Orbits of tori extended by finite groups and their polynomial hulls: the case of connected complex orbits

Abstract

Let $V$ be a complex linear space, $G\subset\GL(V)$ be a compact group. We consider the problem of description of polynomial hulls $\wh{Gv}$ for orbits $Gv$, $v\in V$, assuming that the identity component of $G$ is a torus $T$. The paper contains a universal construction for orbits which satisfy the inclusion $Gv\subset T^\bbC v$ and a characterization of pairs $(G,V)$ such that it is true for a generic $v\in V$. The hull of a finite union of $T$-orbits in $T^\bbC v$ can be distinguished in $\clos T^\bbC v$ by a finite collection of inequalities of the type $\abs{z_1}^{s_1}...\abs{z_n}^{s_n}\leq c$. In particular, this is true for $Gv$. If powers in the monomials are independent of $v$, $Gv\subset T^\bbC v$ for a generic $v$, and either the center of $G$ is finite or $T^\bbC$ has an open orbit, then the space $V$ and the group $G$ are products of standard ones; the latter means that $G=S_nT$, where $S_n$ is the group of all permutations of coordinates and $T$ is either $\bbT^n$ or $\SU(n)\cap\bbT^n$, where $\bbT^n$ is the torus of all diagonal matrices in $\rU(n)$. The paper also contains a description of polynomial hulls for orbits of isotropy groups of bounded symmetric domains. This result is already known, but we formulate it in a different form and supply with a shorter proof.