so recently i've been talking a bit about stuff like invariant contact distributions and other sorts of distributions on partial flag manifolds, and in particular about how to "read off" such things from root systems. now however i'm beginning to think that it'd be nice to put this stuff in a somewhat larger context, where we consider not just "linear distributions" but also "higher-degree" ones...
i first started thinking about this (as far as i know) when thinking about the "rolling" distribution on the short-root grassmanian of g2. it occurred to me that since this distribution is really just a manifestation of the incidence geometry "lines" on the grassmanian, and since the axioms of the g2 incidence geometry are invariant under switching the long and short roots, the long-root grassmanian should display some similar manifestation. but there's no invariant linear distribution of the appropriate dimension (namely 2) on the long-root grassmanian to be read off from the root system, so we have to start looking farther afield for the corresponding manifestation, and my suspicion is that we're now dealing with an invariant "higher-degree distribution"...
but then will there be some reasonable way to "read off from the root system" such invariant higher-degree distributions? of course it's probably more complicated than in the linear case, but is it at least reasonable? maybe you just take the symmetric power of the relevant "weight diagram" and look at the relevant filtration... or something like that...
is a "lorentzian conformal structure" perhaps an example of a higher-degree distribution? that is, is the conformal structure equivalent to the field of light-cones? i think that it is... this is mainly from vague memories of when i thought about the cone-field on an infinite-dimensional smooth manifold of legendrian submanifolds... "lorentzian conformal structure" as equivalent to "causality structure"...
there also seems to be an opportunity to play the "mythical etymology" game here, trying to relate these "higher-degree distributions" with distributions in the "schwartzian" sense, especially in this lie-theoretic context...
so i've been thinking some more about the invariant higher-degree distributions on the g2 long-root grassmanian... i'll try to describe some of this...
all right, so let me try to describe some of my current thoughts about this...
let's consider a g2 "point" p, and then let's consider the variety given as the supremum of all the "lines" through p. so perhaps we can think of this as a schubert sub-variety of the point grassmanian... the zariski closure of a certain bruhat class (or something like that). anyway, it seems to be a non-singular variety, basically a 2-sphere (i'm thinking in terms of real algebraic geometry of course!) or projective plane or something like that, and this (the non-singularness) is related to how the relevant "invariant distribution" is linear. in contrast, if we look at the point/line dual situation, then i think that we're going to get a non-singular schubert variety, because of the lack of a corresponding invariant linear distribution. and i want to get some idea of the nature of the singularity by thinking about the invariant higher-degree distribution and trying to interpret it as the tangent cone of the basepoint (the unique element in the most special bruhat class), and so forth...
now i'm struggling to remember what i'm supposed to understand about "tangent cones" and so forth, though... let's see, one idea is that a "singular point" is one where the _zariski_ tangent space is too big to be the actual tangent cone; the actual tangent cone is of the "right" dimension but isn't shaped like a vector space, and the zariski tangent space is its minimal linear hull. except that i'm not sure whether some part of what i just said might have depended on the assumption that the singularity is "conical" in some sense...
hmm, an idle thought occurs to me.... if hermitian symmetric spaces are the case where the invariant distribution on the grassmanian is as non-linear as possible, then is the schubert singularity of particular interest in that case? or something like that? not sure how much sense that might actually have made... particularly since i'm having trouble seeing _any_ invariant distribution in this case, even higher-degree...
an even more idle thought: i'm wondering about the relationship between various contexts where some mathematical culture has been preoccupied with "conic" things... "conic sections" ... "conical singularities" and "cone of a projective embedding" (or something like that) ... and so forth...
anyway, i seem to be guessing that for the g2 line grassmanian, the zariski tangent space at the basepoint singularity of the 2d schubert variety is the 4d space corresponding to the invariant linear distribution visible from the root system... ??and that the further 2 degrees of constraint will come from... well actually, i seem to have no idea whatsoever where they could come from... i must have something screwed up here...
i guess that one possibility that i should consider here is that the invariant distributions that i've been calling the "visible" ones are maybe just the most photogenic such, and that i've been overlooking some wallflowers...
so is the "nilpotent radical" of a parabolic subalgebra just the "triangular dual" of the killing orthogonal complement of the subalgebra?? or something like that? ...don't know much about standard terminology here... i just mean that wrt the "triangular decomposition" of the root system these things look oppositely placed... or something like that.
how do interpretations of "nilpotent radical" that we might be getting at here relate to other ideas about them, such as "stuff forgotten by residual incidence geometry" or something about classification of irreps?
another possibly interesting example for trying to understand the singular or non-singular nature of a schubert variety is b2, again comparing the 2d schubert variety in the "point" grassmanian to that in the "line" grassmanian...
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