht'h^{-1} = (hg)t(hg)^{-1}$ with $hg \in N_G(T)$, so $t'$ is in the $W$-orbit of $t$. So your question is exactly if characters of $G$ separate conjugacy classes in $G$ (as every $g \in G$ lies in a torus, since $G$ is connected and compact). –  user29720 Jan 12 '13 at 17:37
I should have written $Z_G(t')^0$ rather than $Z_G(t)^0$ above. –  user29720 Jan 12 '13 at 17:53
Many thanks to your comment. Now I feel more secure. As you may know, by Stone-Weierstrass theorem, this would imply that the linear combination of characters would be dense in the space of continuous function on T/W. It is something like Peter-Weyl theorem which says that all continuous functions can be approximated by linear combinations of characters in L^2 norm. –  Jeep Wrangler Jan 12 '13 at 18:08

1 Answer 1

This type of question (separating classes by representations) is natural throughout representation theory of various different flavors. In particular, the systematic treatment of (finite dimensional) irreducible representations of connected compact Lie groups leads to a straightforward affirmative answer: see for instance the textbook Representations of Compact Lie Groups by Brocker and tom Dieck (Springer GTM 98), expecially IV.2 and VI.2. Their treatment of class functions is spread out a bit, but doesn't actually require all details of the Weyl character development.

Briefly, the compact Lie group case involves some structure theory: all maximal tori are conjugate and every element is conjugate to some element in your fixed $T$. Moreover, two elements of $T$ are conjugate iff they are conjugate under the Weyl group $W = N_G(T)/T$. Then one has to pin down the class functions on $G$ (invariant on conjugacy classes) by identifying them with $W$-invariant functions on $T$. (Here the category of groups determines what kind of functions are relevant.) A key fact is that characters of irreducible (necessarily finite dimensional) representations generate the algebra of class functions on $G$, and you can pass to the isomorphic algebra of $W$-invariant functions on $T$. These ingredients are standard but not trivial to develop. As a refinement, when $G$ is semisimple and simply connected, one sees that classes in $G$ are already separated just by the values of the finitely many fundamental characters.

To get more perspective on these ideas, it's worthwhile to look at the closely parallel treatment of a connected semisimple algebraic group (over an algebraically closed field of any characteristic): see especially Theorem 6.1 and its consequences in Steinberg's 1965 paper on regular elements here. For a semisimple algebraic group virtually the same results can be proved as in the compact case, with the important difference that not all elements of $G$ are semisimple (and indeed, characters of representations fail to distinguish elements from their semisimple parts). Here it's more obvious that you don't have to know all the fine details about irreducible highest weight representations (which in fact aren't yet complete in prime characteristic). Broadly speaking, three types of semisimple groups behave similarly: compact Lie groups, complex Lie groups, linear algebraic groups.

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@Allen: Thanks for the edit. I was writing off the top of my head and have now made references more precise. (Is there a better source for the compact groups?) –  Jim Humphreys Jan 13 '13 at 14:23

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