# Richardson Classes and the Bala Carter Theorem

I am interested in trying to understand the following problem. Let $G$ be a connected simple algebraic group of type $D_n$, (with $n\geqslant 4$ even), defined over an algebraically closed field of odd characteristic. Then in such a group there are unipotent classes which are so called degenerate unipotent classes. These are the unipotent classes whose Jordan normal forms are parameterised by partitions $\lambda$ of $2n$ such that all parts of $\lambda$ are even and every even number occurs an even number of times. For example if $G$ is $SO_8(\mathbb{K})$ and $\lambda$ is either the partition $(4,4)$ or $(2,2,2,2)$. They are degenerate as there are two distinct conjugacy classes of unipotent elements in $G$ whose Jordan normal forms are both parameterised by $\lambda$.

Let $\mathcal{O}$ represent one of the two degenerate conjugacy classes of $G$. We can write the partition $\lambda$ as $(2\eta_1,2\eta_1,2\eta_2,2\eta_2,\dots,2\eta_k,2\eta_k)$ for some natural number $k$. It is commented by Carter, (in Finite Groups of Lie Type: Complex Characters and Conjugacy Classes section 13.3), that this class is a Richardson class for a parabolic subgroup $P$ of $G$ whose Levi complement has semisimple type $A_{2\eta_1-1} \times A_{2\eta_2-1} \times \cdots \times A_{2\eta_k -1}$. If we assume the branch point of the Dynkin diagram is on the right of the diagram then there are two such parabolic subgroups arising from the choice to be made over the extremal right hand side node in the construction of the root subsystem of type $A_{2\eta_k-1}$.

EDIT: To be very specific if $G$ is of type $D_4$ let us assume that the simple roots $\Delta = \{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$ are labelled as in Bourbaki, (Groupes et algèbres de Lie: Chapitres 4 à 6). Then the two Levi subgroups of type $A_3$ correspond to the subsets $\{\alpha_1,\alpha_2,\alpha_3\}$ and $\{\alpha_1,\alpha_2,\alpha_4\}$ of the roots and the two Levi subgroups of type $A_1\times A_1$ corespond to the subsets $\{\alpha_1,\alpha_3\}$ and $\{\alpha_1,\alpha_4\}$.

Using the Bala-Carter theorem we can associate to this unipotent class a Levi subgroup of $G$ and a distinguished parabolic subgroup of the Levi subgroup. Now we know from Bala and Carter's classification of the distinguished parabolic subgroups in a simple group of type $D$ that the Levi associated to these classes cannot be $G$ itself. This is because the right extremal nodes of the weighted Dynkin diagrams of this class have values 0 and 2 but in a distinguished parabolic they must either be both 2 or both 0. Therefore we must have that the Levi is a direct product of type $A$ components and the distinguished parabolic is the unique distinguished parabolic in a group of type A, (the Borel).

My question is then the following. Will the Levi subgroup associated to the class $\mathcal{O}$ from the Bala-Carter theorem be conjugate to the Levi complement of the parabolic subgroup $P$. Or alternatively is there a way, say from the weighted Dynkin diagram, that one can determine which Levi subgroup $L$ will be such that $u \in \mathcal{O}$ is distinguished in $L$?

For example if $G$ is $SO_8(\mathbb{K})$ and $\lambda$ is the partition $(4,4)$ then is $L$ a Levi subgroup of type $A_3$?

Thanks for any help anyone may be able to give me with this.

EDIT: Some clarifications of the language, due to the suggestion of Jim.

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A couple of very minor edits. Before attempting to answer this, let me point out that language like "right extremal nodes" of the Dynkin diagram is a little fuzzy for type $D_4$ (but probably without affecting your question). And the initial description "simple connected semisimple" sounds awkward; but I guess you are mainly concerned just with $SO_{2n}$ throughout. –  Jim Humphreys Dec 6 '10 at 16:13
Yes, you're right. I have made it a bit clearer now for $D_4$, especially as this is the running example in my post. Yeah, I always get a bit haphazard with this. I get so used to writing "connected reductive" or "connected semisimple" that I forget when referring to a connected simple algebraic group that this is redundant. Thanks for the suggestion. In fact, no. I'm actually interested in these classes in the half spin group $HSpin_{2n}(\mathbb{K})$. –  Jay Taylor Dec 6 '10 at 17:00

It gets complicated to compare the different ways to parametrize or realize a unipotent class (or equivalently, in good characteristic, a nilpotent orbit in the Lie algebra). But I think the answer to the basic question here is no, unless I'm misreading it. When a class happens to be Richardson (the unique orbit intersecting densely the unipotent radical of a given parabolic subgroup), the dimension of the class is twice the dimension of the unipotent radical in question. The class can also be determined by the Bala-Carter method, starting with a Levi subgroup of $G$ and its Borel subgroup or other distinguished parabolic. Here the class is the Richardson class determined by that distinguished parabolic.

Taking just the example $D_4$ with the given class being one of two determined by the partition $(4,4)$, either class (and a third one as well) has dimension 20. Here the Richardson viewpoint starts with a parabolic subgroup of Levi type $A_2$ whose unipotent radical has dimension 10. But the Bala-Carter viewpoint starts with a Levi subgroup of type $A_3$ together with its distinguished parabolic (Borel) subgroup having a unipotent radical of dimension 10. So the two Levi subgroups in the picture are far from being conjugate.

Generally speaking, the Richardson method starts with a parabolic subgroup having a small Levi subgroup to yield a big class, while the Bala-Carter method starts with a big Levi subgroup to yield the same big class. The former method doesn't usually yield all unipotent classes, whereas the latter method gets them all. But the two methods are roughly dual.

Going back to $D_4$, what you can observe in the closure diagram is a duality between the two situations in which a single partition labels two classes. But there is still the notational task of sorting out which parabolic or Levi goes with which of the two classes in each case. There is a lot to be said here in general for type $D_{2n}$. Let me just add that unipotent classes or nilpotent orbits don't depend on the isogeny type of the group, so special orthogonal groups are convenient to work with.

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Thanks for your answer Jim. Just a couple of questions. Sticking with the example of $D_4$ and the partition $(4,4)$. Do you really mean to say that the Bala-Carter theory gives a parabolic with Levi complement $A_3$ and the Richardson parabolic is of type $A_2$? What I got the impression that Carter was saying was that the parabolic for which the class is a Richardson class has Levi complement $A_3$. This is mentioned in his book (pg. 423 to be precise) in his description of the Springer correspondence. –  Jay Taylor Dec 6 '10 at 18:51
So let's just ignore the parabolic coming from the Richardson theory altogether. Can one determine in a nice way specifically the Levi subgroup attached to the class in the Bala-Carter theory? I had the impression that as the partition determines the Jordan normal form of the matrix that this would determine the Levi subgroup. I guess in my mind in the example in $D_4$ the two Jordan blocks of size 4 fit very nicely in some $A_3$ Levi, which will look like $GL_4\times GL_4$ in $SO_8$, (except one $GL_4$ is determined by the other). Using matrices in $SO_{2n}$ would probably make this concrete. –  Jay Taylor Dec 6 '10 at 19:00
I'll get back into this literature a little later, but at first glance it looks as if your initial concern is about labelling conventions. This gets messy, since one can also look at dual partitions, etc. I don't recall immediately how Bala-Carter connects with partition labels for the classical groups. –  Jim Humphreys Dec 6 '10 at 19:27
P.S. Concerning your first comment, I think you are misreading the complicated page 423 in Carter's book (e.g., how his $\xi_i$ are related to $\lambda$). Concerning Bala-Carter, their original papers may help. But they require pairs: Levi subgroup plus its distinguished parabolic. Class dimensions can be useful in keeping track of partition labels, though in your case the latter by themselves are ambiguous. Too much notation in the whole subject, unfortunately. You need to focus your questions a little more accordingly. –  Jim Humphreys Dec 6 '10 at 21:58
OK, well thanks for all your help and very useful comments Jim. I'll take a look at the Bala-Carter papers to see if I can sort out my labelling issues. Thanks again. –  Jay Taylor Dec 7 '10 at 8:10