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.