Geometric quantizations of mixed polarizations on K\"ahler manifolds with T-symmetry

Abstract

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.