Limit of geometric quantizations on Kähler manifolds with T-symmetry

A compact Kähler manifold $(M,\omega ,J)$ with $T$-symmetry admits a natural mixed polarization ${\mathcal{P}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ whose real directions come from the $T$-action. In \cite{LW1}, we constructed a one-parameter family of Kähler structures $(\omega ,J_t)$'s with the same underlying Kä hler form $\omega$ and $J_0=J$, such that (i) there is a $T$-equivariant biholomorphism between $(M,J_0)$ and $(M,J_t)$ and (ii) Kähler polarizations ${\mathcal{P}}_t$'s corresponding to $J_t$'s converge to ${\mathcal{P}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ as $t$ goes to infinity. In this paper, we study the quantum analog of above results. Assume $L$ is a pre-quantum line bundle on $(M,\omega )$. Let ${\mathcal{H}}_t$ and ${\mathcal{H}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ be quantum spaces defined using polarizations ${\mathcal{P}}_t$ and ${\mathcal{P}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ respectively. In particular, ${\mathcal{H}}_t=H_{{\bar{\mathrm{\partial }}}_t}^0(M,L)$. They are both representations of $T$. We show that (i) there is a $T$-equivariant isomorphism between ${\mathcal{H}}_0$ and ${\mathcal{H}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ and (ii) for regular $T$-weight $\lambda$, corresponding $\lambda$-weight spaces ${\mathcal{H}}_{t,\lambda }$'s converge to ${\mathcal{H}}_{\mathrm{m}\mathrm{i}\mathrm{x},\lambda }$ as $t$ goes to infinity.