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

Let $V$ be a complex linear space, $G\subset \text{GL}(V)$ be a compact group. We consider the problem of description of polynomial hulls $\text{wh}Gv$ 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 $Gv\subset T^{\text{bbC}}v$ 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 $T^{\text{bbC}}v$ can be distinguished in $\text{clos}{T}^{\text{bbC}}v$ by a finite collection of inequalities of the type $\text{abs}{z_1}^{s_1}\mathrm{.}\mathrm{.}\mathrm{.}\text{abs}{z_n}^{s_n}\le c$. In particular, this is true for $Gv$. If powers in the monomials are independent of $v$, $Gv\subset T^{\text{bbC}}v$ for a generic $v$, and either the center of $G$ is finite or $T^{\text{bbC}}$ has an open orbit, then the space $V$ and the group $G$ are products of standard ones; the latter means that $G=S_{n}T$, where $S_n$ is the group of all permutations of coordinates and $T$ is either ${\text{bbT}}^n$ or $\text{SU}(n)\cap {\text{bbT}}^n$, where ${\text{bbT}}^n$ is the torus of all diagonal matrices in $\text{rU}(n)$. 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.