 Injectivity of powers of the Lefschetz operator

The Hermitian Lefschetz operator has injective powers in the indicated range. The commutator formula for powers of the Lefschetz operator proves this by induction on $r+k$: a form in the kernel can be written as $L\alpha$ using injectivity of the smaller power, and the lower-degree form $\alpha$ is then in the kernel of another power covered by the induction hypothesis.