# Characterizations of Jacobson-Morozov parabolics associated to a nilpotent

Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic subalgebra, defined in the Lie algebra as follows. Choose a $\mathfrak{sl}_2$-triple $(x, y,h)$ and define $$\mathfrak{p} = \bigoplus_{i \geq 0} \mathfrak{g}_i$$ where $\mathfrak{g}_i$ is the $i$th eigenspace of $h$. This is independent of the choice of $h$ (and $y$). One can then take a corresponding parabolic subgroup $P$ in the group $G$.

My question is: are there other characterizations of $P$ that don't require us to choose a triple first?

Also, there may be other parabolics of the same type as $P$ (i.e. conjugate to $P$) which contain $x$. In general, I know Springer fibers are hard to describe, but I don't know if something manageable can be said about these particular ones (and for partial flag varieties instead). And, is there something that characterizes the Jacobson-Morozov parabolic amongst these?

I'd be happy to take just $G = SL_n$ so that $G/P$ are different partial flag varieties. For example, for $G = SL_3$, both the regular and subregular orbits have Jacobson-Morozov parabolic conjugate to a Borel $B$. The regular Springer fiber is just a point (so there's only one choice). The subregular nilpotents have $\mathbb{P}^1 \cup_{\text{pt}} \mathbb{P}^1$ as a Springer fiber and the Jacobson-Morozov parabolic is the point in the intersection of the two $\mathbb{P}^1$s.

• Let $\mathfrak{u}$ be the sum $\oplus_{i \geqslant 2} \mathfrak{g}_i$ then I'm pretty sure this is the Lie algebra of the derived subgroup of the unipotent radical of $P$ (it certainly contains it). With some mild restrictions we know the following: $\mathfrak{c_g}(x) \subseteq \mathfrak{p}$, $[x,\mathfrak{p}] = \mathfrak{u}$ and the orbit $(\mathrm{Ad} P)(x)$ is dense in $\mathfrak{u}$. If one can characterise $\mathfrak{u}$ as above then these statements don't involve the grading, i.e., the $\mathfrak{sl}_2$-triple. The last one might be enough to pin down $P$ but I'm not sure. Commented May 15, 2015 at 11:54
• I'll leave the comment there but it's obviously wrong. If the element is regular then the Lie algebra of the derived subgroup is $\oplus_{i \geqslant 3}\mathfrak{g}_i$ and $\oplus_{i \geqslant 2}\mathfrak{g}_i$ is the Lie algebra of the whole radical. Commented May 15, 2015 at 11:59

It's difficult to sort out the various questions you are combining here, so it would be helpful to tighten the formulation. Taken at face value, the answer to your basic question "are there other characterizations of $P$ that don't require us to choose a triple first?" seems to be negative, though I'm not sure exactly what you are looking for. Specific examples might be helpful.

In any case your basic set-up requires characteristic 0 (or large enough characteristic). Typically not all parabolics need arise this way and, moreover, distinct orbits may have the same canonical parabolic: this is seen for example in type $G_2$, where there are 5 nilpotent orbits but only 4 parabolics (up to conjugacy) and where 3 of the orbits lead to the same minimal parabolic but none leads to the other minimal parabolic.

Aside from these complications, the Bala-Carter classification method for the classes or orbits (which has some advantages over the older Dynkin method) doesn't emphasize canonical parabolics. Rather, the distinguished unipotents/nilpotents and the corresponding distinguished parabolics having these as Richardson classes/orbits are essential for their inductive approach.

I'm not sure how easy it is in practice to deal with the canonical parabolics. These differ in subtle ways from the somewhat reverse Richardson method, for example when the group centralizers involved fail to be connected (which doesn't occur in type $A_n$ but often does occur elsewhere).

• Thanks for the answer. I suppose I didn't have anything more specific in mind other to know what the state-of-the-art understanding was, so this is helpful! Could you also elaborate a little on what it means for a nilpotent orbit to be of Ricardson type, and what you mean by the "Richardson method?" I'm not very familiar with these terms.
– hic
Commented Jul 25, 2015 at 20:12
• Also, I want to check a calculation I did for $SL_3$ where I found that not all parabolics arise as JM parabolics. I found that for both the regular and subregular nilpotent orbits, the JM parabolic is a Borel. This seems plausible to me also because the two minimal parabolics for $SL_3$ correspond to choosing a simple root, whereas the JM parabolic should somehow be independent of this choice?
– hic
Commented Jul 25, 2015 at 20:17
• @hic: Concerning Richardson orbits, it's a good idea to look at a textbook like the one by Collingwood-McGovern (or Carter's 1985 book). Basically, to each parabolic subalgebra corresponds a nilpotent orbit determined by the dense orbit in its nilradical, and in type $A_n$ all nilpotent orbits occur this way (but not in general). Concerning $SL_3$, I'll double-check the details; but what you say doesn't sound correct. Commented Jul 25, 2015 at 21:04
• P.S. I've tried to make my answer more precise, but the details here are rather complicated to spell out. Commented Jul 25, 2015 at 21:21
• Is the correspondence that, given a JM parabolic $P$ corresponding to a nilpotent orbit, that nilpotent orbit is dense in $\mathfrak{g}_{\geq 2}$ but not necessarily the nilpotent radical $\mathfrak{g}_{\geq 1}$? I guess the example I have in mind is, take triple $(x, y, h)$ with $x = \left(\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}\right)$ and $h = \left(\begin{array}{ccc}1&0&0\\0&0&0\\0&0&-1\end{array}\right)$. Then, I think the following is a labeling of the eigenspaces (the number in the matrix is the eigenvalue): $\left(\begin{array}{ccc}0&1&2\\-1&0&1\\-2&-1&0\end{array}\right)$
– hic
Commented Jul 25, 2015 at 21:31