C. R. Acad. Sci. Paris, t. 336, Série I, p. 1–??, 2003 - PXMA????.TEX - Théorie des groupes/Group theory (Sous-rubrique/Sub-Heading)

Invariant theory and eigenspaces for unitary reflection groups

G.I. Lehrer^{a}, J. Michel^{b},

a School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia, Fax: +61 2 9351 4534 E-mail: gusl@maths.usyd.edu.au

b LAMFA, Université de Picardie-Jules Verne 33, Rue Saint-Leu 80039 Amiens Cedex France and Institut de Mathématiques, Université Paris VII, 175, rue du Chevaleret 75013 Paris France E-mail: jmichel@math.jussieu.fr

(Reçu le 20 03 2003, accepté après révision le ?? ?? 2003)

Abstract. We prove some variations of formulas of Orlik and Solomon in the invariant theory of finite unitary reflection groups, and use them to give elementary and case-free proofs of some results of [LS1] and [LS2], in particular that a number is regular for a reflection group G if and only if it divides the same number of degrees and codegrees. \begin{matrix}c \\ ◯ \end{matrix} 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Invariants et espaces propres des groupes de réflexion complexes

Résumé. En utilisant des variations d’une formule de Orlik et Solomon relative aux invariants d’un groupe de réflexion complexe G, nous redémontrons de façon élémentaire deux résultats de [LS1] et [LS2], en particulier le fait qu’un entier est régulier pour G si et seulement si il divise le même nombre de degrés et de codegrés. Notre preuve évite l’analyse cas par cas. \begin{matrix}c \\ ◯ \end{matrix} 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Version française abrégée

Soit G un groupe fini engendré par des (pseudo-)réflexions dans un espace vectoriel V sur ℂ de dimension ℓ. L’anneau des fonctions polynomiales sur V est identifié à l’algèbre symétrique S du dual V^{∗}. L’anneau des invariants S^{G} est une algèbre de polynomes à ℓ générateurs. Les degrés d’un ensemble de générateurs homogènes algébriquement indépendants de S^{G} sont uniquement déterminés et sont appelés les degrés de G; nous les noterons d_{1}, …, d_{ℓ}.

Si F est l’idéal de S engendré par leséléments de S^{G} s’annulant en 0 ∈ V, le quotient S_{G} := S/F réalise la représentation régulière de G et est appelé l’anneau des coinvariants de G. Il hérite de la graduation de S; si q est une indéterminée, nous définissons pour tout G-module M le “degré fantôme” comme

∑_{i} q^{i}⟨ (S_{G})_{i}, M ⟩_{G} = ∑_{j} q^{m_{j}(M)}.

Note présentée par Jacques Tits

S0764-4442(00)0????-?/FLA \begin{matrix}c \\ ◯ \end{matrix} 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 1

G.I. Lehrer, J. Michel

Les entiers m_{j}(M) sont appelés les M-exposants de G. Pour M = V on les note simplement m_{j} et ils sont appelés les exposants de G. Pour le contragrédient M = V^{∗} on les note m_{j}^{∗} et ils sont appelés les coexposants de G. On a, si on range les deux listes par ordre croissant, m_{i} = d_{i} − 1. On définit les codegrés de G par d_{i}^{∗} = m_{i}^{∗} −1. Enfin, plus généralement pour σ ∈ Gal(\overline{ℚ}/ℚ) nous notons m_{j}(σ) les V^{σ}-exposants de G.

Nous démarrons par le résultat suivant de [OS, Theorem (3.7)]

Proposition 0.1. – Soient x et y des indéterminées sur ℂ. Pour tout σ ∈ Gal(\overline{ℚ}/ℚ) on a

|G|^{−1} ∑_{g∈G} \frac{det_{V} (1 − yg)^{σ}}{det_{V} (1 − xg)} = \frac{∏_{j=1}^{r}(1 − yx^{m_{j}(σ)})}{∏_{i=1}^{ℓ}(1 − x^{d_{i}})}

Nous en déduisons, en utilisant la même méthode que [PW] qui concerne le cas σ = Id, le Théorème 0.2. – Soit d un entier positif et soit ζ une racine primitive d-ième de l’unité. Soit A(d) =

{i | ζ^{d_{i}} = 1} et soit B_{σ}(d) = {j | ζ^{−σ} = ζ^{m_{j}(σ)}}. Alors, si l’on pose a(d) = #A(d) et b_{σ}(d) = #B_{σ}(d), on a a(d) ≤ b_{σ}(d), et si pour une application linéaire h : V → V, nous notons det′(h) le produit des valeurs propres non nulles de h, nous avons l’égalité polynomiale suivante dans ℂ[T] :

∑_{g∈G} T^{d(g,ζ)}det′(1 − ζ^{−1}g)^{σ−1} =

\begin{cases}∏_{j∈B_{σ}(d)}(T + m_{j}(σ)) ∏_{j∉B_{σ}(d)}(1 − ζ^{−σ−m_{j}(σ)}) ∏_{j∉A(d)} \frac{d_{j}}{1 − ζ^{−d_{j}}} si a(d) = b_{σ}(d) \\ 0 sinon, \end{cases}

où, pour g ∈ G, on a noté V (g, ζ) l’espace propre de g pour la valeur propre ζ et posé d(g, ζ) = dim V (g, ζ). Le cas particulier où σ = Id est le résultat de [PW] :

Corollaire 0.3. – ∑_{g∈G} T^{d(g,ζ)} = ∏_{{j|ζ^{d_{j}} =1}}(T + d_{j} − 1) ∏_{{j|ζ^{d_{j}} ≠1}} d_{j}. Le cas particulier où σ est la conjugaison complexe donne, où ici B(d) = B_{σ}(d) = {i | d divise d_{i}^{∗}} et b(d) = #B(d) :

Corollaire 0.4. –

(−ζ)^{ℓ} ∑_{g∈G} det(g^{−1})(−T)^{d(g,ζ)} = \begin{cases}∏_{j∈B(d)}(T + d_{j}^{∗} + 1) ∏_{j∉B(d)}(1 − ζ^{−d_{j}^{∗}}) ∏_{j∉A(d)}\frac{d_{j}}{1 − ζ^{−d_{j}}} si a(d) = b(d) \\ 0, sinon \end{cases}

En étudiant les termes de degré a(d) dans 0.3 nous redémontrons de façon élémentaire le

Théorème 0.5. – [LS1] Soit g ∈ G tel que E := V (g, ζ) soit maximal parmi les ζ-espaces propres d’éléments de G. Soit N = {x ∈ G | xE = E} et C = {x ∈ G | xe = e pour tout e ∈ E}. Alors N/C est un groupe de réflexion complexe dans son action sur E, qui est fidèle.

L’entier d est régulier pour G (au sens de [Sp]) si E contient un point du complémentaire des hyperplans de réflexion de W. Du théorème [St, 1.5] de Steinberg, on déduit que d est régulier si et seulement si C = {1}. En étudiant les termes de degré a(d) dans 0.4 nous redémontrons de façon élémentaire le

Théorème 0.6. – [LS2] L’entier d est régulier pour G si et seulement si a(d) = b(d).

Invariants and eigenspaces for unitary reflection groups

1. Introduction

Let G be a finite group generated by (pseudo)reflections in a complex vector space V of dimension ℓ > 0. It is well known that if S denotes the coordinate ring of V (identified with the symmetric algebra on the dual V^{∗}) the ring S^{G} of polynomial invariants of G is free; if f_{1}, f_{2}, …, f_{ℓ} is a set of homogeneous free generators of S^{G}, then the degrees d_{i} = deg f_{i} (i = 1, …, ℓ) are determined by G, and are called the invariant degrees of G.

If F is the ideal of S generated by the elements of S^{G} which vanish at 0 ∈ V, then S/F realises the regular representation of G. It is called the ring of coinvariants of G and denoted S_{G}; it inherits a grading from S. Let q be an indeterminate; for any G-module M, we define the “fake degree” of M as

∑_{i} ⟨ (S_{G})_{i}, M ⟩_{G}q^{i} = ∑_{j} q^{m_{j}(M)}.  (1)

Here ⟨ , ⟩_{G} denotes the usual intertwining number for complex representations of G. The integers m_{1}(M) ≤ m_{2}(M) ≤ ⋯ ≤ m_{r}(M), where r = dim M since S_{G} realises the regular representation of G, are called the M-exponents of G. This terminology is sometimes reserved for irreducible M, but applies generally. There are two noteworthy special cases of this definition. When M = V, usually called the reflection representation, the M-exponents are called simply the exponents m_{1}, …, m_{ℓ} of G, so that m_{i} = m_{i}(V). When M = V^{∗}, the contragredient of V, the corresponding exponents m_{i}^{∗} := m_{i}(V^{∗}) (i = 1, …, ℓ) are referred to as the coexponents of G. It is well known that if the degrees d_{1}, …, d_{ℓ} are also written in non-decreasing order, then

d_{i} = m_{i} + 1 for i = 1, …, ℓ.  (2)

In analogy with (2), we define the codegrees d_{i}^{∗} of G by

d_{i}^{∗} = m_{i}^{∗} − 1 for i = 1, …, ℓ.  (3)

For any linear transformation h : V → V and element ζ ∈ ℂ, denote by V (h, ζ) the ζ-eigenspace of h. Consider the following theorems.

Theorem 1.1. – [LS1] Let d be any positive integer, and let ζ ∈ ℂ be a primitive d^{th} root of unity. Suppose g ∈ G is such that E := V (g, ζ) is maximal among ζ-eigenspaces of elements of G. Define subgroups N = {x ∈ G | xE = E} and C = {x ∈ G | xe = e for all e ∈ E}. Then the group N/C acts faithfully as a unitary reflection group on E.

In the notation of Theorem 1.1, the integer d is said (cf. [Sp]) to be regular for G if the maximal eigenspace E contains a vector which is regular for G, i.e., which lies in no reflecting hyperplane. Equivalently, E is not contained in any reflecting hyperplane. By Steinberg’s theorem [St, 1.5], we have in the notation of Theorem 1.1 that d is regular for G if and only if C = {1}.

Theorem 1.2. – [LS2] Let A(d) = {i | d dividesd_{i}} and let B(d) = {i | d dividesd_{i}^{∗}}. Write a(d), b(d) respectively for the cardinalities of A(d), B(d). Then a(d) ≤ b(d), and we have equality if and only if d is regular for G.

Theorem 1.1 was proved in [LS1], using intersection multiplicities of hypersurface intersections, and Bezout’s theorem. Theorem 1.2 was proved in [LS2], where some case by case analysis was needed, which depended on the Shephard-Todd classification of the irreducible reflection groups.

It is our purpose in this note to prove some variations on known formulae in the invariant theory of G, and to show how they may be used to give case-free elementary proofs of Theorems 1.1 and 1.2.

2. Invariant theory

We begin by recalling some results of Solomon and Orlik-Solomon, which may be found in [OS].

For any ℂG-module M, consider the ℂG-module S⊗M^{∗}, which is canonically isomorphic to Hom(M, S). To determine the subspace of G-invariant elements, recall that by Chevalley’s theorem, S ≅ S^{G} ⊗ S_{G} as G-module, whence (S ⊗ M^{∗})^{G} ≅ S^{G} ⊗ (S_{G} ⊗ M^{∗})^{G}, which is a free S^{G}-module of rank r = dim M (since ⟨ S_{G}, M ⟩_{G} = r), which is generated over S^{G} by elements {u_{i} = ∑_{j=1}^{r} h_{ij} ⊗ y_{j} | i = 1, …, r}, where {y_{j} | j = 1, …, r} is a ℂ-basis of M^{∗} and the h_{ij} are homogeneous elements of S_{G} whose degrees are the M^{∗}-exponents m_{j}(M^{∗}) of G. Define Q_{M} ∈ S by Q_{M} = det(h_{ij}). The following properties of Q_{M} may be found in [OS, pp. 79–82], where Q_{M} is denoted by j_{M}.

Lemma 2.1. – (i) The polynomial Q_{M} is uniquely defined by M up to multiplication by a non-zero constant, and has degree q(M) := ∑_{j} m_{j}(M).

(ii) We have gQ_{M} = det_{M}(g)Q_{M} for any element g ∈ G.

(iii) We have Q_{M} = P Q_{Λ^{r}M} for some P ∈ S^{G}.

(iv) If M is a Galois conjugate of the reflection representation V, then Q_{M} ∈ ℂ^{×}Q_{Λ^{r}M}.

Proposition 2.2. – Let G be a unitary reflection group and let M be a ℂG module such that Q_{M} ∈ ℂ^{×}Q_{Λ^{r}M}, where r = dim M. Let x and y be indeterminates over ℂ. Then

|G|^{−1} ∑_{g∈G} \frac{det_{M}(1 − yg)}{det_{V} (1 − xg)} = \frac{∏_{j=1}^{r}(1 − yx^{m_{j}(M)})}{∏_{i=1}^{ℓ}(1 − x^{d_{i}})}

Proof . – This is essentially [OS, Theorem (3.7)], which is the statement for Galois conjugates of V. The proof of the general case is the same; it uses the obvious bigrading of (S ⊗ Λ^{∗}M^{∗})^{G} and op. cit. Theorem (3.1). □

It is proved in [OS, Theorem 2.13] that the Galois conjugates V^{σ}, where σ ∈ Gal(\overline{ℚ}/ℚ), satisfy the conditions of (2.2), and hence that the formula holds for M = V^{σ}. We shall exploit this to prove the following generalisation of [OS, Theorem 3.3].

Theorem 2.3. – Let G be a unitary reflection group in V, with basic degrees {d_{1}, …, d_{ℓ}}. If σ ∈ Gal(\overline{ℚ}/ℚ), denote by m_{i}(σ) the V^{σ}-exponents of G. Let d be a fixed positive integer and let ζ be a primitive d^{th} root of unity in ℂ. Let A(d) = {i | ζ^{d_{i}} = 1} and let B_{σ}(d) = {j | ζ^{−σ} = ζ^{m_{j}(σ)}}. Write a(d), b_{σ}(d) respectively for the cardinalities of A(d), B_{σ}(d). Then a(d) ≤ b_{σ}(d), and if, for any linear transformation h : V → V, we denote by det′(h) the product of the non-zero eigenvalues of h, we have the following polynomial identity in ℂ[T].

∑_{g∈G} T^{d(g,ζ)}det′(1 − ζ^{−1}g)^{σ−1} =

\begin{cases}∏_{j∈B_{σ}(d)}(T + m_{j}(σ)) ∏_{j∉B_{σ}(d)}(1 − ζ^{−σ−m_{j}(σ)}) ∏_{j∉A(d)} \frac{d_{j}}{1 − ζ^{−d_{j}}} if a(d) = b_{σ}(d) \\ 0 otherwise, \end{cases}  (4)

where d(g, ζ) = dim V (g, ζ) for g ∈ G.

Proof . – We start with the formula (2.2) for M = V^{σ}; writing λ_{1}(g), …, λ_{ℓ}(g) for the eigenvalues of g ∈ G on V, we get

|G|^{−1} ∑_{g∈G} ∏_{i=1}^{ℓ} \frac{(1 − yλ_{i}(g)^{σ})}{(1 − xλ_{i}(g))} = ∏_{j=1}^{ℓ} \frac{(1 − yx^{m_{j}(σ)})}{(1 − x^{d_{j}})}.  (5)

The term corresponding to g ∈ G on the left side has a zero of order d(g, ζ) at x = ζ^{−1} in the denominator. We therefore introduce the change of variable y = ζ^{−σ}(1 − T (1 − xζ)), so that 1 − yζ^{σ} = T (1 − xζ), which ensures that the same is true of the numerator. The left side then becomes a rational function in x and T with no pole at x = ζ^{−1}. Now the factor 1 − x^{d_{j}} of the denominator of the right side has a zero at x = ζ^{−1} exactly when j ∈ A(d), and this zero is simple. Similarly, the factor 1 − yx^{m_{j}(σ)} of the numerator has a zero at x = ζ^{−1} exactly when j ∈ B_{σ}(d), and this zero is simple. Since the pole at x = ζ^{−1} is removable, a(d) ≤ b_{σ}(d), and the right side is zero unless we have equality, as claimed.

Evaluating the left side at x = ζ^{−1} we get

|G|^{−1} ∑_{g∈G} ∏_{\{i|λ_{i}(g)=ζ\}} T ∏_{\{i|λ_{i}(g)≠ζ\}} \frac{1 − ζ^{−σ}λ_{i}(g)^{σ}}{1 − ζ^{−1}λ_{i}(g)} = |G|^{−1} ∑_{g∈G} T^{d(g,ζ)}det′(1 − ζ^{−1}g)^{σ−1} (6)

For the right side, we find, by differentiation with respect to x, that if j ∈ B_{σ}(d) and i ∈ A(d), then the limit as x → ζ^{−1} of \frac{1−yx^{m_{j} (σ)}}{1−x^{d_{i}}} is \frac{T +m_{j}(σ)}{d_{i}}. Hence the value of the right side at x = ζ^{−1} is

∏_{j∈B_{σ}(d)}(T + m_{j}(σ)) ∏_{j∈A(d)} d_{j}^{−1} ∏_{j∉B_{σ}(d)}(1 − ζ^{−σ−m_{j}(σ)}) ∏_{j∉A(d)}(1 − ζ^{−d_{j}})^{−1}  (7)

The Theorem now follows immediately by comparing (6) and (7), taking into account that |G| = d_{1} … d_{ℓ}. □

We next state four corollaries of Theorem 2.3, the first two of which are well known; we believe the others are new.

Corollary 2.4. – [OS, Theorem 3.3] We have ∑_{g∈G} T^{d(g,1)}det′(1 − g)^{σ−1} = ∏_{j=1}^{ℓ}(T + m_{j}(σ)). This is simply the case d = 1 (whence ζ = 1) of the Theorem.

Corollary 2.5. – [PW] We have ∑_{g∈G} T^{d(g,ζ)} = ∏_{{j|ζ^{d_{j}} =1}}(T + d_{j} − 1) ∏_{{j|ζ^{d_{j}} ≠1}} d_{j}. This is the case σ = Id of the Theorem. Note that here the integers m_{j}(σ) = m_{j} = d_{j} − 1 are the usual

exponents.

We believe the next result is new.

Corollary 2.6. – Maintaining the above notation, we have a(d) ≤ b(d), and

(−ζ)^{ℓ} ∑_{g∈G} det(g^{−1})(−T)^{d(g,ζ)} = \begin{cases}∏_{j∈B(d)}(T + d_{j}^{∗} + 1) ∏_{j∉B(d)}(1 − ζ^{−d_{j}^{∗}}) ∏_{j∉A(d)} \frac{d_{j}}{1 − ζ^{−d_{j}}} ifa(d) = b(d) \\ 0 ifa(d) < b(d) & . \end{cases}

where the integers d_{j}^{∗} = m_{j}^{∗} − 1 are the codegrees of G.

Proof . – Take σ to be complex conjugation in (4). Then by definition, m_{j}(σ) = m_{j}^{∗} = d_{j}^{∗} − 1 and σ acts as inversion on roots of unity, thus for a root of unity λ ≠ 1 we have (1 − λ)^{σ−1} = −λ^{−1}; so for g ∈ G, det′(1 − ζ^{−1}g)^{σ−1} = (−ζ)^{ℓ}(−1)^{d(g,ζ)} det(g^{−1}). The formula (2.6) now follows directly from (4).

□

There are of course many other special cases which might be explored. We give one other.

Corollary 2.7. – With the above notation, for σ ∈ Gal(\overline{ℚ}/ℚ) we have

∑_{g∈G} T^{d(g,−1)}det′(1 + g)^{σ−1} = \begin{cases}0, unless #\{j | m_{j}(σ) is odd\} = #\{j | d_{j} is odd\} \\ ∏_{\{j|m_{j}(σ) is odd\}}(T + m_{j}(σ)) ∏_{\{j|d_{j} is odd\}} d_{j}, otherwise & . \end{cases}

This is obtained from (4) by putting d = 2 and hence ζ = −1.

3. Eigenspaces

In this section we show how the results above may be applied to questions concerning maximal eigenspaces of elements of G. We begin with the Proof of Theorem 1.1. It follows from [Sp, Theorem 3.4] (whose proof is elementary) that the maximal ζ-eigenspaces V (g, ζ) all have dimension a(d) and are conjugate under the action of G. We compute the coefficient of T^{a(d)} on both sides of (2.5), obtaining in the notation of 1.1 the equality \frac{|G||C|}{|N|} = ∏_{j:d|̸d_{j}} d_{j}, the left side being the number |C| of elements g ∈ G such that V (g, ζ) = E multiplied by the number \frac{|G|}{|N|} of distinct conjugates of E. Since |G| = ∏_{j=1}^{ℓ} d_{j}, it follows that |\frac{N}{C} | = ∏_{j:d|d_{j}} d_{j}, which by [Sp, Proposition 3.5(ii)] (whose proof is also elementary) proves Theorem 1.1.

The next result includes Theorem 1.2.

Theorem 3.1. – (i) Let σ ∈ Gal(\overline{ℚ}/ℚ). In the notation of Theorem 4, if d is regular for G, then b_{σ}(d) = a(d).

(ii) When σ is complex conjugation, so that b_{σ}(d) = b(d), then the converse of (i) holds. That is, d is regular for G if and only if a(d) = b(d).

Proof . – Maintain the notation of the proof of Theorem 1.1 above. Clearly the set {g ∈ G | V (g, ζ) = E} is a coset g_{0}C of the pointwise stabiliser C = G_{E} of E in G. Hence if C = 1, which is the regular case, there is a unique element g with V (g, ζ) = E. If we write g_{h} for the corresponding element when E is replaced by a conjugate hE (h ∈ G) of E, we have clearly, that g_{h} = hgh^{−1}, so that in particular, all the elements g_{h} have the same eigenvalues.

Let us compare the coefficients of T^{a(d)} in both sides of (4) when d is regular. The left hand side is |G/N| det′(1 − ζ^{−1}g), which is clearly non-zero, while the right-hand side is zero unless a(d) = b_{σ}(d), which proves (i).

To prove (ii), assume that d is not regular for G. Then by the observations above, the coefficient of T^{a(d)} in the left side of (2.6) has a factor ∑_{h∈C} det^{−1}(h) which is zero if C is non-trivial, since by Steinberg’s theorem C is a reflection subgroup of G. Hence the right side of (2.6) is zero, so b(d) ⪈ a(d). □

Note that generally whenever d is regular for G, σ ≠ Id and g ∈ G is such that V (g, ζ) is maximal for some primtive d-th root of unity ζ, evaluating the coefficient of T^{d(g,ζ)} in both sides of (4) gives information on the eigenvalues of g. The particular case when σ is complex conjugation yields

Corollary 3.2. – Let ζ be a primitive d^{th} root of unity, where d is regular for G, and let g ∈ G be such that V (g, ζ) is maximal. Then

det(g) = ζ^{ℓ} ∏_{j∉B(d)}(ζ^{−d_{j}^{∗}} − 1)^{−1} ∏_{j∉A(d)}(1 − ζ^{−d_{j}}) = ζ^{− ∑_{j=1}^{ℓ} m_{j}}.

The first equality is immediate from (2.6). The second, which follows from [Sp, Theorem 4.2(v)] may also be written ∏_{j∉B(d)}(1 − ζ^{−d_{j}^{∗}}) = ∏_{j∉A(d)}(1 − ζ^{d_{j}}), which permits the simplification of 2.6 above.

References

[LS1] G.I. Lehrer and T.A. Springer, “Intersection multiplicities and reflection subquotients of unitary reflection groups I”, Geometric group theory down under (Canberra, 1996), 181–193, de Gruyter, Berlin, 1999.

[LS2] G.I. Lehrer and T.A. Springer, “Reflection subquotients of unitary reflection groups.” Canad. J. Math. 51 (1999), 1175–1193.

[OS] P. Orlik and L. Solomon, “Unitary reflection groups and cohomology”, Inv. Math. 59 (1980), 77-94.

[PW] A. Pianzola and A. Weiss, “Monstrous E_{10}’s and a generalization of a theorem of L. Solomon”, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 189–194.

[Sp] T. Springer, “ Regular elements of finite reflection groups”, Invent. Math. 25 (1974), 159-198.

[St] R. Steinberg, “Differential equations invariant under finite reflection groups”, Trans. Amer. Math. Soc. 112 (1964), 392-400.