# $\mathbb R$-points of semisimple real algebraic groups, connectivity, and Cartan involutions: some questions

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 $$G$$?

• 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.