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 $G$.