MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

A compact connected semisimple Lie Group $G$ has an essentially unique maximal torus $T$, a maximal abelian subgroup of maximum dimension (the rank of $G$ is the dimension of this torus). Although $G$ has lots of such torii (in fact any element of $G$ is contained in at least one), any two are conjugate to one another by some element of $G$.

In a similar vein, one can break a given maximal torus $T$ up into congruent pieces (the images of Weyl chambers under the exponential map applied to the Lie algebra of $T$), any two of which are equivalent to one another by an element of the Weyl Group of $G$. The value of any class function on $G$ is then completely determined on all of $G$ by its values on a single one of these pieces.