  We describe the components of the moduli space of conjugacy classes of
commuting pairs and triples of elements in a compact Lie group. This
description is in terms of the extended Dynkin diagram of the simply connected
cover, together with the coroot integers and the action of the fundamental
group. In the case of three commuting elements, we compute Chern-Simons
invariants associated to the corresponding flat bundles over the three-torus,
and verify a conjecture of Witten which reveals a surprising symmetry involving
the Chern-Simons invariants and the dimensions of the components of the moduli
space.
