# Classification of adjoint orbits for orthogonal and symplectic Lie algebras?

This might be standard, but I have not seen it before:

Let $K$ be an algebraically closed field (of characteristic 0 if necessary). Let $G$ be the orthogonal group ${\bf O}(m)$ or the symplectic group ${\bf Sp}(2n)$, and let $\mathfrak{g}$ be its Lie algebra. Is there a classification of the adjoint orbits of $G$ acting on $\mathfrak{g}$? An answer for the special orthogonal group ${\bf SO}(m)$ is also welcome. I am aware of the classification of nilpotent and semisimple orbits.

I am looking for an answer kind of like Jordan normal form in the case that $G$ is the general linear group. My guess is that the classification is Jordan normal form plus some other invariant.

-
Not quite spot on, but still worth mentioning: en.wikipedia.org/wiki/Orbit_method – Steve Huntsman May 25 '10 at 17:25
I think this is discussed in Carter's book. You can get the answer yourself using Jordan decomposition in $\mathfrak{g} (X=X_{ss}+X_n, [X_{ss},X_n]=0)$ and the classification of the nilpotents that you already know. – Victor Protsak May 25 '10 at 18:03
This seems similar to this question: mathoverflow.net/questions/1876/… and perhaps the references there might be useful. – José Figueroa-O'Farrill May 25 '10 at 18:35
José: Thanks! The reference that Pasha Zusmanovich gives seems to be exactly what I wanted (lines 2,3,5,6 of Table III). – Steven Sam May 25 '10 at 19:39

## 2 Answers

"Classification" can mean more than one thing, but it's useful to be aware of the extensive development of adjoint quotients by Kostant, Steinberg, Springer, Slodowy, and others. This makes sense for all semisimple (or reductive) groups and their Lie algebras over any algebraically closed field, perhaps avoiding a few small prime characteristics. Older sources include Steinberg's 1965 IHES paper on regular elements and the Springer-Steinberg portion of the 1970 Lecture Notes in Math. vol. 131. (For an overview with references, based partly on Steinberg's Tata lectures, see Chapter 3 of my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups.) While the general focus has been on developing a picture of the collection of all classes or orbits as some kind of "quotient", quite a few special features of the classical types are also brought out in the Springer-Steinberg notes. As suggested by Victor, Roger Carter's book Finite Groups of Lie Type also has a lot of related material but with special emphasis on nilpotent orbits. The Jordan decomposition does reduce many classification questions to the nilpotent case, at least in principle, if you are willing to deal with various centralizers along the way.

[ADDED] The paper in J. Math. Physics gives a nice concrete answer to the original question, building on some of the older theory but using mainly tools from linear algebra and basic group theory. This is the traditional approach of most physicists, though papers in this mixed journal are sometimes unreliable and contain mathematics of the sort probably not usable in physics but also not publishable in math journals. Djokovic and his collaborators are more reliable than most, fortunately, and he has written many papers using parts of Lie theory as well. One downside is the narrower perspective than found in the notes of Springer-Steinberg, for instance. But it all depends on whether you want to work over other fields or want to organize the classes/orbits more conceptually.

Here is a MathSciNet reference: MR708648 (85g:15018) 15A21 (17B99 17C99) Djokovic´,D. Zˇ . [¯Dokovic´, Dragomir Zˇ .] (3-WTRL); Patera, J. [Patera, Jirˇ´ı] (3-MTRL-R); Winternitz, P. (3-MTRL-R); Zassenhaus, H. (1-OHS) Normal forms of elements of classical real and complex Lie and Jordan algebras. J. Math. Phys. 24 (1983), no. 6, 1363–1374 (review by R.C. King). Their references include the work by Burgoyne-Cushman, Milnor, Springer-Steinberg mentioned by Bruce and me.

-
I wasn't even aware that you wrote a book on the subject! – Victor Protsak May 25 '10 at 19:40

There's a paper by Burgoyne and Cushman in the Journal of Algebra which answers this for the real field, the complex field and finite fields. My recollection is that there is a paper by Milnor for perfect fields.

Here are the references:
MR0432778 (55 #5761) Burgoyne, N. ; Cushman, R. Conjugacy classes in linear groups. J. Algebra 44 (1977), no. 2, 339--362.

MR0249519 (40 #2764) Milnor, John . On isometries of inner product spaces. Invent. Math. 8 1969 83--97.

-