Suppose G is a semi-simple adjoint group over complex numbers. Suppose T is a maximal torus in G. Does one know what are the W invariant (non-trivial) elements in T? Perhaps I might give a few examples to motivate the question. For PGL(2), there is one non-trivial element which is diagonal (-1,1), but none for PGL(n), n>2. For SO(2n+1) there is one: represented by the diagonal element (-1.,,,-1, 1, -1,...-1). For PSp(2n) there is none.... I do not know the general answer which I am sure is well-known.
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$\begingroup$ There is some confusion here. Recall that $W$ is defined to be the normalizer of $T$ modulo the centralizer, while the centralizer is just $T$ itself. So under the action of $W$ induced by conjugation only the identity element of $T$ is invariant. Your examples are mostly incorrect. $\endgroup$– Jim HumphreysCommented Jan 15, 2014 at 11:34
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$\begingroup$ The action of the Weyl group for PGL(2) is t--->t^{-1} so indeed -1 is left invariant under the Weyl group? $\endgroup$– anonymousCommented Jan 15, 2014 at 11:59
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$\begingroup$ The problem with this example is that the equivalence class of this diagonal matrix in the adjoint group PGL is the identity element. And for the odd special orthogonal group, the negative of the identity matrix is not in this group. $\endgroup$– Jim HumphreysCommented Jan 15, 2014 at 12:22
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1$\begingroup$ There was a 1 in the (n+1)st place! $\endgroup$– anonymousCommented Jan 15, 2014 at 12:34
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$\begingroup$ @Jim: I don't understand your implication "So under the action of W induced by conjugation only the identity element of T is invariant" in your first comment. Could you clarify this? $\endgroup$– Tom De MedtsCommented Jan 15, 2014 at 12:37
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Let us first assume that $G$ is a simple adjoint group and fix a maximal torus $T \leqslant G$. Let $W = N_G(T)/T$ be the corresponding Weyl group. What you're asking is, for a semisimple element $s \in T$, when does $W(s) = W$? Here $W(s) = \{w \in W \mid s^w = s\}$ is the centraliser of $s$ in $W$. As $G$ is adjoint this group is typically larger than the Weyl group of the reductive group $C_G(s)^{\circ}$.
Such a semisimple element is known as quasi-isolated, which means the group $C_G(s)$ is not contained in any proper Levi subgroup of $G$. This is equivalent to saying that $W(s)$ is not contained in any proper parabolic subgroup of $W$. Up to conjugacy such semisimple elements have been classified by Bonnafé in his truly beautiful article "Quasi-Isolated Elements in Reductive Groups", Communications in Algebra, 2005.
Outside of the identity element I think the ones you have given provide an exhaustive list. [Edit: I meant to say that from this one can deduce a list for all adjoint groups. The example in $SO_{2n+1}$ arises from the exceptional situation where $C_G(s)^{\circ}$ is of type $D_n$. You should check Tables 2 and 3 in Bonnafé's article to confirm what I claim here.]
answered Jan 15, 2014 at 12:35
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1$\begingroup$ Great! The reference to Bonnafe I believe does say that I got all the examples. It is of course clear that the elements s that I was looking at had a centralizer whose connected component must be a semi-simple group containing the maximal torus, so I was in what's called the Borel-Siebenthal situation with the extra knowledge that the Weyl group of C(s) is W. I knew among classical groups this was possible only for O(2n) < SO(2n+1). However, I was not sure on the Exceptional groups. The paper of Bonnafe I believe shows this phenomenon does not happen for exceptional groups. Thank you! $\endgroup$ Commented Jan 15, 2014 at 13:11