Parabolic subgroup of a Lie group (as oppose to algebraic group) Let $G$ be a semisimple Lie group (real or complex). What does it mean for $P\le G$ to be a parabolic subgroup?
I only know that if $G$ is an algebraic group then $P\le G$ means that $G/P$ is a complete variety. But I don't see how to define it for a semisimple Lie group
Please see for example, the first page of paper "recurrence properties of random walks on finite volume homogeneous manifolds" by Eskin and Margulis for the notion of a parabolic subgroup of a Lie group (they didn't give the definition, though).
 A: Most likely, they will use geometrically defined parabolic subgroups. You can find a detailed definition for instance here:
Eberlein, Patrick B., Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics. Chicago, IL: The University of Chicago Press. 449 p. (1996). ZBL0883.53003.
The upshot is: Let $G$ be a connected semisimple Lie  group, without  compact factors, and finite center; let $K<G$ be a maximal compact subgroup. Then $X=G/K$ has natural structure of a symmetric space of noncompact type on which $G$ acts isometrically. The space $X$ has a compactification $X\cup \partial X$ defined using equivalence classes of geodesic rays in $X$. The visual boundary $\partial X$ has two natural topologies; the relevant one is the Tits topology, giving $\partial X$ structure of a Tits building ${\mathcal B}$ (this building is noncompact, the compactification of $X$ you obtain using another topology). Then parabolic subgroups of $G$ are stabilizers of simplices in ${\mathcal B}$. (With this definition, the group $G$ itself does not count as parabolic.)
To relate this to an algebraic definition: The group $G$ itself might not be algebraic, but, if you replace it with the corresponding adjoint group (divide by $G$ its finite center) $Ad(G)$, then $Ad(G)$ is commensurable to the group of real points $\underline{G}({\mathbb R})$ of an algebraic group $\underline{G}$. The parabolic subgroups of $\underline{G}({\mathbb R})$ (defined algebraically) then will be commensurable to parabolic subgroups of $Ad(G)$ defined geometrically as above: The Tits building of $\underline{G}({\mathbb R})$ will be equivariantly isomorphic to the Tits building ${\mathcal B}$ above. Commensuration amounts to passing to finite index subgroups and does not change the dynamical features that Eskin and Margulis (and many others doing homogeneous dynamics) are interested in.
