Let (X, J, ω) be a complete 2n-dimensional almost Kähler manifold. First part of this article, we construct some identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of ker ∆ ∂ ∩ ker ∆ ∂ on pure bidegree. In the second part, as some applications of those identities, we establish some vanishing theorems on the spaces of L 2 -harmonic (p, q)-forms on X under some growth assumptions on the Käher form ω. We also give some L 2 -estimates to sharpen the vanishing theorems in two specific cases. At last of the article, as an application, we study the topology and geometry of the compact almost Kähler manifold with negative sectional curvature.