\(G\)-invariant Bergman kernel and geometric quantization on complex manifolds with boundary (Q6634451)

The setting for this article is a domain \(M\) with smooth boundary \(X\) in an ambient complex manifold that admits a holomorphic action by a compact Lie group \(G\). Assuming that the group action preserves \(X\), and the reduced boundary space is strongly pseudoconvex (or nondegenerate), the authors obtain an asymptotic expansion for the \(G\)-invariant Bergman kernel as well as a \(G\)-invariant version of \textit{C. Fefferman}'s theorem on boundary regularity of biholomorphic maps [Invent. Math. 26, 1--65 (1974; Zbl 0289.32012)]. Additionally, the authors prove in this setting a version of the principle ``quantization commutes with reduction'' in the sense of \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 515--538 (1982; Zbl 0503.58018)]. Some related results about quantization of CR manifolds can be found in an article by the first author et al. [Commun. Contemp. Math. 25, No. 10, Article ID 2250074, 73 p. (2023; Zbl 1535.53086)].