Primitive decomposition of Bott–Chern and Dolbeault harmonic (k, k)-forms on compact almost Kähler manifolds

We consider the primitive decomposition of?,?, Bott-Chern and Aeppli-harmonic(k,k)-forms on compact almost K & auml;hler manifolds(M,J,?). For any D?{?,?,BC,A}, it is known that theLkP0,0component of??Hk,kDis a constant multiple of?kup to real dimension 6. In this paper we generalise this result to every dimension. We also deduce information on the componentsLk-1P1,1andLk-2P2,2of the primitive decomposition. Focusing on dimension 8, we give a full description of the spacesH2,2BCandH2,2A, from which followsH2,2BC?H2,2?andH2,2A?H2,2?.Wealso provide an almost K & auml;hler 8-dimensional example where the previous inclusions are strict and the primitive components of a harmonic form??Hk,kDare not D-harmonic, showing that the primitive decomposition of(k,k)-forms in general does not descend to harmonic forms