Conjugacy classes of reductive groups defined over local commutative rings - MathOverflow

Conjugacy classes of reductive groups defined over local commutative rings

2010-01-10T01:31:07Z

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)?.

Answer by Ben Webster for Conjugacy classes of reductive groups defined over local commutative rings

2010-01-10T02:39:25Z

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.

Answer by A Stasinski for Conjugacy classes of reductive groups defined over local commutative rings

2010-04-07T11:17:44Z

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.