Deformations of Lagrangian NQ-submanifolds

In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian NQ-submanifolds of degree n symplectic NQ-manifolds. Using Weinstein's Lagrangian tubular neighbourhood Theorem, we attach to every Lagrangian NQ-submanifold an $L_{\infty }$-algebra, which controls its deformation theory. The main examples are coisotropic submanifolds of Poisson manifolds and (higher) Dirac structures with support in (higher) Courant algebroids.