Neron-Severi Lie algebra, group of autoequivalences of the derived category, and monodromy

Let X be a smooth complex projective manifold, $G^{eq}(X)\subset GL(H(X))$ – the image of the group $Aut(D(X))$ in the group of automorphisms of $H(X)$. First I will explain about the relation of the group $G^{eq}(X)$ and the Neron-Severi Lie algebra. Then I plan to discuss the conjecture of Kontsevich on the relation of the group $G^{eq}(X)$ (if $X$ is CY) with the monodromy group of the mirror symmetric family.