I'm interested in the following question: let $k$ be a field of characteristic zero (just for simplicity), $G$ a connected semi-simple $k$-group, $P\subsetneq G$ a parabolic $k$-subgroup, and $H\subsetneq G$ a connected $k$-subgroup. Assume that $H\nsubseteq P$, then does the intersection $H\cap P$ becomes a parabolic $k$-subgroup for $H$? One might even assume that $H$ is reductive.

veryrarely the case. – Torsten Ekedahl Dec 27 '10 at 15:56`$H$`

(reductive or not) is that they can be extremely small relative to`$G$`

and not well-placed in terms of Lie-theoretic structure. Think of`$G$`

as a huge special linear group, for instance. In other words, your question is too loosely formulated to have an interesting answer. – Jim Humphreys Dec 27 '10 at 16:11