Weyl group invariants in a maximal torus 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.  
 A: 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.]
