Invariant CR mappings between hyperquadrics. (English) Zbl 1432.32045

Grundmeier, Dusty; Linsuain, Kemen; Whitaker, Brendan

Ill. J. Math. 62, No. 1-4, 321-340 (2018).

A hyperquadric \(Q(a,b)\) is the set \[ Q(a,b) = \left\{ z \in {\mathbb C}^{a+b} : \sum_{j=1}^a |z_j|^2 - \sum_{j=a+1}^{a+b} |z_j|^2 = 1 \right\}. \] The sphere is the set \(Q(a,0)\). These are the model hypersurfaces with Levi-form of a given signature (the sphere is the strictly pseudoconvex hypersurface). Furthermore, these hypersurfaces posses a large group of automorphisms. It is therefore interesting to study the set of possible CR mappings from \(Q(a,b)\) to \(Q(A,B)\) invariant under certain subgroups of automorphisms, and such mappings have found applications in a variety of related fields, in addition to several complex variables, such as number theory, combinatorics, representation theory, and others.
In particular, \(f : Q(a,b) \to Q(A,B)\) is invariant under \(\Gamma\), a finite subgroup of \(\operatorname{SU}(a,b)\) or \(\operatorname{U}(a,b)\), if \(f \circ \gamma = f\) for all \(\gamma \in \Gamma\). The canonical construction of such an \(f\) given by D’Angelo and Lichtblau is to average over the group in the right way. Consider \(\langle z,w \rangle_b\) be the inner product with \(b\) negative eigenvalues. Then let \(\Phi_\Gamma(z,\bar{z}) = 1 - \prod_{\gamma \in \Gamma} (1-\langle \gamma z,z \rangle_b)\). Expanding \(\Phi_\Gamma(z,\bar{z}) =\|F(z)\|^2 - \|G(z)\|^2\) gives a group invariant map \((F,G)\). Let the numbers \(N^+(\Gamma)\) and \(N^-(\Gamma)\) be the number of components of \(F\) and \(G\), which give the \(A\) and \(B\) of the target hyperquadric \(Q(A,B)\).
The purpose of the present paper is to compute the numbers \(N^+(\Gamma)\) and \(N^-(\Gamma)\) (the signature pair of \(\Gamma\)) for all finite subgroups of \(\operatorname{SU}(1,1)\). That is, to determine the signature pair for all the canonical group invariant hyperquadric maps from \(Q(1,1)\). The authors extend combinatorial results of Loehr, Warrington, and Wilf to give more precise information on the canonical invariant maps for cyclic subgroups of \(\operatorname{U}(1,1)\), in particular, they compute the asymptotic proportion of positive eigenvalues as the order goes to infinity.

Reviewer: Jiri Lebl (Stillwater)

MSC:

32V20	Analysis on CR manifolds
32V10	CR functions

Keywords:

hyperquadrics; CR mappings invariant under groups

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