Lefschetz decompositions for eigenforms on a Kähler manifold

Abstract: We show that the eigenspaces of the Laplacian $\Delta_k$ on $k$-forms on a compact K\"ahler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of $\Delta_k$ lies in the spectrum of $\Delta_{k+1}$.