  It is known that the hard Lefschetz action, together with K\"ahler identities
for K\"ahler (resp. hyperk\"ahler) manifolds, determines a
$\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra
action on differential forms. In this paper, we explain the geometric origin of
this action, and we also generalize it to manifolds with other holonomy groups.
For semi-flat Calabi-Yau (resp. hyperk\"ahler) manifolds, these symmetries can
be enlarged to a $\mathfrak{so}(2,2)_{sup}$ (resp. $\mathfrak{su}(2,2)_{sup}$)
action.
