On the almost algebraicity of groups of automorphisms of connected Lie groups

Let $G$ be a connected Lie group, $C$ be the maximal compact connected subgroup of the center of $G$, and let Aut$(G)$ denote the group of Lie automorphisms of $G$, viewed, canonically, also as a subgroup of GL$(\mathfrak{G})$, where $\mathfrak{G}$ is the Lie algebra of $G$. It is known (see Dani (1992) and Previts-Wu (2001)) that when $C$ is trivial Aut$(G)$ is almost algebraic, in the sense that it is open in an algebraic subgroup of GL$(\mathfrak{G})$, and in particular has only finitely many connected components. In this paper we analyse the situation further in this respect, with $C$ possibly nontrivial, and identify obstructions for almost algebraicity to hold; the criteria are in terms of the group of restrictions of automorphisms of $G$ to $C$, and the abelian quotient Lie group $G\mathrm{/}\overline{[G,G]}C$ (see Theorem 1.1 for details). For the class of Lie groups which admit a finite-dimensional representation with discrete kernel (called class $\mathcal{C}$ groups) this yields a more precise description as to when Aut$(G)$ is almost algebraic (see Corollary 1.3), while in the general case a variety of patterns are observed (see §6). Along the way we also study almost algebraicity of subgroups of Aut$(G)$ fixing each point of a given torus in $G$, containing $C$ (see in particular Theorem 1.5), which also turns out to be of independent interest.