I am reading about Cartan involutions on semisimple real Lie groups and have a point of confusion I am trying to reconcile with linear algebraic groups. Let $\mathbf G$ be a linear algebraic group over $\mathbb R$. Assume that $\mathbf G$ is semisimple, i.e. $\mathbf G \times_{\mathbb R} \operatorname{Spec} \mathbb C$ is semisimple as an algebraic group (is Zariski-connected and has trivial radical). This implies that $\mathbf G$ is also connected in the Zariski topology.

Let $G = \mathbf G(\mathbb R)$, which is a real Lie group.

  • $G$ is in general not connected, right? For example, $\mathbf G = \operatorname{SO}(p,q)$. Does $G$ have finitely many components?

  • A real Lie group is defined to be semisimple if it is connected and if its Lie algebra is semisimple (has nondegenerate Killing form). The connected component $G^0$ of $G$ is a semisimple real Lie group, right?

  • Let $\mathfrak g$ be the Lie algebra of the algebraic group $\mathbf G$. Then do we have $\mathfrak g(\mathbb R) = \operatorname{Lie}(G)$?

  • Let $\theta$ be a Cartan involution on $\operatorname{Lie}(G)$. There is a corresponding involution Lie group automorphism $\Theta$ of $G^0$ whose differential is $\theta$, as in the Wikipedia article on Cartan decomposition. Does $\Theta$ extend to an automorphism of $G$?

  • Let $K$ be the maximal compact subgroup of $G^0$ obtained as the fixed points of $\Theta$. Is there a way to realize $K$ as the intersection with $G^0$ of a canonical maximal compact subgroup of $G$?


1 Answer 1


In a slightly different order:

  • For future visitors to this problem, here an answer about connectivity on MO.

  • Yes, the functor from real linear algebraic groups to real Lie groups respects Lie algebras. See these notes by Milne, section III.2.

  • In characteristic zero, a linear algebraic group is semisimple iff its Lie algebra is semisimple. By the above point, $G^0$ is a semisimple real Lie group. (See the same notes, section II.4)

  • One can include a fixed choice of Cartan involution in the defintion of reductive real Lie group, and hence in the definition of semisimple real Lie group (as a reductive group with finite centre). A good choice is inverse-conjugate transpose, and then $K$ is essentially just the intersection of $G^0$ with the relevant orthogonal or unitary group, all as subgroups of some ambient $\mathrm{GL}(n,\mathbb{R})$. This is the approach taken by Knapp in his book Representation Theory of Semisimple Groups. This simplification might help answer the last part of your question.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .