On subgroups of an automorphism group of an irreducible symplectic manifold

Let X be an irreducible symplectic manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost abelian. We give a formula of the rank of Aut(X,L) in terms of MBM divisors. We also prove that the nef cone of X cut out MBM classes, which is a generalization of Kovac's structure theorem of nef cones of K3 surfaces