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Background: I'm trying a problem on representations of reductive groups over various finite rings towards which this is very relevant (what I want to do is a very specialized case of this problem, and I want to know what background theory has been done for this situation in the literature and what is known about this). In characteristic $0$ over an algebraically closed field, and over finite fields, classifying conjugacy classes in reductive groups over field is a very well-known and well-studied problem (sometimes in preparation for studying representations of these).

Question: Let $R$ be a local commutative ring, either in characteristic $0$, algebraically closed, or in characteristic $p$ (algebraically closed OR finite field). If you want to complicate matters, and have an answer for non-commutative rings as well, I would be happy to see it, but I think the problem is non-trivial enough as is - and as far as I know, only $GL$ can be easily defined over non-commutative rings). Edit: Also specify that $R$ is an algebra over its residue field, and has an identity (which I believe is necessary to make the following argument work; again if you don't need this restriction feel free to not use it).

Let $G$ be a reductive group (if you want, feel free to restrict to just the classical groups, $GL$, $SL$, $Sp$, and $SO$) defined over $R$. What can be said about classifying conjugacy classes in $G$? What is clear is the Levi decomposition of $G$, as the semi-direct product of the reductive group defined over the residue field of $R$, and $N$, the set of all matrices that are congruent entry-wise to the identity matrix, modulo the maximal ideal of $R$ (the latter is the normal subgroup). Using the semi-direct product, one can say something implicit about the conjugacy classes; first by studying the conjugation action of the reductive group on the unipotent algebraic group $N$, then studying the conjugacy classes in $N$, then extending this to the whole group.

Are there any special cases of this problem that have been studied in the literature? Is there something more than can be said in general (further to what I have said above about the semi-direct product)?.

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It's not a semi-direct product; it's a more interesting extension than that. –  Ben Webster Jan 10 '10 at 2:20
    
Ah right I think I understand my mistake: in my calculations I was assuming that the local ring $R$ is actually an algebra (over the residue field $F$) containing the identity. In that case would you agree that it is a semi-direct product? In my arguments for this case, $N$ and $G(F)$ (the reductive group over the field) are disjoint, $N$ is normal, and together they generate the group, so I thought it was a semi-direct product. –  Vinoth Jan 10 '10 at 5:38
    
To add a bit more to Ben's comment, by taking $R = W_ 2(k)$ (length-2 Witt vectors) for an algebraically closed field $k$ of characteristic $p > 0$ one proves via root-group arguments for any Chevalley group $G$ that $G(R)$ "viewed as a $k$-group" (to be rigorous use Greenberg functor, akin to Weil restriction from $k[t]/(t^2)$ to $k$) is a smooth connected affine $k$-group with no Levi subgroup; e.g., ${\rm{SL}}_ 2(W_ 2(k))$ as $k$-group. So in this way one gets lots of examples of groups without Levi subgroups over alg. closed fields of nonzero char. (in contrast char. 0). –  BCnrd Apr 7 '10 at 11:45

2 Answers 2

The short answer is that over local PIRs which are not fields, all the conjugacy classes are known explicitly for $\mathrm{SL}_2$, $\mathrm{GL}_2$, and $\mathrm{GL}_3$, some can be explicitly described for $\mathrm{GL}_n$, and in general the problem is hopelessly complex (in the sense that it is wild).

The longer answer is as follows. The cases $\mathrm{SL}_2$ and $\mathrm{GL}_2$ over finite local PIRs have been known for a long time, and can easily be worked out directly. The case of $\mathrm{GL}_2$ and $\mathrm{GL}_3$ over arbitrary local PIRs has been treated more recently by Avni, Onn, Prasad and Vaserstein in Similarity classes of 3x3 matrices over a local principal ideal ring.

Trying to tackle the $\mathrm{GL}_4$ case, one runs into the problem of classifying pairs of $2\times2$ matrices over the residue field, under simultaneous conjugation. More generally, the problem of describing the conjugacy classes in $\mathrm{GL}_{4n}$, for all $n$ contains the matrix pair problem over the residue field. This is a wild classification problem (see the above link) and hence one cannot expect a useful explicit classification in general. I think (but I have not checked it carefully) that if you do not assume that the ring is a PIR, you run into the matrix pair problem already for the groups $\mathrm{GL}_{2n}$.

On the other hand, despite the lack of a general canonical form for matrices over local rings, there is a canonical form for a large subset of matrices, notably the regular ones. Regular elements were defined by Hill (for matrices over local PIRs) in Regular Elements and Regular Characters of $\mathrm{GL}_n(\mathcal{O})$, and are characterised by the property of being conjugate to their respective companion matrix, i.e., the canonical form consists of a single companion matrix block. One can of course generalise this to a canonical form with several companion matrix blocks. This is discussed in detail in Guralnick, Similarity of Matrices over Local Rings. Over fields this is the familiar rational canonical form.

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I don't have a complete answer or a reference, but I have a principle:

(*) a conjugacy class in GL(n) is the same as an isomorphism class of n-dimensional representation of $\mathbb{Z}$.

Similarly, one in SO(n) is an isomorphism class of n-dimensional representation with symmetric form, etc.

So, over an algebraically closed field, the Jordan decomposition says that indecomposable representations of $\mathbb{Z}$ are a single Jordan block, for example.

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hmm that's interesting, I hadn't seen that perspective before - so it's conceivable that in this case, a conjugacy class could be an isomorphism class of $n$-dimensional representations of some other ring, which corresponds to the local ring $R$, like $\mathbb{Z}$ does to $\mathbb{C}$ (or any other algebraically closed field of char $0$), possibly in some very vague sense the "ring of integers" of a local ring. Now I think I have reason to believe that my question above has no entirely explicit answer; but a correspondence like this would be extremely helpful. –  Vinoth Jan 10 '10 at 5:43
    
No, it's still Z. The representation Ben is talking about is the one which sends an integer n to g^n, where g is an element of GL(n). When you change the base ring you just change the thing you're looking for representations in, not the thing you're looking for representations of. –  Qiaochu Yuan Jan 10 '10 at 6:33
    
Sorry, those n's shouldn't be the same. –  Qiaochu Yuan Jan 10 '10 at 6:35
    
Ah right, thanks! Sorry for my silly comment.. So each conjugacy class of $G=G(R)$ defines a representation $\mathbb{Z}^{+} \rightarrow GL_n(R)$ via $m \rightarrow g^m$, and conversely every such representation arises in this way clearly. That's certainly good food for thought: one (slightly sloppy) approach from there seems this: fix a representation $GL_n(R) \rightarrow GL_m(\mathbb{C})$ , then compose to get a representation $ \mathbb{Z}^{+} \rightarrow GL_m(\mathbb{C})$ ; we can classify the latter and then we have to track back. I'll try and see where that leads me –  Vinoth Jan 10 '10 at 7:14

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