# An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if

1. $P$ is Lagrangian

2. P involutive

3. dim$P\cap\bar P \cap TM$ is constant

This definition shows that every complex polarization induces a real isotrpic distribution $D:=P\cap\bar P \cap TM$ which is also involutive by Frobenius theorem. Moreover the complexification of distribution $D$ is $D^{\mathcal{C}}=P\cap \bar P$ and it is called isotropic distribution. Now we define the subbundle $E:=(P+\bar P)\cap TM$ and $E^{\mathcal{C}}=P+ \bar P$. Notice that orthogonal symplectic complement of $D$ is $E$, i.e., $D^{\perp}=E$ and $E$ is called coisotrpic distribution. We know that polarized sections are covariantly constant along the leaves of $D$,.

I am looking for an example to show that when these leaves be non-compact then these polarized sections are not square integrable with respect to volume form $\omega^n$.

The standard example is the case when $M=T^*Q$ is a cotangent bundle over a manifold $Q$ with its usual symplectic form, and $P$ is the vertical polarization. Then the isotropic and coisotropic polarizations agree (both equaling the vertical polarization), and their integral manifolds are the fibers of the bundle projection $T^*Q\rightarrow Q$. For any polarized section, the integration with respect to $\omega^n$ along these fibers will give an infinite contribution. Most books on geometric quantization include explicit formulas for the case $Q=\mathbb{R}^n$.