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If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here), there is a simple description of the quotients $G/P$ by parabolic subgroups of $G$ as grassmannians or partial flag varieties:

  • $\mathit{SL}_n/P$ parametrizes flags $V_1 \subseteq \cdots \subseteq V_r$ of vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r < n$ in a fixed vector space of dimension $n$;

  • $\mathit{SO}_n/P$ parametrizes flags $V_1 \subseteq \dotsb \subseteq V_r$ of totally isotropic vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r \leq \frac{n}{2}$ in a fixed vector space of dimension $n$ endowed with a nondegenerate quadratic form, except that when $d_r = \frac{n}{2}$ (for $n$ even, of course), we can also impose that $V_r$ belongs to one or the other of the two families of $(n/2)$-dimensional subspaces;

  • $\mathit{Sp}_n/P$ parametrizes flags $V_1 \subseteq \cdots \subseteq V_r$ of totally isotropic vector subspaces of fixed specified dimensions $0 < d_1 < \dotsb < d_r \leq n$ in a fixed vector space of dimension $2n$ endowed with a nondegenerate alternating form.

(Of course, the above can be more clearly worded to specify the $(d_i)$ in function of the nodes chosen in the Dynkin diagram of $G$ to define $P$.)

Question: Is there a similar description, when $G$ is an exceptional semisimple group, of its parabolic quotients $G/P$ as parametrizing subspaces or partial flags of some space endowed with additional structure on which parts of that structure are “isotropic” in a certain sense?

The kind of answer I am looking for is something like this:

  • $G_2/P$ parametrizes $1$-dimensional, or $2$-dimensional, or nested $V_1\subseteq V_2$ (with $\dim V_i = i$) subspaces of the $7$-dimensional purely imaginary part $A^0$ of the octonion algebra $A$ (meaning split, as we are over an algebraically closed field) such that the octonion product is identically zero on the subspace

(I think this is correct, but I don't really have a reference clearly stating this.)

What about $F_4$, $E_6$, $E_7$, $E_8$? (Of course, this all depends on appropriate representations having been constructed. For example, I expect the descriptions of $E_8/P$ to be something like parametrizing subspaces or partial flags in the Lie algebra $\mathfrak{e}_8$ since the adjoint representation generates every other fundamental representation by taking alternating powers. But what kind of subspaces, exactly?)

Such a description doesn't seem to be in J. F. Adams's Lectures on Exceptional Groups. I think B. Rosenfeld's Geometry of Lie Groups §7.6 may be supposed to answer the question (under the name “fundamental figures” and “parabolic figures”), but it is extremely difficult to decipher (and the fact that it treats the real, not-necessarily-split, case, of course makes it even more difficult). An answer might also be in Landsberg & Manivel's paper “The Geometry of Freudenthal's Magic Square”, but I was unable to find where exactly (and even then, it doesn't seem to be couched in the language of subspaces and flags).

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    $\begingroup$ The $G_2$ case is descibred in "Élie Cartan and geometric duality" by Robert Bryant. Freely available at dukespace.lib.duke.edu/dspace/handle/10161/12696 $\endgroup$ Mar 24, 2022 at 14:09
  • $\begingroup$ (But the homogeneous spaces are not described in terms of octonions.) $\endgroup$ Mar 24, 2022 at 14:12

1 Answer 1

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Edit 2: A good discussion of the lowest dimensional pieces of each of the flags below is found in Geometries, the principle of duality, and algebraic groups by Carr and Garibaldi. In particular for the $E_7$ example we can realise these as "inner ideals" for the symmetric trilinear form $t$ built out of the symplectic form and quartic form $f(x,y,z,w) = \omega(x,t(y,z,w))$. This means that for each of those subspaces $X \leq W$, $t(X,W,X)$ is contained in $X$ and moreover for everything except $\alpha_1$ the map $y \mapsto t(x,x,y)$ has rank 1 ($x \in X$, $y \in W$).

Definitely not a full answer (and rather late to the party) but I am interested in the $E_7$ case myself so I have been doing a little computing of the dimensions of these flags that I thought was worth sharing. I also cannot find a good source for these so hopefully this is helpful for someone.

Number the nodes of $E_7$ as LiE does it, so that node 1 is at the end closest to the branch, node 2 is the node on the branch and node 7 is at the end furthest from the branch. Then the parabolic subalgebras corresponding to crossing those nodes give flags (in the 56-dimensional representation) with dimensions as follows:

$$\begin{array}{|c|c|c|} \hline \text{Simple Root} & \text{Flag dimensions} & \text{Associated Grading dimensions} \\\hline \alpha_1 & 12\leq 44 & 12,32,12\\ \alpha_2 & 7\leq 28 \leq 49& 7,21,21,7 \\ \alpha_3 & 6\leq 18 \leq 38 \leq 50 & 6,12,20,12,6\\ \alpha_4 & 4\leq 10 \leq 22 \leq 34\leq 46\leq 52& 4,6,12,12,12,6,4 \\ \alpha_5 & 3\leq 13 \leq 28\leq 43\leq 53& 3,10,15,15,10,3\\ \alpha_6 & 2\leq 18 \leq 38 \leq 54 & 2,16,20,16,2\\ \alpha_7 & 1\leq 28 \leq 55 & 1,27,27,1\\\hline \end{array}$$

Of course this isn't the full information since we need to identify exactly which subspaces of these dimensions appear. I'm not sure how to tackle this in general but I note that we have two $E_7$-invariant polynomial functions in this representation. A symplectic form $\omega$ and a symmetric quartic form $f$. It is not clear to me, however, how these flags interact with these forms. I suspect that the lowest dimensional subspace in each case will be isotropic for $\omega$ (a maximal isotropic here has dimension 28 so perhaps all $\dim \leq 28$ subspaces will be) but they will also likely need to be "isotropic" for $f$ as well whatever that means — perhaps $f(v,v,v,v) =0$. Then I believe that the higher end of each flag will be the polars with respect to $\omega$ of the lower end as we see for the $\operatorname{SO}_n$, $\operatorname{Sp}_{2n}$ cases. In particular, each of the 28-dimensional subspaces appearing (there are 3 different types according to the table) will be maximal isotropic. I don't quite understand how the quartic form then tells us the rest of the information unfortunately.

To see this in practice take the $\alpha_7$ example. This is the symmetric R-space in our list and is also projective variety in this representation since its smallest subspace is a line $L$. Of course, every line is isotropic for $\omega$ and it is also true that $f$ vanishes when restricted to $L\times L\times L\times L$. I haven't managed to prove that these are the only lines with that property but I believe it is true. It is less clear to me what the 28-dimensional subspace $H$ is although it is certainly isotropic and we have $f\rvert_{L\times L \times H \times H} = 0$ although I am less confident with my calculations here and I'm not sure if this defines $H$. Indeed using the setup from J. Adams's Lectures on Exceptional Groups the representation is $W \cong \bigwedge^2 V \oplus \bigwedge ^2 V^*$ for $V \cong \mathbb{C}^8$. Then decomposing $V = \Pi \oplus U$ for $\dim \Pi = 2$ and correspondingly $V^* = \Pi^* \oplus U^*$ for $\dim \Pi^* = 2$ we can write our flag in $W$ as:

$$ \left(\bigwedge\nolimits^2 \Pi \oplus 0 \right) \leq \left(\Pi \wedge V \oplus \bigwedge\nolimits^2 U^* \right) \leq \left(\bigwedge\nolimits^2 V \oplus U^* \wedge V^* \right) \leq W. $$

Edit: here are the tables for the other exceptional simple Lie groups/algebras in their lowest dimensional representations. Simple roots are numbered in line with how the program LiE numbers the nodes.

$E_6$ in its 27-dimensional rep (the one corresponding to node 1): $$\begin{array}{|c|c|c|} \hline \text{Simple Root} & \text{Flag dimensions} & \text{Associated Grading dimensions} \\\hline \alpha_1 & 1\leq 17 & 1,16,10\\ \alpha_2 & 6\leq 21 & 6,15,6 \\ \alpha_3 & 2\leq 12 \leq 22 & 2,10,10,5\\ \alpha_4 & 3\leq 9 \leq 18 \leq 24 & 3,6,9,6,3 \\ \alpha_5 & 5\leq 15 \leq 25 & 5,10,10,2\\ \alpha_6 & 10 \leq 26 & 10,16,1 \\\hline \end{array}$$

$E_8$ in its 248-dimensional (adjoint) rep: $$\begin{array}{|c|c|c|} \hline \text{Simple Root} & \text{Flag dimensions} & \text{Associated Grading dimensions} \\\hline \alpha_1 & 14\leq 78\leq 170\leq 234 & 14,64,92,64,14\\ \alpha_2 & 8\leq 36 \leq 92\leq 156 \leq 212 \leq 240 & 8,28,56,64,56,28,8 \\ \alpha_3 & 7 \leq 21 \leq 56 \leq 98 \leq 150 \leq 192 \leq 227 \leq 241 & 7,14,35,42,52,42,35,14,7\\ \alpha_4 & 5 \leq 11 \leq 26 \leq 46 \leq 76 \leq 106 \leq 142 \leq 172 \leq 202 \leq 222 \leq 237 \leq 243 & 5,6,15,20,30,30,36,30,30,20,15,6,5 \\ \alpha_5 & 4 \leq 14 \leq 34 \leq 64 \leq 104 \leq 144 \leq 184 \leq 214 \leq 234 \leq 244 & 4,10,20,30,40,40,40,30,20,10,4\\ \alpha_6 & 3 \leq 19 \leq 49 \leq 97 \leq 151 \leq 199 \leq 229 \leq 245 & 3,16,30,48,54,48,30,16,3\\ \alpha_7 & 2 \leq 29 \leq 83 \leq 165 \leq 219 \leq 246 & 2,27,54,82,54,27,2\\ \alpha_8 & 1 \leq 57 \leq 191 \leq 247 & 1,56,134,56,1\\\hline \end{array}$$

$F_4$ in its 26-dimensional representation: $$\begin{array}{|c|c|c|} \hline \text{Simple Root} & \text{Flag dimensions} & \text{Associated Grading dimensions} \\\hline \alpha_1 & 6 \leq 20 & 6,14,6\\ \alpha_2 & 3 \leq 9 \leq 17 \leq 23 & 3,6,8,6,3\\ \alpha_3 & 2 \leq 5 \leq 11 \leq 15 \leq 21 \leq 24 & 2,3,6,4,6,3,2\\ \alpha_4 & 1 \leq 9 \leq 17 \leq 25 & 1,8,8,8,1\\\hline \end{array}$$

$G_2$ in its 7-dimensional representation: $$\begin{array}{|c|c|c|} \hline \text{Simple Root} & \text{Flag dimensions} & \text{Associated Grading dimensions} \\\hline \alpha_1 & 1 \leq 3 \leq 4 \leq 6 & 1,2,1,2,1\\ \alpha_2 & 2\leq 5 & 2,3,2 \\\hline \end{array}$$

In each case I computed these by obtaining a full list of weights (with multiplicities but only 0 has multiplicity higher than 1 in each of these examples) of the representation and converted this into a basis of the simple roots. Then the coefficient of a weight in one of the simple roots is the eigenvalue of that weight under the corresponding grading element. That is, we get a grading of the representation which splits the flag and these coefficients tell us in which graded piece that weight space lies. Then we can convert the grading back into the flag.

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    $\begingroup$ I didn't want to make the change without your approval, but, if you put your {array}s in display mode ($$ $$ rather than $ $), then they become scrollable independently (rather than having to drag the rest of the page along). Would you be willing to make that change, or mind if I made it? $\endgroup$
    – LSpice
    Dec 11, 2022 at 23:28
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    $\begingroup$ Ah thanks for the tip. I've edited them now but am always happy for others to make formatting improvements. $\endgroup$
    – Callum
    Dec 12, 2022 at 13:57

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