THE ORDER OF THE CENTRALIZER OF A REGULAR ELEMENT

DAVID G. A. JACKSON

Abstract. In [6] the truncation operation was defined which, given a finite subgroup H of a group K, associates to a class function ψ on K a class function T_{K/H}(ψ) on N_{K}(H). Let G be an F -split connected reductive algebraic group defined over a finite field of good characteristic and suppose G has connected centre. Let ϕ(w) be the number of rational points on a w-twisted rational maximal torus of G. The purpose of this paper is to prove a formula expressing the order of the centralizer in G^{F} of a regular element of 𝔤^{F} in terms of the truncation of ϕ to a certain parabolic subgroup of the Weyl group of G.

Introduction and Notation

We will assume the reader is familiar with the basic theory of linear algebraic groups. Most of what is needed here can be found in Borel [1] or Borel & Tits [2]. Let G be a connected reductive algebraic group defined over a finite field whose characteristic is a good prime for G with corresponding Frobenius endomorphism F : G → G. We will here suppose that G is F -split and that the centre Z(G) of G is connected. We get an induced action of Frobenius on 𝔤, the Lie algebra of G, which we shall also denote by F. We will used the term rational for F -stable. Fix a maximally split rational maximal torus T and a rational Borel subgroup B containing T. We will denote by Π the set of simple roots defined with respect to the pair (T, B). For J ⊆ Π let L_{J} and B_{J} be as defined in Carter [4, §2.6]. Let W = N_{G}(T)/T be the Weyl group of G with respect to T. If w ∈ W, a rational maximal torus T′ will be said to be w-twisted if T′ = ^{g}T for some g ∈ G with g^{−1}F (g) = ẇ, a representative of w in N = N_{G}(T). A regular element X of 𝔤 is one for which dim C_{G}(X) = rk G.

The Main Theorem of this paper occurs in the author’s PhD thesis [7, §5.4]. Whilst studying the Fourier transforms of generalized Gel’fand-Graev functions it was observed that several results involving regular elements could be obtained in a canonical way using the notion of truncation from the corresponding results for regular semisimple elements. The Main Theorem was obtained by applying this process to the formula which expresses the order of the connected centralizer of a regular semisimple element as the number of rational points of a w-twisted rational maximal torus.

The author would like to ackowledge the support and encouragement of Bob Howlett and Gus Lehrer.

The Main Theorem

First let us recall the definition of the truncation operation.

Definition ([6]). Let H be a finite subgroup of a group K and suppose that ϕ is a class function on K. Define the truncation T_{K/H}(ϕ) of ϕ to H to be the class function on N_{K}(H) given by

T_{K/H}(ϕ)(x) = |H|^{−1} ∑_{y∈H} ϕ(xy).

It is a simple matter to verify that T_{K/H}(ϕ) is indeed a class function on N_{K}(H). It was shown in [6] that if M is a K-module with character ϕ then M^{H}, the space of elements of M fixed by H, is an N_{K}(H)-module with character T_{K/H}(ϕ).

It was also shown in [6] that, under the given conditions on the characteristic, if X ∈ 𝔤^{F} with semisimple part X_{s} then C_{G}(X_{s})^{∘} is of the form ^{x}L_{J} for some J ⊆ Π and x ∈ G with x^{−1}F (x) = ẇ_{1} and w_{1} ∈ N_{W} (J) = {w ∈ W | w(J) = J}. Moreover the conjugacy class C of w_{1} in N_{W} (J) is uniquely determined. In [6] the pair (J, C) is called the genus of X.

Main Theorem. Let G be an F -split connected reductive algebraic group defined over a finite field whose characteristic is a good prime for G and suppose G has connected centre. Let X be a regular element of genus (J, C) in 𝔤^{F} and ϕ be the class function on W given by w ↦ |T_{w}^{F} | where T_{w} is a w-twisted F -stable maximal torus of G. Then |C_{G^{F}} (X)| = T_{W/W_{J}} (ϕ)(C).

Note. We have J = ∅ if and only if X is a regular semisimple element and in this case T_{W/W_{J}} (ϕ) = ϕ. We see that using the truncation operation provides a canonical means of generalizing results concerning regular semisimple elements to regular elements in general: simply replace all occurences of ϕ(w) by T_{W/W_{J}} (ϕ)(C).

First we prove a twisted version of Carter [4, Theorem 3.7.6]. We assume w ∈ W acts on T from the right via t · w = t^{w}.

Lemma 1. Let G be a connected reductive group with connected centre. Let T be an maximally split F -stable maximal torus of G, J ⊆ Π and w_{1} ∈ N_{W} (W_{J}). Then

\frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T^{w^{−1}w_{1}^{−1}∘F} | = q^{|J|}|Z(L_{J})^{w_{1}^{−1}∘F} |.

Proof. Let g ∈ G be such that g^{−1}F (g) = ẇ_{1}. Then ^{g}T is an F -stable maximal torus contained in the F -stable Borel subgroup ^{g}B_{J} of ^{g}L_{J} and is hence a maximally split maximal torus in ^{g}L_{J}. Let T′ = ^{^{g}hg}T = ^{gh}T be an F -stable maximal torus of ^{g}L_{J} where h ∈ L_{J}. Suppose that T′ is twisted from T_{1} = ^{g}T by ^{g}(w_{1}ww_{1}^{−1}) ∈ N(T_{1})/T_{1} so that (^{g}h)^{−1}F (^{g}h) = ^{g}(ẇ_{1}ẇẇ_{1}^{−1}). Then h^{−1}ẇ_{1}F (h) = ẇ_{1}ẇ and |T′^{F} | is the number of t ∈ T such that F (t) = t^{w_{1}w}, that is, |T^{w^{−1}w_{1}^{−1}∘F} |. On the other hand, since T′ is twisted from T_{1} by ^{g}(w_{1}ww_{1}^{−1}), we have

|T′^{F} | = |T_{1}^{^{g}(w_{1}w^{−1}w_{1}^{−1})∘F} |.

Hence

\begin{aligned}\frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T^{w^{−1}w_{1}^{−1}∘F} | & = \frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T_{1}^{^{g}(w_{1}w^{−1}w_{1}^{−1})∘F} | \\ & = \frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T_{1}^{^{g}w^{−1}∘F} | \end{aligned}

upon change of variable. Now, according to the proof of Carter [4, Theorem 3.7.6] applied to the group^{g}L_{J}, this equals q^{|J|}|Z(^{g}L_{J})^{F} | = q^{|J|}|Z(L_{J})^{w_{1}^{−1}∘F} | as required.

We now prove what is essentially the Lie algebra analogue of Digne & Michel [5, Proposition 14.24].

Lemma 2. Let G be a connected reductive algebraic group. If X ∈ 𝔤^{F} is regular nilpotent then, regardless of the characteristic,

|C_{G^{F}} (X)| = q^{l}|Z(G)^{∘}^{F} ||H^{1}(F, Z(G))|

where l is the semisimple rank of G and H^{1}(F, Z(G)) is the set of F -conjugacy classes in Z(G).

Proof. According to Springer [10, 5.9(c)], all the regular nilpotent elements of 𝔤 are conjugate under Ad G. Hence, according to Digne & Michel [5, 3.21], the Ad G^{F} -orbits of regular nilpotent elements in 𝔤^{F} which are Ad G-conjugate to X are parametrized by H^{1}(F, A(X)) where A(X) = C_{G}(X)/C_{G}(X)^{∘} and moreover these Ad G^{F} -orbits have the same cardinality. By Springer & Steinberg [11, 3.7(a)], we have C_{G}(X) = Z(G) · C_{U}(X) (one does not need the restriction that p be good for this). By Springer & Steinberg [11, 3.7(b)] and Keny [8, p. 197, line 8], C_{U}(X) is connected regardless of the characteristic and hence A(X) ≅ Z(G)/Z(G)^{∘}. Thus H^{1}(F, A(X)) ≅ H^{1}(F, Z(G)/Z(G)^{∘}) ≅ H^{1}(F, Z(G)) according to Springer & Steinberg [11, 2.6]. The proof of Digne & Michel [5, 14.23] carries across directly to the Lie algebra and we find that the number of regular nilpotent elements in 𝔤^{F} is |G^{F} |/|Z(G)^{∘}^{F} |q^{l} (that regular nilpotent elements exist is proved by Springer [10, 5.9(b)] for good characteristic and by Keny [8] for bad characteristic). This completes the proof.

Proof of Main Theorem. Let C_{G}(X_{s})^{∘} = ^{g}L_{J} with g^{−1}F (g) = ẇ_{1}. Since X is regular, X_{n} is regular in C_{𝔤}(X_{s}) and hence, according to Lemma 2,

|C_{G^{F}} (X)| = q^{|J|}|Z(^{g}L_{J})^{∘F} ||H^{1}(F, Z(^{g}L_{J}))|.

But, according to Carter [4, Prop. 8.1.4], L_{J} has connected centre and hence

|C_{G^{F}} (X)| = q^{|J|}|Z(^{g}L_{J})^{F} |.

On the other hand,

T_{W/W_{J}} (ϕ)(w_{1}) = \frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T_{w_{1}w}^{F}|

= \frac{1}{|W_{J}|} ∑_{w∈W_{J}}|T^{w^{−1}w_{1}^{−1}∘F} |

= q^{|J|}|Z(^{g}L_{J})^{F} |

by Lemma 1. The result then follows.

We have the following interesting corollary of the main Theorem.

Corollary. Let V = Y (T) ⊗_{ℤ} ℝ. For J ⊆ Π, let V_{J} be the subspace of V spanned by J. If X is a regular element of 𝔤^{F} of genus (J, C) with w_{1} ∈ C then

|C_{G^{F}} (X)| = |Z(G)^{F} | ∑_{i=0}^{l}(−1)^{l−i}q^{i} Tr(w_{1}, Λ^{l−i}(V_{J}^{⊥}))

where Λ^{i}(V_{J}^{⊥}) is the ith exterior power of V_{J}^{⊥}.

Before giving the proof of this corollary we need the following lemma.

Lemma 3. With notation as above, we have (Λ^{i}V)^{W_{J}} = Λ^{i}(V_{J}^{⊥}) for 0 ⩽ i ⩽ l.

Proof. We have a W_{J}-stable decomposition V = V_{J} ⊕ V_{J}^{⊥} in which W_{J} acts trivially on V_{J}^{⊥} and as the reflection representation on V_{J}. Now

Λ^{i}V ≅ ⨁_{j=0}^{i} Λ^{j}V_{J} ⊗ Λ^{i−j}V_{J}^{⊥}.

So let v ∈ (Λ^{i}V)^{W_{J}} and write v = ∑_{j=0}^{i} v_{j} ∧ v′_{i−j} where v_{k} ∈ Λ^{k}V_{J} and v′_{k} ∈ Λ^{k}V_{J}^{⊥}. Then for any w ∈ W_{J} we have

∑_{j=0}^{i} v_{j} ∧ v′_{i−j} = ∑_{j=0}^{i} wv_{j} ∧ v′_{i−j}

and hence v_{j} ∧ v′_{i−j} = wv_{j} ∧ v′_{i−j} for all j ∈ {0, …, i}. Hence v_{j} ∈ (Λ^{j}V_{J})^{W_{J}} and according to Bourbaki [3, Chapitre V, Exercice §2, 3(a)] this implies that v_{j} = 0 if j ⩾ 1. Since (Λ^{0}V_{J})^{W_{J}} ≅ ℝ the result follows.

Proof of Corollary. Since G is F -split, the action of Frobenius on Y (T) ⊗_{ℤ} ℝ is just multiplication by q. Let S = T ∩ G′. Then T = SZ and V = Y (S) ⊗_{ℤ} ℝ is an ldimensional W -module affording the reflection representation. According to Carter [4, 3.3.8], we then have

\begin{aligned}|T_{w}^{F} | & = |Z(G)^{F} | det_{V} (q − w) \\ & = |Z(G)^{F} | ∑_{i=0}^{l}(−1)^{l−i}q^{i}ρ_{l−i}(w) \end{aligned}

where ρ_{i} is the character of the ith exterior power of V (cf. Lehrer [9, Corollary 2.3]). Hence

|C_{G^{F}} (X)| = |Z(G)^{F} | ∑_{i=0}^{l}(−1)^{l−i}q^{i}T_{W/W_{J}} (ρ_{l−i})(w_{1}).

Since ρ_{l−i} is the character of the W -module Λ^{l−i}V, T_{W/W_{J}} (ρ_{l−i}) is the character of the N_{W} (W_{J})-module (Λ^{l−i}V)^{W_{J}}. The result then follows from Lemma 3.

References

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[2] Armand Borel and Jacques Tits. Groupes réductifs. Inst. Hautes Études Sci. Publ. Math., 27:55– 152, 1965.

[3] N. Bourbaki. Groupes et Algèbres de Lie. Éléments de Mathématique. Masson, Paris, 1981.

[4] Roger W. Carter. Finite Groups of Lie Type. Conjugacy Classes and Complex Characters. John Wiley and Sons, Chichester, second edition, 1993.

[5] François Digne and Jean Michel. Representations of Finite Groups of Lie Type. Number 21 in London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1991.

[6] D. G. A. Jackson. Values of the Fourier transforms of Green functions at regular elements. In preparation.

[7] D. G. A. Jackson. On Nilpotent Orbits of Type G_{2}. PhD thesis, The University of Sydney, 1997.

[8] Sharad V. Keny. Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics. J. Algebra, 108:195–201, 1987.

[9] G. I. Lehrer. Rational tori, semisimple orbits and the topology of hyperplane complements. Commentarii Math. Helv., 67:226–251, 1992.

[10] T. A. Springer. Some arithmetical results of semi-simple Lie algebras. Inst. Hautes Études Sci. Publ. Math., 30:115–141, 1966.

[11] T. A. Springer and R. Steinberg. Conjugacy classes. In Seminar on Algebraic Groups and Related Finite Groups, number 131 in Lecture Notes in Mathematics, part E, pages 167–266. Springer-Verlag, Berlin-Heidelberg, 1986. Second Corrected Printing.

School of Mathematics and Statistics, The University of Sydney, Sydney, Australia 2006

E-mail address: davidj@maths.usyd.edu.au