Automorphisms of nondegenerate CR quadrics and Siegel domains. Explicit description.
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.
scientific article; zbMATH DE number 1777176