Suspension splittings and self-maps of flag manifolds

If $G$ is a compact connected Lie group and $T$ is a maximal torus, we give a wedge decomposition of $\mathrm{\Sigma }G\mathrm{/}T$ by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of $G\mathrm{/}T$.