  On a compact K\"ahler manifold there is a canonical action of a
Lie-superalgebra on the space of differential forms. It is generated by the
differentials, the Lefschetz operator and the adjoints of these operators. We
determine the asymptotic distribution of irreducible representations of this
Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because
of the high degree of symmetry the Laplace-Beltrami operator on forms can not
be quantum ergodic. We show that after taking these symmetries into account
quantum ergodicity holds for the Laplace-Beltrami operator and for the
Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions
for our theorem are known to be satisfied for instance for negatively curved
K\"ahler manifolds of odd complex dimension.
