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Automorphisms of nondegenerate CR quadrics and Siegel domains. Explicit description. (English) Zbl 1042.32014
Let \(z= (z^1,\dots, z^n)\), \(w= (w^1,\dots, w^k)\) be coordinates in \(\mathbb{C}^{n+k}\), \(k\geq 1\), and \[ \langle z,z\rangle= (\langle z,z\rangle^1,\dots, \langle z,z\rangle^k) \] be a \(\mathbb{C}^k\)-valued Hermitian form on \(\mathbb{C}^n\). Let \(C\) be the interior of the convex hull of \(\{\langle z,z\rangle: z\in\mathbb{C}^n\}\) and suppose,that \(C\) is an acute cone, i.e. \(C\) does not contain any entire line. Let \(V\supset C\) be an open acute cone in \(\mathbb{R}^k\). The domain \(\Omega_V= \{(z,w)\in \mathbb{C}^{n+k}: \text{Im\,}w- \langle z,z\rangle\in V\}\) is called a Siegel domain of the second kind associated with \(V\), while the quadric \(Q= \{(z, w)\in \mathbb{C}^{n+k}: \text{Im\,}w= \langle z,z\rangle\}\) forms the Silov boundary of \(\Omega_V\).
After recalling the non-degenerateness of quadrics , the authors give an explicit formula for one-parameter groups of automorphisms of arbitrary nondegenerate quadrics and for the automorphisms of Siegel domains of second kind. The authors also introduce a family of \(k\)-dimensional chains, which are analogs of one-dimensional Chern-Moser chains for hyper-quadrics and clarify their structure, which is used for the proof of the main results.

32V20 Analysis on CR manifolds
32N05 General theory of automorphic functions of several complex variables
Full Text: DOI
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