ADJOINT SPACES AND FLAG VARIETIES OF p-COMPACT GROUPS

For a compact Lie group G with maximal torus T, Pittie and Smith showed that the flag variety G/T is always a stably framed boundary. We generalize this to the category of p-compact groups, where the geometric argument is replaced by a homotopy theoretic argument showing that the class in the stable homotopy groups of spheres represented by G/T is trivial, even G-equivariantly. As an application, we consider an unstable construction of a G-space mimicking the adjoint representation sphere of G inspired by work of the second author and Kitchloo. This construction stably and G-equivariantly splits off its top cell, which is then shown to be a dualizing spectrum for G.