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?

  • 4
    $\begingroup$ Hm. How is a manifold the set of real points of a scheme? $\endgroup$ – Mariano Suárez-Álvarez May 13 '12 at 4:23
  • $\begingroup$ 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 $M$. $\endgroup$ – Earthliŋ May 13 '12 at 7:50
  • $\begingroup$ @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. $\endgroup$ – 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.

  • $\begingroup$ Thanks for such a clear answer. Is there any chance of studying proper actions of real Lie groups in the language of group schemes? The obvious candidate, proper actions of schemes, seems to be out the window... $\endgroup$ – Earthliŋ May 14 '12 at 2:10
  • $\begingroup$ Do you happen to have a reference for your first sentence? $\endgroup$ – Earthliŋ May 14 '12 at 8:20
  • $\begingroup$ See SGA 1, Exposé XII. $\endgroup$ – user1688 May 15 '12 at 19:57

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