# Proper morphisms: Lie groups vs. group schemes

A Lie group can (often) be recovered as the $\mathbb{R}$-points of a group scheme. I am wondering if this parallelism carries over to proper actions.

In particular, let $G$ be a Lie group acting on a manifold $M$. Let $\underline G$ be the associated group scheme acting on the scheme $\underline M$, where the $\mathbb{R}$-points of $\underline M$ can be identified with $M$.

Could it be true that the map $G\times M\to M\times M$ is proper (in the topological sense) if and only if $\underline G\times \underline M\to \underline M\times \underline M$ is a proper morphism of schemes?

• Hm. How is a manifold the set of real points of a scheme? – Mariano Suárez-Álvarez May 13 '12 at 4:23
• For example when $M$ is itself a Lie group. (This should work, no?) Otherwise I could try to reformulate the question for a complex Lie group acting on a complex manifold $M$ and require that $\underline M$ be smooth, so that the $\mathbb{C}$-points of $\underline M$ can be identified with $M$. – Earthliŋ May 13 '12 at 7:50
• @Earthliŋ, "when $M$ is itself a Lie group": the adjective for Lie groups that arise in this way is 'linear'. Some Lie groups, like the metaplectic groups, are only covers of linear Lie groups. – LSpice Mar 29 at 23:21

A morphism of schemes over $\mathbb C$ is proper iff the map on topological spaces is proper, so the whole thing works for complex points if the fibred products are over $\mathbb C$ as well. Over the reals it goes wrong as the following example shows: Let $G$ be the group scheme $SO(2)$ and let $M$ be a single point. The real points of $G$ form a compact group, so the map $G({\mathbb R})\times pt\to pt\times pt$ is proper, but the corresponding map of schemes is not, as $G$ is affine of positive dimension.