rupert's question

coincidentally, someone named rupert posted a question to the newsgroup sci.math yesterday or so:

Suppose that G is an almost simple algebraic group. What would the
unipotent radical of a parabolic subgroup look like, in general?

(this qualifies as a coincidence because this past week or so i've been thinking about such unipotent radicals, even though it didn't occur to me that that's what they are until after i read rupert's question- i'd been thinking about them from a different point of view.

i guess that i'll try to compose an answer to the question here. obviously it's not ready for prime time yet and probably has a fair amount of mistakes, though, so i don't know whether i'll get around to posting it to the newsgroup.)

can you give some idea of what kind of answer you're fishing for? i can try to give you my current take on it, based mostly on stuff that i've been thinking about in the past week or so, but i could imagine some pretty different sorts of answers as well.

(i'll probably be somewhat careless here about the distinction between lie groups and lie algebras, in part because a lot of what i want to talk about involves root systems. for example i'll probably be talking more directly about "nilpotent radicals of parabolic subalgebras" than about "unipotent radicals of parabolic subgroups".)

first of all, notice that as the parabolic subalgebra gets bigger it gets "closer to being semi-simple", so its nilpotent radical actually gets smaller. (for example the biggest parabolic subalgebra is the simple lie algebra itself.) another nilpotent subalgebra that gets smaller as the parabolic subalgebra p gets bigger is the killing-orthogonal complement of p, which in fact is not only isomorphic to the nilpotent radical of p, but is itself the nilpotent radical of the isomorphic-but-"opposite" parabolic subalgebra to p.

thus instead of talking about the nilpotent radical of p, i can equivalently talk about the killing-orthogonal complement of p, and i'll do that because that's the way that i've been thinking about these subalgebras, as killing-orthogonal complements of parabolic subalgebras rather than as unipotent radicals of them.

the killing-orthogonal complement of p is naturally identified with the tangent space of the partial flag variety associated with p. and my basic answer to your question is this: the structure of the killing-orthogonal complement of p as a multi-graded nilpotent lie algebra tells you about the invariant distributions (in the differential geometric sense) on the partial flag variety, and conversely, the nature of those invariant distributions tells you the structure of the multi-graded nilpotent lie algebra.

(the relationship between graded nilpotent lie algebras and distributions is discussed in for example:

N. Tanaka, On the differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ. , v. 10, 1-82, 1970.

actually i'm not sure how useful a reference this is, though.)

for example, when the partial flag variety is a hermitian symmetric space, its geometry is "isotropic" and so there are no invariant distributions; thus this is the case where the killing-orthogonal complement of p is not just a nilpotent lie algebra but an abelian one.