# Adjoint spaces and flag varieties of $p$-compact groups

### Abstract

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. We replace the geometric argument by a homotopy-theoretic one, showing that the framed bordism class represented by $G/T$ is trivial, even $G$-equivariantly. As an application and inspired by work by the second author and Kitchloo, we consider an unstable construction of a $G$-space mimicking the adjoint representation sphere of $G$. Stably and $G$-equivariantly, this construction splits off its top cell, which we then shown to be a dualizing spectrum for $G$.

Publication
Preprint
Date