Do absolutely irreducible group actions have odd dimensional fixed point spaces?

In his volume [5] on "Symmetry Breaking for Compact Lie Groups" Mike Field quotes a private communication by Jorge Ize claiming that any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. The proof should come from the fact that any absolutely irreducible representation possesses an odd dimensional fixed point space. In this paper we show that there are many examples of groups which have absolutely irreducible representations but no odd dimensional fixed point space. This observation may be relevant also for some degree theoretic considerations concerning equivariant bifurcation. Moreover we show that our examples give rise to some interesting Hamiltonian dynamics and we show that despite some complications we can go a long way towards doing explicit computations and providing complete proofs. For some of the invariant theory needed we will depend on some computer aided computations. The work presented here greatly benefited from the computer algebra program GAP [6], which is an indispensable aid for doing the required group theory computations.