  Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative
if and only if $n=3$. We generalize this result to that the loop spaces of all
irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy
commutative. The computation also applies to determining the homotopy
nilpotency of the loop spaces of flag manifolds.
