Signature Pairs for Group-Invariant Hermitian Polynomials

Abstract

Abstract. We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of SU (2). We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of U (2). We calculate the signature pair for dihedral subgroups of U (2).

A multiplicative analogue of the Reynolds operator and construction of invariants

“…where ω is a sixth root of unity. This matrix group was examined by Grundmeier in this doctoral thesis [6].…”

Heisenberg quasiregular ellipticity

“…provides a bijection between H and S 3 \ {(0, −1)}, and is furthermore well-known to be both contactomorphic and conformal, see for instance [49, p. 315] More generally, if Γ is any group of isometries of the sub-Riemannian S 3 such that S 3 /Γ is a smooth manifold and π : S 3 → S 3 /Γ denotes the quotient map, then the composition π • ι is a quasiregular mapping from H to S 3 /Γ with its standard contact structure and sub-Riemannian metric. See [19] and [33] for examples of finite isometry groups of S 3 arising in the study of proper holomorphic mappings between balls and CR representation theory.…”

Invariant CR mappings between hyperquadrics

“…CR mappings invariant under finite group actions yield many interesting connections with other areas of mathematics, including number theory, combinatorics, algebraic geometry, and representation theory (see for instance [3,4,6,7,11,12,14] and the references therein). In this case, let Γ be a finite subgroup of an indefinite unitary group SU (a, b) or U (a, b) (defined precisely in Section 2).…”

“…The problem of constructing group-invariant CR mappings has attracted substantial interest over the years (see [2][3][4][6][7][8][9][10][11][12][13] and the references therein). D'Angelo and Lichtblau [5] gave a canonical construction of invariant-polynomial CR mappings, and they used this construction to study the CR spherical space form problem.…”

“…D'Angelo and Lichtblau [5] gave a canonical construction of invariant-polynomial CR mappings, and they used this construction to study the CR spherical space form problem. The first author [11,12] found the target hyperquadric for this canonical construction for all subgroups of SU (2). In the present paper we build on this work in the case of the source hyperquadric Q(1, 1).…”