Suppose I have a symplectic manifold $M$, and have a deformation quantization of it, i.e. an associative product $\ast:C(M)[[\hbar]]\otimes C(M)[[\hbar]]\to C(M)[[\hbar]]$ so that $f\ast g=fg+\{f,g\}\hbar+O(\hbar^2)$, where $\{\cdot,\cdot\}:C(M)\otimes C(M)\to C(M)$ is the Poisson bracket coming from the symplectic structure.

According to Lu (http://www.ams.org/mathscinet-getitem?mr=1244874 page 395), a Lagrangian submanifold $L\subseteq M$ should correspond to a left-module over $C(M)[[\hbar]]$, that is, we have a deformed module structure $\ast:C(M)[[\hbar]]\otimes C(L)[[\hbar]]\to C(L)[[\hbar]]$, which reduces to the standard module structure when $\hbar=0$, and is compatible with the start product on $C(M)[[\hbar]]$. (Actually, Lu thought of the left-ideal $I_L\subseteq C(M)[[\hbar]]$ so that $C(L)[[\hbar]]=C(M)[[\hbar]]/I_L$, but I perfer to think of the deformation of the module structure.)

Then we can write, for $f\in C(M)$ and $g\in C(L)$, that $f\ast g=fg+\{f,g\}\hbar+O(\hbar^2)$, for some brackets $\{\cdot,\cdot\}:C(M)\otimes C(L)\to C(L)$.

What are these brackets? How are the related to the symplectic structure?

It seems that in general the existence of a deformed module structure on $C(L)[[\hbar]]$ is nontrivial (perhaps not even known for arbitrary $L$?), but I am hoping that perhaps the deformation to first order is something that one can understand more easily.