  We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild
cohomology of generalised Grassmannians, i.e. partial flag varieties associated
to maximal parabolic subgroups in a simple algebraic group. We explain how the
decomposition is concentrated in global sections for so-called (co)minuscule
and (co)adjoint generalised Grassmannians, and we suggest that for (almost) all
other cases the same vanishing of the higher cohomology does not hold. Our
methods give an explicit partial description of the Gerstenhaber algebra
structure for the Hochschild cohomology of cominuscule and adjoint generalised
Grassmannians. We also consider the case of adjoint partial flag varieties in
type A, which are associated to certain submaximal parabolic subgroups.
