Let $M$ be a compact K\"ahler manifold equipped with a pre-quantum line bundle $L$. In [9], using $T$-symmetry, we constructed a polarization $\mathcal{ P}_{ \mathrm{ mix}}$ on $M$, which generalizes real polarizations on toric manifolds. In this paper, we obtain the following results for the quantum space $\mathcal{ H}_{ \mathrm{ mix}}$ associated to $\mathcal{ P}_{ \mathrm{ mix}}$. First, $\mathcal{ H}_{ \mathrm{ mix}}$ consists of distributional sections of $L$ with supports inside $\mu^{ -1}(\mathfrak{ t}^{ *}_{ \mathbb{ Z}})$. This gives $\mathcal{ H}_{ \mathrm{ mix}}=\bigoplus_{ \lambda \in \mathfrak{ t}^{ *}_{ \mathbb{ Z}} } \mathcal{ H}_{ \mathrm{ mix}, \lambda}$. Second, the above decomposition of $\mathcal{ H}_{ \mathrm{ mix}}$ coincides with the weight decomposition for the $T$-symmetry. Third, an isomorphism $\mathcal{ H}_{ \mathrm{ mix}, \lambda} \cong H^{ 0}( M//_{ \lambda}T, L//_{ \lambda}T)$, for regular $\lambda$. Namely, geometric quantization commutes with symplectic reduction.
Categories: math.SG math-ph math.MP