Limit of geometric quantizations on Kähler manifolds with T‐symmetry

Abstract

A compact Kähler manifold with ‐symmetry admits a natural mixed polarization whose real directions come from the ‐action. In Leung and Wang [Adv. Math. 450 (2024), 109756], we constructed a one‐parameter family of Kähler structures ’s with the same underlying Kähler form and , such that (i) there is a ‐equivariant biholomorphism between and and (ii) Kähler polarizations ’s corresponding to ’s converge to as goes to infinity. In this paper, we study the quantum analog of above results. Assume that is a prequantum line bundle on . Let and be quantum spaces defined using polarizations and , respectively. In particular, . They are both representations of . We show that (i) there is a ‐equivariant isomorphism between and and (ii) for regular ‐weight , corresponding ‐weight spaces ’s converge to as goes to infinity.