The parallel Bismut torsion condition and the isotropy representation of flag manifolds

A classification of all invariant Hermitian structures with parallel Bismut torsion on most flag manifolds is given. More precisely, we prove that on a flag manifold $M=G\mathrm{/}H$ such that $G$ is simple and different from ${\mathrm{E}}_6$, ${\mathrm{E}}_7$, ${\mathrm{E}}_8$ if $H$ is non-abelian, a $G$-invariant Hermitian structure on $M$ has parallel Bismut torsion if and only if the metric is either Kähler or a multiple of the Killing or standard metric.