  We define a spectral flow for paths of selfadjoint Fredholm operators that
are equivariant under the orthogonal action of a compact Lie group as an
element of the representation ring of the latter. This $G$-equivariant spectral
flow shares all common properties of the integer valued classical spectral
flow, and it can be non-trivial even if the classical spectral flow vanishes.
Our main theorem uses the $G$-equivariant spectral flow to study bifurcation of
periodic solutions for autonomous Hamiltonian systems with symmetries.
