Geometric quantizations of mixed polarizations on Kähler manifolds with T-symmetry

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