SUBGROUP GROWTH IN SOME PROFINITE CHEVALLEY GROUPS

arXiv:1511.04333v4 [] 2 Nov 2016

INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN
Abstract. In this article we improve the known uniform bound for subgroup growth of Chevalley groups G(Fp [[t]]). We introduce a new parameter, the ridgeline number v(G), and give new
bounds for the subgroup growth of G(Fp [[t]]) expressed through v(G). We achieve this by deriving a new estimate for the codimension of [U, V ] where U and V are vector subspaces in the
Lie algebra of G.

For a finitely generated group G, let an (G) be the number of subgroups of G of index n
and sn (G) the number of subgroups of G of index at most n. The “subgroup growth” of G is
the asymptotic behaviour of the sequence (an (G))n∈N . It turns out that the subgroup growth and
structure of G are not unrelated, and in fact the latter can sometimes be used as a characterisation
of G. (For complete details we refer the reader to a book by Lubotzky and Segal [LSe03].)
For example, Lubotzky and Mann [LM91] show that a group G is p-adic analytic if and only
if there exists a constant c > 0 such that an (G) < nc . This inspiring result is followed by Shalev
who proves that if G is a pro-p group for which an (G) ≤ nc logp n for some constant c < 81 , then G
is p-adic analytic.
Answering a question of Mann, Barnea and Guralnick investigate the subgroup growth of
SL12 (Fp [[t]]) for p > 2, and show that the supremum of the set of all those c that a pro-p group G
is p-adic analytic provided that an (G) < nc logp n , is no bigger than 21 . Thus one may see that not
only the growth type, but also the precise values of the constants involved are important when
studying the connection between subgroup growth and the structure of a group.
Later on Lubotzky and Shalev pioneer a study of the so-called Λ-standard groups [LSh94]. A
particular subclass of these groups are Λ-perfect groups for which they show existence of a constant
c > 0 such that
an (G) < nc logp n .
An important subclass of those groups are the congruence subgroups of Chevalley groups over
Fp [[t]]. Let G be a simple simply connected Chevalley group scheme, G(1) the first congruence
subgroup of G(Fp [[t]]). Abért, Nikolov and Szegedy show that if m is the dimension of G, then
7

spk (G(1)) ≤ p 2 k

2

+mk

,

7

that is, sn (G(1)) ≤ n 2 logp n+m [ANS03].
In this article we improve their estimates (cf. [ANS03]). The Lie algebra gZ of G is defined
over integers. Let K be a field of characteristic p, which could be either zero or a prime. To state
the results we now introduce a new parameter of the Lie algebra g := gZ ⊗Z K. We fix an invariant
bilinear form η = h , i on g of maximal possible rank. Let g0 be its kernel. Notice that the nullity
r := dim g0 of η is independent of the choice of η.
Definition 0.1. Let l be the rank of g, m its dimension and s the maximal dimension of the
centraliser of a non-central element x ∈ g. We define the ridgeline number of g as
l
v(G) = v(g) :=
.
m−s−r
We discuss ridgeline numbers in Section 2. The values of v(g) can be found in the table in
Appendix A.
Date: November 2, 2016.
1991 Mathematics Subject Classification. Primary 20E07; Secondary 17B45, 17B70.
Key words and phrases. subgroup growth, Lie algebras, Chevalley groups.
1

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INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN

Definition 0.2. The positive characteristic p of the field K is called good if p does not divide the
coefficients of the highest root. The positive characteristic p of the field K is called very good if p
is good and g is simple. We call the positive characteristic p tolerable if any proper ideal of g is
contained in its centre.
We discuss these restrictions in Section 2. We may now state our main result.
Theorem 0.3. Let G be a simple simply connected Chevalley group scheme of rank l ≥ 2. Suppose
p is a tolerable prime for G. Let G(1) be the first congruence subgroup of G(Fp [[t]]), that is
G(1) = ker(G(Fp [[t]]) ։ G(Fp )). If m := dim G, then
apk (G(1)) ≤ p

(3+4v(g))
2

k2 +(m− 23 −2v(g))k

.

If l = 2 and p is very good, then a stronger estimate holds:
3

apk (G(1)) ≤ p 2 k

2

+(m− 32 )k

.

Notice that as one can see from the table in Appendix A, with one exception (when G = Al
and p divides l + 1) the biggest possible value of v(g) is 12 (v(g) ≤ 32 in that special case). This
makes 3+4v(g)
≤ 52 ( 3+4v(g)
≤ 17
2
2
6 correspondingly).
Our proof of Theorem 0.3 in many ways follows the ones of Barnea and Guralnick and of Abért,
Nikolov and Szegedy. The improvement in the result is due to the following new estimates.
Theorem 0.4. Let a be a Lie algebra over a field K. Suppose that the Lie algebra g = a ⊗K K is
a Chevalley Lie algebra of rank l ≥ 2 and that the characteristic of K is zero or tolerable. Then
for any two subspaces U and V of a, we have
codim([U, V ]) ≤ (1 + v(g))(codim(U ) + codim(V )).
If l = 2 and the characteristic of K is zero or very good, a stronger result holds:
codim([U, V ]) ≤ codim(U ) + codim(V ).
We conjecture that the second estimate holds for any Lie algebra g (if the characteristic of K
is zero or very good).
1. Proof of Theorem 0.3
The proof of Theorem 0.3 relies on Theorem 0.4 that will be proved later. We follow Abért,
Nikolov and Szegedy [ANS03, Theorem 2] and Barnea, Guralnick [BG01, Theorem 1.4].
Suppose that hypotheses of Theorem 0.3 hold. We start with the following observation (cf.
[ANS03, Corollary 1 and Lemma 1] and [LSh94, Lemma 4.1] ).
Lemma 1.1. If H is an open subgroup of G(1) and d(H) is the minimal number of generators of
H, then
d(H) ≤ m + (3 + 4v(g)) logp |G(1) : H|.
Moreover, if l = 2 and p is very good, then
d(H) ≤ m + 3 logp |G(1) : H|.
Notice that in the second case g = A2 , C2 or G2 and m = 8, 10 or 14 correspondingly.
Proof. First of all recall that d(H) = logp |H : Φ(H)| ≤ logp |H : H ′ | where Φ(H) is the Frattini
subgroup. Because of the correspondence between the open subgroups of G(1) and subalgebras of
its graded Lie algebra L = L(G(1)) (see [LSh94]), logp |H : H ′ | ≤ dim H/H′ where H = L(H) is
the corresponding subalgebra of L. Hence it suffices to show that
dim H/H′ ≤ m + (3 + 4v(g)) dim L/H
in the general case, and that
dim H/H′ ≤ m + 3 dim L/H
in the very good rank 2 case.

SUBGROUP GROWTH

3

Recall that the graded Lie algebra L is isomorphic to g ⊗F tF[t] where F = Fp . Since every
i
element a ∈ L can be uniquely written as a = Σ∞
i=1 ai ⊗ t with ai ∈ g, one can define l(a) := as
where s is the smallest integer such that as 6= 0, and in this case s := deg(a). Now set
Hi := hl(a) | a ∈ H with deg(a) = ii.
Observe that Hi = {l(a) | a ∈ H with deg(a) = i} ∪ {0}. Then dim L/H = Σ∞
i=1 dim g/Hi , and
this sum is finite as the left hand side is finite. Then
′
[Hi ⊗ ti , Hj ⊗ tj ] ⊆ [Hi , Hj ] ⊗ ti+j ⊆ Hi+j
⊗ ti+j
′
′
:= hl(a) | a ∈ H′ with deg(a) = i + ji, and so dim g/[Hi , Hj ] ≥ dim g/Hi+j
. Adding
where Hi+j
up these inequalities for i = j and i = j + 1 we get
∞
′
dim L/H′ = Σ∞
i=1 dim L/Hi ≤ dim g + Σ1≤i≤j≤i+1 dim g/[Hi , Hj ].

Now we use the estimates of Theorem 0.4:
dim L/H′ ≤ m + Σ∞
1≤i≤j≤i+1 α(dim g/Hi + dim g/Hj ) ≤ m + 4α dim L/H,
where α = 1 + v(g) or 1 depending on the rank of g and p. The result follows immediately.
Now we apply an estimate [LSh94, Lemma 4.1]: apk (G(1)) ≤ p

g1 +...+gpk−1



where
i

gpi = gpi (G(1)) = max{d(H) | H ≤open G(1), |G(1) : H| = p }.
Using Lemma 1.1, in the general case (l ≥ 2) we have
i=k−1

apk (G(1)) ≤ pΣi=0

m+(3+4v(g))i

=p

(3+4v(g)) 2
k +(m− 23 −2v(g))k
2

.

For l = 2 and very good p, Lemma 1.1 gives us
i=k−1

apk (G(1)) ≤ pΣi=0

m+3i

3

= p2k

2

+(m− 32 )k

.

This finishes the proof of the theorem.
2. Ridgeline numbers and small primes
We adopt the notations of Definition 0.1. We prove that m − s = 2(h∨ − 1) where h∨ is the
dual Coxeter number of g (see Proposition 3.4). Therefore,
v(g) =

2(h∨

l
.
− 1) − r

We present the values of v(g) in Appendix A. We include only Lie algebras in tolerable characteristics (see Definition 0.2) where our method produces new results.
Let us remind the reader that the very good characteristics are p ∤ l + 1 in type Al , p 6= 2
in types Bl , Cl , Dl , p 6= 2, 3 in types E6 , E7 , F4 , G2 , and p 6= 2, 3, 5 in type E8 . If p is very
good, the Lie algebra g behaves as in characteristic zero. In particular, g is simple, its Killing
form is non-degenerate, etc. Let us contemplate what calamities betide the Lie algebra g in small
characteristics.
Suppose that p is tolerable but not very good. If p does not divide the determinant of the
Cartan matrix of g, the Lie algebra g is simple. This covers the following primes: p = 2 in types
E6 and G2 , p = 3 in types E7 and F4 , p = 2, 3, 5 in type E8 . In this scenario, the g-modules g
and g∗ are isomorphic, which immediately gives us a non-degenerate invariant bilinear form on g
[H95, 0.13].
If p divides the determinant of the Cartan matrix of g, there is more than one Chevalley Lie
algebra. We study the simply connected Lie algebra g, i.e., [g, g] = g and g/z is simple (where z is
the centre). There is a canonical map to the adjoint Lie algebra g♭ :
M
M
ϕ:g=h⊕
gα → g♭ = h♭ ⊕
gα .
α

α

The map ϕ is the identity on the root spaces gα . Let us describe it on the Cartan subalgebras.
The basis of the Cartan subalgebra h are the simple coroots hi = α∨
i = [eαi , e−αi ]. The basis of

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INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN

the Cartan subalgebra h♭ are the fundamental coweights yi = ̟i∨ defined by αi (yj ) = δi,j . Now
the map ϕ on the Cartan subalgebras is given by
X
cj,i yj
ϕ(hi ) =
j

where cj,i are entries of the Cartan matrix of the coroot system of g. The image of ϕ is [g♭ , g♭ ].
The kernel of ϕ is the centre z. From our description z is the subspace of h equal to the null space
of the Cartan matrix. It is equal to g0 , the kernel of η. The dimension of z is at most 2 (see the
values of r in Appendix A).
The key dichotomy now is whether the Lie algebra g/z is simple or not. If g is simply-laced,
the algebra g/z is simple. This occurs when p | l + 1 in type Al , p = 2 in types Dl and E7 , p = 3
in type E6 . Notice that A1 in characteristic 2 needs to be excluded: g/z is abelian rather than
simple. In this scenario the g-modules g/z and (g/z)∗ are isomorphic. This gives us an invariant
bilinear form with the kernel z [H95, 0.13].
Let us look meticulously at g of type Dl when p = 2. The standard representation gives a
homomorphism of Lie algebras


ρ11 (x) ρ12 (x)
ρ : g → so2l (K), x 7→ ρ(x) =
,
ρ21 (x) ρ22 (x)
where ρ22 (x) = ρ11 (x)t , while ρ12 (x) and ρ21 (x) are skew-symmetric l × l-matrices, which for
p = 2 is equivalent to symmetric with zeroes on the diagonal. The Lie algebra so2l (K) has a
1-dimensional centre spanned by the identity matrix. If l is odd, ρ is an isomorphism, and g has a
1-dimensional centre. However, if l is even, ρ has a 1-dimensional kernel, and g has a 2-dimensional
centre.
It is instructive to observe how the standard representation ρ equips g with an invariant form.
A skew-symmetric matrix Z can be written uniquely as a sum Z = Z L + Z U , where Z L is strictly
lower triangular and Z U is strictly upper triangular. Then the bilinear form is given by
η(x, y) := hρ(x), ρ(y)i := Tr (ρ11 (x)ρ11 (y) + ρ12 (x)L ρ21 (y)U + ρ21 (x)L ρ12 (y)U ).
This form η is a reduction of the form 12 Tr (ϕ(x)ϕ(y)) on so2l (Z), hence it is invariant.
Finally we suppose that p is not tolerable. This happens when p = 2 in types Bl , Cl and F4
or p = 3 in type G2 . In all these cases g is not simply-laced and the quotient algebra g/z is not
simple. The short root vectors generate a proper non-central ideal I. This ideal sits in the kernel
of any non-zero invariant form. Consequently, our method fails to produce any new result.
3. Proof of Theorem 0.4: the General Case
Let a be an m-dimensional Lie algebra over a field K of characteristic p (prime or zero). We
consider it as a topological space in the Zariski topology. We also consider a function dim ◦c : a → R
that for an element x ∈ a computes the dimension of its centraliser c(x).
Lemma 3.1. The function dim ◦c is upper semicontinuous, i.e., for any number n the set
{x ∈ a | dim(c(x)) ≤ n} is Zariski open.
Proof. Observe that c(x) is the kernel of the adjoint operator ad(x). Thus, dim(c(x)) ≤ n is
equivalent to rank(ad(x)) ≥ m − n. This is clearly an open condition, given by the non-vanishing
of one of the (m − n)-minors.

Now we move to K, the algebraic closure of K. Let ā = a ⊗K K. To distinguish centralisers in
a and ā we denote c(x) := ca (x) and c̄(x) := cā (x). Now we assume that ā is the Lie algebra of a
connected algebraic group A. Let Orb(x) be the A-orbit of an element x ∈ ā.
Lemma 3.2. Let x and y be elements of ā such that x ∈ Orb(y). Then dim c̄(x) ≥ dim c̄(y).
Proof. The orbit Orb(y) intersects any open neighbourhood of x, and, in particular, the set
X = {z ∈ ā | dim(c̄(z)) ≤ dim(c̄(x))}, which is open by Lemma 3.1. If z ∈ Orb(y) ∩ X, then
dim c̄(x) ≥ dim c̄(z) = dim c̄(y).


SUBGROUP GROWTH

5

The stabiliser subscheme Ax is, in general, non-reduced in positive characteristic. It is reduced
(equivalently, smooth) if and only if the inclusion c(x) ⊇ Lie(Ax ) is an equality (cf. [H95, 1.10]).
If Ax is smooth, the orbit-stabiliser theorem implies that
dim(a) = m = dim Ax + dim Orb(x) = dim c̄(x) + dim Orb(x).
In particular, Lemma 3.2 follows from the inequality dim Orb(x) ≤ dim Orb(y).
Let us further assume that A = G is a simple connected simply-connected algebraic group
and ā = g is a simply-connected Chevalley Lie algebra. Let us fix a triangular decomposition
g = n− ⊕ h ⊕ n. An element x ∈ g is called semisimple if Orb(x) ∩ h 6= ∅. An element x ∈ g
is called nilpotent if Orb(x) ∩ n 6= ∅. We call a representation x = xs + xn a quasi-Jordan
decomposition if xs ∈ g(h) (image of h under g) and xn ∈ g(n) for the same g ∈ G.
Recall that a Jordan decomposition is a quasi-Jordan decomposition x = xs + xn such that
[xs , xn ] = 0. A Jordan decomposition exists and is unique if g admits a non-degenerate bilinear
form [KW, Theorem 4].
Notice that part (1) of the following lemma cannot be proved by the argument that the Lie subalgebra Kx is contained in a maximal soluble subalgebra: in characteristic 2 the Borel subalgebra
b = h ⊕ n is not maximal soluble.
Lemma 3.3. Assume that p 6= 2 or G is not of type Cl (in particular, this excludes C2 = B2 and
C1 = A1 ). Then the following statements hold.
(1) Every x ∈ g admits a (non-unique) quasi-Jordan decomposition x = xs + xn .
(2) xs belongs to the orbit closure Orb(x).
(3) If Orb(x) is closed, then x is semisimple.
(4) dim c̄(xs ) ≥ dim c̄(x).
Proof. (cf. [KW, Section 3].) (1) Our assumption on g assures the existence of a regular semisimple
element h ∈ h, i.e., an element such that c̄(h) = h. The differential d(e,h) a : g ⊕ h → g of the
action map a : G × h → g is given by the formula
d(e,h) a(x, k) = [x, h] + k.
Since the adjoint operator ad(h) is a diagonalizable operator whose 0-eigenspace is h, the kernel
of d(e,x) a is h ⊕ 0. Hence, the image of a contains an open subset of g. Since the set ∪g∈G g(b)
contains the image of a, it is a dense subset of g.
Let B be the Borel subgroup of G whose Lie algebra is b. The quotient space F = G/B is a
flag variety. Since F is projective, the projection map π : g × F → g is proper. The Springer
variety S = {(x, g(B)) | x ∈ g(b)} is closed in g × F . Hence, ∪g∈G g(b) = π(S) is closed in g. Thus,
∪g∈G g(b) = g. Choosing g such that x ∈ g(b) gives a decomposition.
(2) Suppose xs ∈ g(h). Let T be the torus whose Lie algebra is g(h). We decompose x over the
roots of T :
X
xα .
x = xs + xn = x0 +
α∈Y (T )

We can choose a basis of Y (T ) so that only positive roots appear. Hence, the action map a : T → g,
a(t) = t(x) extends alone the embedding T ֒→ Kl to a map b
a : Kl → g. Observe that xs = b
a(0).
−1
Let U ∋ xs be an open subset of g. Then b
a (U ) is open in Kl and T ∩ b
a −1 (U ) is not empty.
Pick t ∈ T ∩ b
a −1 (U ). Then a(t) = t(x) ∈ U , thus, xs ∈ T (x) ⊆ Orb(x).
(3) This immediately follows from (1) and (2).
(4) This immediately follows from (2) and Lemma 3.2.

If α is a long simple root, its root vector eα ∈ g = ā is known as the minimal nilpotent. The
dimension of Orb(eα ) is equal to 2(h∨ − 1) (cf. [W99]).
Proposition 3.4. Suppose that l ≥ 2 and that the characteristic p of the field K is tolerable for
g. Then for any noncentral x ∈ a
dim c(x) ≤ dim c̄(eα ) = m − 2(h∨ − 1).

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INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN

Proof. Let x ∈ a (y ∈ g) be a noncentral element with c(x) (c̄(y) correspondingly) of the largest
possible dimension. Observe that dim c(x) ≤ dim c̄(x) ≤ dim c̄(y).
Let us examine a quasi-Jordan decomposition y = ys +yn . Since ys ∈ Orb(y), we conclude that
dim c̄(ys ) ≥ dim c̄(y). But dim c̄(y) is assumed to be maximal. There are two ways to reconcile
this: either dim c̄(ys ) = dim c̄(y), or ys is central.
Suppose ys is central. Then y and yn have the same centralisers. We may assume that y = yn
is nilpotent. Lemma 3.2 allows us to assume without loss of generality that the orbit Orb(y) is
minimal, that is, Orb(y) = Orb(y) ∪ {0}. On the other hand, the closure Orb(y) contains a root
vector eβ .
Let us prove the last statement. First, observe that K× y ⊆ Orb(y). If p is good, this immediately follows from Premet’s version of Jacobson-Morozov Theorem [P95]. If Orb(λy) 6= Orb(y) in
an exceptional Lie algebra in a bad tolerable characteristic, then we observe two distinct nilpotent
orbits with the same partition into Jordan blocks. It never occurs: all the partitions are listed in
the VIGRE paper [V05, section 6]. The remaining case of p = 2 and g is of type Dl is also settled
in the VIGRE paper [V05]. Now let y ∈ g(n), and T0 be the torus whose Lie algebra is g(h).
:= T0 × K× with the second factor acting on g via the vector space structure. Write
Consider
P T
y = β∈Y (T0 ) yβ using the roots of T0 . The closure of the orbit T (y) is contained in Orb(y). Let
us show that T (y) contains one of yβ . Let us write T0 = Gm × Gm × . . . × Gm and decompose
y = yk + yk+1 + . . . + yn using the weights of the first factor Gm with yk 6= 0. Then
T (y) ⊇ {(λ, 1, 1 . . . , 1, λ−k ) · y|λ ∈ K× } = {yk + λ1 yk+1 + . . . + λn−k yn |λ ∈ K× }.
Hence, yk ∈ T (y). Repeat this argument with yk instead of y for the second factor of T0 , and so
on. At the end we arrive at nonzero yβ , hence, eβ ∈ Orb(y).
Without loss of generality we now assume that y = eβ for a simple root β. If p is good, then
dim(c̄(eβ )) does not depend on the field:
M
c̄(eβ ) = ker(dβ : h → K) ⊕
gγ .
γ+β is not a root
In particular, it is as in characteristic zero: the long root vector has a larger centraliser then the
short root vector and dim c̄(y) = dim c̄(eα ) = m − 2(h∨ − 1) [W99]. If p = 2 and g is of type
Dl , then a direct calculation gives the same formula for dim c̄(eα ). In the exceptional cases in
bad characteristics the orbits and their centralisers are computed in the VIGRE paper [V05]. One
goes through their tables and establishes the formula for dim c̄(y) in all the cases.
Now suppose dim c̄(ys ) = dim c̄(y). We may assume that y = ys is semisimple. Then y is in
some Cartan subalgebra g −1 (h) and dim c̄(g(y)) = dim c̄(y). Moreover,
M
gα
c̄(g(y)) = h ⊕
{α|α(g(y))=0}

is a reductive subalgebra. If ϕ : g → g♭ is the canonical map (see Section 2), then dim c̄(g(y)) =
dim cg♭ (ϕ(g(y))). It remains to examine the Lie algebras case by case and exhibit a non-zero
element in h♭ with the maximal dimension of centraliser. This is done in Appendix A.

Now we can give a crucial dimension estimate for the proof of Theorem 0.4.
Proposition 3.5. Let a be an m-dimensional Lie algebra with an associative bilinear form η,
whose kernel a0 is the centre of a. Suppose r = dim(a0 ) and k ≥ dim(c(x)) for any non-central
element x ∈ a. Finally, let U, V be subspaces of a such that dim(U ) + dim(V ) > m + k + r. Then
[U, V ] = a.
Proof. Suppose not. Let us consider the orthogonal complement W = [U, V ]⊥ 6= a0 under the
form η. Observe that U ⊆ [V, W ]⊥ since η is associative. But W admits a noncentral element
x ∈ W so that dim(c(x)) ≤ k. Hence
dim([V, W ]) ≥ dim(V ) − k and dim([V, W ]⊥ ) ≤ m + k + r − dim(V ).
Inevitably, dim(U ) ≤ m + k + r − dim(V ).



SUBGROUP GROWTH

7

We may now prove the first part of Theorem 0.4. We use m, l, r and s as in Definition 0.1. If
dim(U ) + dim(V ) > m + s + r, we are done by Proposition 3.5:
codim([U, V ]) = 0 ≤ (1 + v(g))(codim(U ) + codim(V )).
Now we assume that dim(U ) + dim(V ) ≤ m + s + r. It is known [ANS03] that
codim([U, V ]) ≤ l + codim(U ) + codim(V ).
It remains to notice that l = v(g)(m−s−r) ≤ v(g)(codim(U )+codim(V )). The theorem is proved.
4. Proof of Theorem 0.4: Rank 2
In this section G is a Chevalley group scheme of rank 2. The characteristic p of the field K is
zero or very good for g. Let {α, β} be the set of simple roots of g with |β| ≤ |α|. If g is of type A2
then α and β have the same length. The group G = G(K) acts on on g via the adjoint action. By
c(x) we denote the centraliser cg (x) in this section. Let us summarise some standard facts about
this adjoint action (cf. [H95]).
If x ∈ g, the stabiliser Gx is smooth, i.e., its Lie algebra is the centraliser c(x).
The dimensions dim(Orb(x)) = dim(G) − dim(c(x)) and dim(c(x)) are even.
If x 6= 0 is semisimple, dim(c(x)) ∈ {2, 4}. Hence, dim(Orb(x)) ∈ {m − 2, m − 4}.
A truly mixed element x = xs + xn (with non-zero semisimple and nilpotent parts) is
regular, i.e., dim(c(x)) = 2 (cf. Lemma 3.3).
(5) x is nilpotent if and only if Orb(x) contains 0.
(6) There is a unique orbit of regular nilpotent elements Orb(er ) where er = eα + eβ . In
particular, dim(c(er )) = 2 and dim(Orb(er )) = m − 2.
(7) For two nilpotent elements x and y we write x  y if Orb(x) ⊇ Orb(y). The following are
representatives of all the nilpotent orbits in g (in brackets we report [dim(Orb(x)), dim(c(x))]):
(a) If G is of type A2 , then

(1)
(2)
(3)
(4)

er [6, 2]  eα [4, 4]  0 [0, 8] .
(b) If G is of type C2 , then eα and eβ are no longer in the same orbit and so we have
er [8, 2]  eβ [6, 4]  eα [4, 6]  0 [0, 10] .
(c) If G is of type G2 , there is an additional subregular nilpotent orbit of an element
esr = e2α+3β + eβ . In this case we have
er [12, 2]  esr [10, 4]  eβ [8, 6]  eα [6, 8]  0 [0, 14] .
We will now prove Theorem 0.4 for groups of type A2 , C2 and G2 . We need to show that if U
and V are subspaces of g, then
(1)

dim([U, V ]) ≥ dim(U ) + dim(V ) − dim g.

We will use the following device repeatedly:
Lemma 4.1. The inequality
(2)

dim([U, V ]) ≥ dim(V ) − dim(V ∩ c(x))

holds for any x ∈ U . In particular, if there exists x ∈ U such that dim(U )+dim(V ∩c(x)) ≤ dim g,
then inequality (1) holds.
.
Proof. It immediately follows from the observation [U, V ] ⊇ [x, V ] ∼

= V (V ∩ c(x)).
Now we give a case-by-case proof of inequality (1). Without loss of generality we assume that
1 ≤ dim(U ) ≤ dim(V ) and that the field K is algebraically closed.

8

INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN

4.1. G = A2 . Using the standard facts, observe that if x ∈ g \ {0}, then dim(c(x)) ≤ 4. Moreover,
if dim(c(x)) = 4, then either x ∈ Orb(eα ), or x is semisimple. Since dim g = 8, we need to
establish that
dim([U, V ]) ≥ dim(U ) + dim(V ) − 8
Now we consider various possibilities.
Case 1: If dim(U ) ≤ 4, then dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 4 ≤ 8 − dim(U ) for any nonzero
x ∈ U . We are done by Lemma 4.1.
Case 2: If dim(U ) + dim(V ) > 12, then the hypotheses of Proposition 3.5 hold. Hence,
[U, V ] = g that obviously implies the desired conclusion.
Therefore we may suppose that dim(U ) + dim(V ) ≤ 12 and dim U ≥ 5. This leaves us with the
following two cases.
Case 3: dim(U ) = 5 and dim(V ) ≤ 7. We need to show that
dim([U, V ]) ≥ dim(U ) + dim(V ) − 8 = dim(V ) − 3.
As dim(Orb(eα )) = 4, we may pick x ∈ U with x 6∈ Orb(eα ). If x is regular, we are done by
Lemma 4.1 since dim(V ∩ c(x)) ≤ dim(c(x)) = 2. If x is not regular, then dim(c(x)) = 4 and x is
semisimple. In particular, its centraliser c(x) contains a Cartan subalgebra g(h) of g.
Let us consider the intersection V ∩ c(x). If dim(V ∩ c(x)) ≤ 3, we are done by Lemma 4.1.
Otherwise, V ⊇ c(x) and V contains a regular semisimple element y ∈ g(h) ⊆ V . If U ⊇ c(y) =
g(h), then U ∋ y and we are done by Lemma 4.1 as in the previous paragraph. Otherwise,
dim(U ∩ c(y)) ≤ 1 and we finish the proof using Lemma 4.1:
dim([U, V ]) ≥ dim(U ) − dim(U ∩ c(y)) ≥ 5 − 1 = 4 ≥ dim(V ) − 3.
Case 4: dim(U ) = dim(V ) = 6. This time we must show that
dim([U, V ]) ≥ 4 = dim(V ) − 2.
By Lemma 4.1 it suffices to find a regular element in x ∈ U (or in V ) since dim(V ∩ c(x)) ≤
dim(c(x)) = 2. Observe that
dim(U ∩ V ) ≥ dim(U ) + dim(V ) − 8 = 4 = dim(Orb(eα )).
Since Orb(eα ) is an irreducible algebraic variety and not an affine space, there exists x ∈ U ∩ V
such that x 6∈ Orb(eα ). If x is regular, we are done. If x is not regular, x is semisimple and its
centraliser c(x) = Kx ⊕ l, a direct sum of Lie algebras Kx ∼
= K and l ∼
= sl2 (K).
Consider the intersection V ∩ c(x). If dim(V ∩ c(x)) ≤ 2, we are done by Lemma 4.1 as before.
Assume that dim(V ∩ c(x)) ≥ 3. If dim(V ∩ c(x)) = 4, V contains c(x) and consequently a regular
semisimple element y.
Finally, consider the case dim(V ∩ c(x)) = 3. Let π2 be the natural projection π2 : c(x) → l and
set W := π2 (V ∩ c(x)). Since Kx ⊆ V ∩ c(x), the subspace W of sl2 (K) is 2-dimensional. Clearly,
V ∩ c(x) ⊆ Kx ⊕ W . Since both spaces have dimension 3, V ∩ c(x) = Kx ⊕ W . Then W = a⊥
(with respect to the Killing form), where 0 6= a ∈ sl2 (K) is either semisimple or nilpotent. In
both cases W contains a nonzero nilpotent element z. Thus, we have found a regular element
x + z ∈ V ∩ c(x). This finishes the proof for A2 .
4.2. G = C2 . Notice that this time dim(c(x)) ≤ 6 for all 0 6= x ∈ g. Moreover, if dim(c(x)) = 6,
x ∈ Orb(eα ). Finally, the set Orb(eα ) = Orb(eα ) ∪ {0} is a 4-dimensional cone, and the set
Orb(eβ ) = Orb(eβ ) ∪ Orb(eα ) ∪ {0} is a 6-dimensional cone.
As dim g = 10, this time we need to show that
dim([U, V ]) ≥ dim(U ) + dim(V ) − 10 = dim(V ) − (10 − dim(U )).
Case 1: dim(U ) ≤ 4. We are done by Lemma 4.1 since for any 0 6= x ∈ U ,
dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 6 ≤ 10 − dim(U ).

SUBGROUP GROWTH

9

Case 2: 5 ≤ dim(U ) ≤ 6. Hence, we may choose x ∈ U such that x 6∈ Orb(eα ). We are done
by Lemma 4.1 since
dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 4 ≤ 10 − dim(U ).
Case 3: If dim(U ) + dim(V ) > 16, then then the hypotheses of Proposition 3.5 hold. Hence,
[U, V ] = g, which implies the desired conclusion.
Therefore, we may assume that dim(U ) + dim(V ) ≤ 16 and dim(U ) ≥ 7. This leaves us with
the remaining two cases.
Case 4: dim(U ) = 7, dim(V ) ≤ 9. Now we must show that dim([U, V ]) ≥ dim(V ) − 3. By
Lemma 4.1 it suffices to pick x ∈ U with dim(V ∩ c(x)) ≤ 3. In particular, a regular element will
do.
Let us choose x ∈ U such that x 6∈ Orb(eβ ). If x is regular, we are done. If x is not regular, x
is semisimple. Hence, its centraliser c(x) contains a Cartan subalgebra g(h). Let us consider the
intersection V ∩ c(x). If dim(V ∩ c(x)) ≤ 3, we are done again. Assume that dim(V ∩ c(x)) = 4.
Consequently, V ⊇ c(x) and V contains a regular semisimple element y ∈ g(h) ⊆ V . Now if
U ⊇ c(y) = g(h), then we have found a regular element y ∈ U . Otherwise, dim(U ∩ c(y)) ≤ 1,
and so, as y ∈ V , we finish using inequality (2) of Lemma 4.1:
dim([U, V ]) ≥ dim(U ) − dim(U ∩ c(y)) ≥ 7 − 1 = 6 ≥ dim(V ) − 3.
Case 5: dim(U ) = dim(V ) = 8. Let us observe that
dim(U ∩ V ) ≥ dim(U ) + dim(V ) − 10 = 6 = dim(Orb(eβ )).
Since Orb(eβ ) is an irreducible algebraic variety and not an affine space, there exists x ∈ U ∩ V
such that x 6∈ Orb(eβ ). If x is regular, we are done by Lemma 4.1:
dim([U, V ]) ≥ dim(V ) − dim(V ∩ c(x)) ≥ 8 − 2 = 6 = dim(U ) + dim(V ) − 10.
If x is not regular, then x is semisimple and its centraliser c(x) = Kx ⊕ l, a direct sum of Lie
algebras K and l ∼
= sl2 (K). If dim(V ∩ c(x)) ≤ 2, then by Lemma 4.1
dim([U, V ]) ≥ dim(V ) − dim(V ∩ c(x)) ≥ 8 − 2 = 6.
Thus we may assume that dim(V ∩c(x)) ≥ 3. We now repeat the argument from the last paragraph
of § 4.1. This concludes § 4.2.
4.3. G = G2 . In this case dim(c(x)) ≤ 8 for all 0 6= x ∈ g. Moreover, if dim(c(x)) = 8, then
x ∈ Orb(eα ). The centre of c(eα ) is Keα . Finally, the set Orb(eα ) = Orb(eα ) ∪ {0} is a 6dimensional cone, the set Orb(eβ ) = Orb(eβ ) ∪ Orb(eα ) ∪ {0} is an 8-dimensional cone and the
set Orb(esr ) = Orb(esr ) ∪ Orb(eβ ) ∪ Orb(eα ) ∪ {0} is a 10-dimensional cone.
As dim g = 14, our goal now is to show that
dim([U, V ]) ≥ dim(U ) + dim(V ) − 14
In order to do so, as before, we are going to consider several mutually exclusive cases.
Case 1: dim(U ) ≤ 6. We are done by Lemma 4.1 since for any 0 6= x ∈ U ,
dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 8 ≤ 14 − dim(U ).
Case 2: 7 ≤ dim(U ) ≤ 8. In this case we may choose x ∈ U such that x 6∈ Orb(eα ). We are
done by Lemma 4.1 since
dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 6 ≤ 14 − dim(U ).
Case 3: 9 ≤ dim(U ) ≤ 10. Now we may pick x ∈ U such that x 6∈ Orb(eβ ). Again we are
done by Lemma 4.1 since
dim(V ∩ c(x)) ≤ dim(c(x)) ≤ 4 ≤ 14 − dim(U ).
Case 4: If dim(U ) + dim(V ) > 22, then [U, V ] = g by Proposition 3.5. This leaves us with a
single last possibility.
Case 5: dim(U ) = dim(V ) = 11. It remains to show that
dim([U, V ]) ≥ 8 = dim(V ) − 3.

10

INNA CAPDEBOSCQ, KARINA KIRKINA, AND DMITRIY RUMYNIN

By dimension considerations we can choose x ∈ U such that x 6∈ Orb(esr ). Then dim(c(x)) ≤ 4.
If dim(V ∩ c(x)) ≤ 3, we are done by by Lemma 4.1. Thus we may assume that dim(c(x)) = 4
and c(x) ⊆ V . Since x is not nilpotent, x must be semisimple. Hence, c(x) ⊆ V contains a
Cartan subalgebra g(h) and, therefore, a regular semisimple element y ∈ g(h). We are done by
Lemma 4.1:
dim(U ∩ c(y)) ≤ dim(c(y)) ≤ 2.
We have finished the proof of Theorem 0.4.
Appendix A. Ridgeline numbers and maximal dimensions of centralisers
Column 3 contains the nullity r of an invariant form. It is equal to dim z. Column 4 contains
the dual Coxeter number. Column 5 contains the ridgeline number. Column 6 contains dimension
of the centraliser of the minimal nilpotent. Column 7 contains a minimal non-central semisimple
element in g♭ , using simple coweights yi and the enumeration of roots in Bourbaki [Bo68]. Column
8 contains the dimension of the centraliser of the minimal semisimple element in g♭ .
type of g

p

r

h∨

v(g)

m − 2(h∨ − 1)

y

dim c(y)

Al , l ≥ 2

(p, l + 1) = 1

0

l+1

1
2

l2

y1

l2

Al , l ≥ 2

p | (l + 1)

1

l+1

l
2l−1

l2

y1

l2

Bl , l > 3

p 6= 2

0 2l − 1

2l2 − 3l + 4

y1

2l2 − 3l + 2

Cl , l ≥ 2

p 6= 2

0

2l2 − l

y1

2l2 − 3l + 2

Dl , l ≥ 4

p 6= 2

0 2l − 2

1
4 (1

+

3
2l−3 )

2l2 − 5l + 6

y1

2l2 − 5l + 4

Dl , l = 2l0 ≥ 4

2

2 2l − 2

1
4 (1

+

2
l−2 )

2l2 − 5l + 6

y1

2l2 − 5l + 4

Dl , l = 2l0 + 1 ≥ 4

2

1 2l − 2

1
4 (1

+

7
4l−7 )

2l2 − 5l + 6

y1

2l2 − 5l + 4

G2

p>3

0

4

1
3

8

y1

4

G2

p=2

0

4

1
3

8

y1

6

F4

p 6= 2

0

9

1
4

36

y1

22

E6

p 6= 3

0

12

3
11

56

y1

46

E6

3

1

12

2
7

56

y1

46

E7

p 6= 2

0

18

7
34

99

y7

79

E7

p=2

1

18

7
33

99

y7

79

E8

p 6= 2

0

30

4
29

190

y8

134

E8

p=2

0

30

4
29

190

y3

136

1
4 (1

1
l−1 )

+
1
2

l+1

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E-mail address: I.Capdeboscq@warwick.ac.uk, K.Kirkina@warwick.ac.uk, D.Rumynin@warwick.ac.uk

