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\(G\)-invariant Szegő kernel asymptotics and CR reduction. (English) Zbl 1467.32016

Authors’ abstract: Let \((X, T^{1,0}X)\) be a compact connected orientable CR manifold of dimension \(2n+1\) with non-degenerate Levi curvature. Assume that \(X\) admits a connected compact Lie group \(G\) action. Under certain natural assumptions about the group \(G\) action, we show that the \(G\)-invariant Szegő kernel for \((0, q)\) forms is a complex Fourier integral operator, smoothing away \(\mu^{-1}(0)\) and there is a precise description of the singularity near \(\mu^{-1}(0)\), where \(\mu\) denotes the CR moment map. We apply our result to the case when \(X\) admits a transversal CR \(S^1\) action and deduce an asymptotic expansion for the \(m\) th Fourier component of the \(G\)-invariant Szegő kernel for \((0, q)\) forms as \(m\rightarrow +\infty\) and when \(q=0\), we recover Xiaonan Ma and Weiping Zhang’s result about the existence of the \(G\)-invariant Bergman kernel for ample line bundles. As an application, we show that if \(m\) large enough, quantization commutes with reduction.

MSC:

32V20 Analysis on CR manifolds
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