On the linearization of the automorphism groups of algebraic domains

Let $D$ be a domain in $C^n$ and $G$ a topological group which acts effectively on $D$ by holomorphic automorphisms. In this paper we are interested in projective linearizations of the action of $G$, i.e. a linear representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$ into $\P^N$ with respect to this representation. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. Assume that the group $G$ acts by birational automorphisms. Our main result is the equivalence of the following conditions: 1) there exists a projective linearization, i.e. a linear representation of $G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $G$-equivariant. 2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ and has finitely many connected components; 3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$; 4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times D\to D$ extends to a Nash action $\hat G\times D\to D$; 5) the degree of the automorphism $\phi_g\colon D\to D$