Brane involutions on irreducible holomorphic symplectic manifolds

In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists' language, i.e. a submanifold which is either complex or lagrangian submanifold with respect to each of the three K\"ahler structures of the associated hyperk\"ahler structure. Starting from a brane involution on a K3 or abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier--Mukai transform. Later, we recall the lattice-theoretical approach to Mirror Symmetry. We provide two ways of obtaining a brane involution on the mirror and we study the behaviour of the brane involutions under both mirror transformations, giving examples in the case of a K3 surface and $K3^{[2]}$-type manifolds.