Phases of Lagrangian-invariant objects in the derived category of an abelian variety

We continue the study of Lagrangian-invariant objects (LI-objects for short) in the derived category $D^b(A)$ of coherent sheaves on an abelian variety, initiated in arXiv:1109.0527. For every element of the complexified ample cone $D_A$ we construct a natural phase function on the set of LI-objects, which in the case $\dim A=2$ gives the phases with respect to the corresponding Bridgeland stability (see math.AG/0307164). The construction is based on the relation between endofunctors of $D^b(A)$ and a certain natural central extension of groups, associated with $D_A$ viewed as a hermitian symmetric space. In the case when $A$ is a power of an elliptic curve, we show that our phase function has a natural interpretation in terms of the Fukaya category of the mirror dual abelian variety. As a byproduct of our study of LI-objects we show that the Bridgeland's component of the stability space of an abelian surface contains all full stabilities.