Enveloppes polynômiales d'ensembles compacts invariants

We consider the following problem: let V ℂ be a finite dimensional vector space, and U be a compact group of ℂ‐linear automorphisms of V ℂ . The polynomial envelope $ \hat Q $ of a compact set Q ⊂ V ℂ is defined as\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document} $$ {\hat Q} = \bigg \{ z \in V^{\mathbb C} \bigg | \quad {\rm for \, all} \quad p \in {\mathcal P} (V^{\mathbb C}), \quad \left \vert p(z) \right \vert \le {\mathop {\rm sup} \limits _{\zeta \in Q}} \left \vert p (\zeta) \right \vert \bigg \} \, , $$ \end{document}where ( V ℂ ) denotes the space of holomorphic polynomial functions on V ℂ . The problem is to determine the polynomial envelope of a compact set which is U ‐invariant. We solve the problem when U is the isotropy subgroup at the origin of the automorphism group of a bounded symmetric domain of tube type. The case of a domain of type II has been solved by C. Sacré [1992], and, for a domain of type IV, it has been solved by L. Bou Attour [1993]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)