Aspects of the conjugacy class structure of simple algebraic groups
J Algebr Comb (2010) 31: 319–353
DOI 10.1007/s10801-009-0187-y
Aspects of the conjugacy class structure of simple
algebraic groups
Martin Cook
Received: 11 July 2006 / Accepted: 2 June 2009 / Published online: 26 June 2009
© Springer Science+Business Media, LLC 2009
Abstract Let G be an adjoint simple algebraic group over an algebraically closed
field of characteristic p; let be the root system of G, and take t ∈ N. Lawther has
proven that the dimension of the set G[t] = {g ∈ G : g t = 1} depends only on and t.
In particular the value is independent of the characteristic p; this was observed for
t small and prime by Liebeck. Since G[t] is clearly a disjoint union of conjugacy
classes the question arises as to whether a similar result holds if we replace G[t] by
one of those classes. This paper provides a partial answer to that question. A special
case of what we have proven is the following. Take p, q to be distinct primes and
Gp and Gq to be adjoint simple algebraic groups with the same root system and over
algebraically closed fields of characteristic p and q respectively. If s ∈ Gp has order
q then there exists an element u ∈ Gq such that o(u) = o(s) and dim uGq = dim s Gp .
Keywords Algebraic groups · Conjugacy classes · Characteristic independent
1 Background and main results
The present paper comes from the PhD thesis of the author; the main results of which
are given below. However first we must make some definitions. So let X and Y be
reductive groups (which for us means they are connected) over algebraically closed
fields with (possibly different) prime characteristics. A reductive group X is said to
have type (, n) if rank X = n and the commutator subgroup, [X, X], is semisimple
with root system . If X is simple we will often abuse this notation and refer to
as the type; furthermore if is a family of irreducible root systems and X is simple,
we say X has type if ∈ . We write X ∼t Y if X and Y have the same type;
The author gratefully acknowledges the support of the EPSRC during the preparation of this work.
M. Cook ()
Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK
e-mail:
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J Algebr Comb (2010) 31: 319–353
this is clearly an equivalence relation. Next let be a root system, p a prime and let
G() be an adjoint simple algebraic group of type (we write G()p if the group
is over Fp ). It is worth stating at this point that throughout this paper all groups will
be over algebraically closed fields (in fact all fields are of the form Fp ); we are not
considering rationality properties here. Now set
(X, ) = {s : s is a semisimple element of the group G()p ,
for some prime p, and CG()p (s)◦ ∼t X}.
(Recall that since s is a semisimple element, CG()p (s)◦ is reductive.) Next set
omin (X, ) = min{o(s) : s ∈ (X, )}
(here o(s) is the order of the group element s) and for a prime number q set
q (X,) = min{q e : q e ≥ omin (X, )}.
Now given a reductive subgroup X ≤ G()p we say that a prime q is admissible
with respect to X if for some prime r and nonnegative integer i there is a semisimple
element s ∈ G()r such that o(s) = q i and CG()r (s)◦ ∼t X. Note that the identity
element is semisimple and of order 1 = q 0 . Thus every prime q is admissible with
respect to X = G()p ∼t G()r = CG()r (1)◦ . Finally we come to the main results.
Theorem 1 Let Gp = G()p be an adjoint simple algebraic group of type
over Fp . Let X ≤ Gp be the connected centralizer of a semisimple element; if
= Bn , Cn or Dn for some n let q be admissible with respect to X, otherwise let q
be any prime. Then there exists a unipotent conjugacy class C ⊂ Gq = G()q such
that for any u ∈ C
dim CGq (u) = dim X
and
q (X,) = o(u) or q.o(u).
Equality between q (X,) and o(u) holds in most cases and holds in all but one
case if q is a good prime. Full details are given in Theorem 4 at the end of the paper.
There is an immediate corollary to Theorem 1.
Corollary 2 Given a semisimple element s ∈ Gp = G()p and a prime q which is
admissible with respect to X = CGp (s)◦ if = Bn , Cn or Dn for some n, there is a
unipotent element u ∈ Gq = G()q such that dim CGq (u) = dim X and o(u) divides
min{q e : q e ≥ o(s)}.
Proof Take any u in the class C given by Theorem 1 and observe that q (X,) divides
min {q e : q e ≥ o(s)}, since o(s) ≥ omin (X, ).
If we have a semisimple element of prime order we can make a stronger statement.
Theorem 3 Let p, q be distinct primes and let Gp and Gq be as above. If s ∈ Gp
has order q then there exists an element u ∈ Gq such that o(u) = o(s) and dim uGq =
dim s Gp .
J Algebr Comb (2010) 31: 319–353
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In [5] it was proven that given a natural number t and an adjoint simple algebraic
group G over an algebraically closed field of characteristic p, the dimension of the
set G[t] = {g ∈ G : g t = 1} is independent of p. Clearly G[t] is a union of conjugacy
classes. The author’s thesis was an attempt to see what could be said if G[t] was
replaced by one of those classes. The statement of Theorem 3 above is in the style
of [5].
The proof of Theorem 1 will be given by case analysis in the following sections.
The individual proofs are quite straightforward once certain functions are defined.
These functions may appear to come from nowhere, but can be characterized combinatorially. This will be done in future papers and hints that a more uniform approach
may be possible. For the moment though, this possibility has not been realized and
we must stick with case analysis. So Section 2 recalls an important algorithm for finding all connected centralizers of semisimple elements in a simple algebraic group and
uses this to prove Theorem 1 for the exceptional cases. Section 3 lays out our strategy
for proving Theorem 1. Sections 4 to 7 deal with types A, C, D and B respectively.
Theorem 3 is proven in Section 8.
2 Known results and exceptional groups
This section recalls some well known results of Steinberg and a useful result from
[2] and then applies them to the exceptional groups. Recall that if s is a semisimple
element, T a maximal torus containing s and {Xα : α ∈ } the root subgroups with
respect to T , then it can be shown that CG (s)◦ = T , Xα : α(s) = 1 . Furthermore
CG (s)◦ is reductive with root system = {α ∈ : α(s) = 1}. Finally if G is simplyconnected then we have that CG (s)◦ = CG (s). This material is due to Steinberg and
can be found in section 3.5 of [1]. Now given t ∈ N and a simple simply-connected
group Gsc , let Gad be the adjoint group in the isogeny class of Gsc and let φ : Gsc →
Gad be an isogeny. Also let s ∈ Gsc be a semisimple element such that the order
of φ(s) is t. Note that if Gad is over a field of characteristic p then p must not
divide t. An algorithm is given in [2] for finding all possible centralizers of s ∈ Gsc .
The following lemma is essentially taken from [5] and shows that the algorithm can
actually be used to determine all possible connected centralizers of a (...truncated)
