Lie algebras and non-smoothness of centralisers in bad characteristic

Question

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$ denote the centraliser, considered as a group scheme over $k$. If $p$ is a very good prime for $G$ it is known that $C_{G}(x)$ is smooth (over $ $$k$). Assume that $p$ is not very good for $G$.

Is there a general description of the elements $x\in G$ for which the centraliser $C_{G}(x)$ is not smooth?

It does not seem too difficult to do this via a direct case by case check, so I'm mainly asking whether there is a uniform argument and whether there are any relevant references.

Let $\mathfrak{c}_{\mathfrak{g}}(x)$ denote the centraliser in the Lie algebra $\mathfrak{g}$ of $G$. We have $\dim C_{G}(x)=\dim\mathfrak{c}_{\mathfrak{g}}(x)$ if $C_{G}(x)$ is smooth.

In the cases where $C_{G}(x)$ is not smooth, are the values $\dim\mathfrak{c}_{\mathfrak{g}}(x)-\dim C_{G}(x)$ known?

The second question is related to Problem 12 of Steinberg's 1966 ICM contribution, which asks for a proof of the statement that $\mathfrak{c}_{\mathfrak{g}}(x)$ is always the sum of the Lie algebra of $C_{G}(x)$ and the centre of $\mathfrak{g}$. Steinberg's problem is mentioned in Humphreys's book Conjugacy classes in semisimple algebraic groups (1.10) but there is no indication as to whether Steinberg's problem has been solved or not.

Answer 1

For all groups of exceptional types the answer can be deduced from the paper "Jordan block sizes of unipotent elements in exceptional algebraic groups" by Ross Lawther, published in Comm. Algebra, 23, Issue 11, 1995, 4125-4156. In order to determine the Jordan block sizes of ${\rm Ad}\ u$, where $u\in G$ is unipotent, Lawther used comuter-aided computations. The case of a general $x\in G$ reduces quickly to the case where $x$ is unipotent by using the Jordan-Chevalley decomposition in $G$.

Of course, the number of Jordan blocks is just $\dim\ \mathfrak{c}_{\mathfrak g}(u)$, or rather $\dim\ \mathfrak{g}^{{\rm Ad}\ u}$, which is one of the numbers you are interested in. This number could then be compared with $\dim\ C_G(u)$ which is much easier to find in the literature (Spaltenstein's book on unipotent classes might have an answer).

Regarding Steinberg's Problem 12, when $p=2$ the Lie lagebra of type $E_8$ is simple, hence has zero centre, but there are about 20 instances when the value of $\dim\ \mathfrak{g}^{{\rm Ad}\ u}$ jumps and becomes bigger than that for the counterpart of $u$ in good characteistic (these cases are starred in Lawther's paper). Since the number of "new" unipotent classes in characteristic $2$ is far less than 20, I think, there will be instances when $\dim\ \mathfrak{g}^{{\rm Ad}\ u}>\dim\ C_G(u)$. The orbit labelled $E_7$ could do the trick, but I didn't check.

There is also a version of Lawther's results for nilpotent elements in exceptional Lie lagebras in the literature; see "Varieties of nilpotent elements for simple Lie algebras II: Bad primes" by University of Georgia VIGRE Algebra Group, published in J. Algebra, 292, 2005, 65–99. This paper also relies on extensive computer-aided computations.

As far as I'm aware there is no computer-free proof of the above results at the present time, hence there is no uniform proof either. One could even say, on a more philosophical note, that nothing is uniform when $p$ is bad.

Answer 2

[EDIT] This replaces my less focused (and more optimistic) attempt at answering both questions. One remark about terminology: it's more standard to refer to the "smoothness" of a conjugacy class than to the "smoothness" of a centralizer. In the algebraic group context, study of a tangent space (Lie algebra) and adjoint action of the group is a natural tool for investigating smoothness.

1) NOTATION: $G$ is a simply connected simple algebraic group over an algebraically closed field $K$ of characteristic $p>0$. Its Lie algebra $\mathfrak{g}$ is then the reduction mod $p$ of Chevalley's $\mathbb{Z}$-form of the corresponding simple Lie algebra over $\mathbb{C}$. If $x \in G$, write $C_G(x)$ for its centralizer in $G$ and $\mathfrak{c}_{\mathfrak{g}}(x)$ for the fixed point space of $\text{Ad} x$ on $\mathfrak{g}$. Similarly, if $X \in \mathfrak{g}$, write $C_G(X)$ for the centralizer of $X$ under the adjoint action of $G$ and $\mathfrak{c}_{\mathfrak{g}} (X)$ for the Lie algebra centralizer of $X$. In each situation the two types of centralizer may be called "global" and "infinitesimal".

2) For semisimple elements of $G$ or $\mathfrak{g}$, global and infinitesimal centralizers always agree in dimension (see Borel, Linear Algebraic Groups, 9.1). In general, one has natural inclusions $\text{Lie}\, C_G(x) \subset \mathfrak{c}_{\mathfrak{g}}(x)$ and $\text{Lie}\, C_G(X) \subset \mathfrak{c}_{\mathfrak{g}}(X)$ With this set-up, the Lie algebra of $\text{SL}_n(K)$ has a 1-dimensional center $\mathfrak{z}$ (consisting of scalar matrices) precisely when $p|n$; otherwise $p$ is "very good" for type $A_{n-1}$. For other simple types, $p$ is "bad" just when $p=2$ for types $B, C, D$ or $p=2,3$ for types $E_6, E_7, F_4, G_2$ or $p=2,3,5$ for type $E_8$. For good (or very good) $p$, the behavior of all centralizers follows the characteristic 0 pattern. In general, induction on dimension reduces the remaining hard problems to the case when $x=u$ is unipotent or $X$ is nilpotent: in a simply connected group like $G$, the centralizer of a semisimple element is reductive and also connected.

3) The center $\mathfrak{z}$ is well understood (Hurley) along with the full ideal structure and its behavior under the adjoint action of $G$ (Hogeweij, Hiss). Beyond type $A$ one always has $\dim \mathfrak{z} \leq 1$ (with equality sometimes for bad primes) except for type $D_{2n}$ where $\dim \mathfrak{z} =2$ if $p=2$.

4) The answer to the first question is almost certainly no: all known methods for the study of the various centralizers when $p$ is not very good for $G$ involve case-by-case study of the simple types. What is known so far makes it unlikely that more uniform methods will emerge.

5) In type $A_{n-1}$, comparison with the well-behaved centralizers in $\text{GL}_n(K)$ shows that centralizer dimensions go up by 1 in $\text{Lie}\, \text{SL}_n(K)$ for unipotent or nilpotent elements just when $\mathfrak{z} \neq 0$ (i.e., $p|n$).

6) In other types, the literature seems inconclusive about further unusual contributions to $\mathfrak{c}_{\mathfrak{g}} (u)$. (The new AMS monograph here by Martin Liebeck and Gary Seitz promises to deal systematically but still case-by-case with many fine points about classes and centralizers for unipotent elements in the group and nilpotent elements in the Lie algebra.) So far the emphasis has been on classification and counting of unipotent classes or nilpotent orbits for bad primes, along with explicit descriptions of centralizers in $G$. In fact, it's unclear to me what consequences a resolution of the problem posed by Steinberg in his (12) might have (rephrased here as the second question). Steinberg himself looked directly at the case of a regular unipotent element in section 4 of his 1965 IHES paper on regular elements (see 4.3), where global and infinitesimal centralizers differ at most by $\mathfrak{z}$; but this requires delicate ad hoc arguments.

7) In his 1966 ICM survey, Steinberg's format is stimulating and unusual: he gives a concise list of 28 items, including summaries of recent work along with 18 items labelled "Problem". These are of varying breadth or precision but are by now mostly resolved one way or the other. Problem (10), later resolved by Lusztig, involves the finiteness of the number of unipotent classes in all simple algebraic groups; this isn't as "modest" as Steinberg expected it to be. On the other hand, problem (22) was over-optimistic. No doubt it would be useful for someone to write a short follow-up survey on the status of all those problems. (In my own 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups I aimed mostly for the uniform results in good characteristic. The book lists most of the relevant earlier references.)