Reduction of symmetries of Dirac structures

We consider a Dirac structure D on a manifold Q invariant under a proper action of a Lie group G on Q. Our aim is to describe the structure of the orbit space D/G in terms of the structure of Q/G. If the action of G on Q is free, then Q is a left principal fibre bundle with structure group G and base manifold Q/G. We denote by Q[g] the adjoint bundle and by Q[g*] the coadjoint bundle of Q. We show that for a free and proper action, D/G is a maximal isotropic subbundle of the direct sum Q[g] circle plus Q[g*] circle plus T(Q/G) circle plus T*(Q/G). If the action of G on Q is proper but not free, then the orbit space Q/G is stratified. For each stratum of Q/G, the restriction of D/G to the stratum can be described in the same way as for a free and proper action.