TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 12, December 1998, Pages 5047–5066 S 0002-9947(98)02025-X

CHARACTER SUMS ASSOCIATED TO FINITE COXETER GROUPS

JAN DENEF AND FRANÇOIS LOESER

Abstract. The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald’s conjecture proved by Opdam.

0. Introduction

Throughout this paper F will denote a finite field of characteristic p different from 2.

We first slightly reformulate Macdonald’s conjecture in a form which has a direct analogue over F. Let G be a finite subgroup of GL_{n}(R) generated by reflections and let q be a positive definite quadratic form which is invariant under G. Let A_{G} be the associated arrangement consisting of the reflection hyperplanes. Let ℓ_{1}, …, ℓ_{N} be equations for the N different reflection hyperplanes. Set Δ(x) = (∏_{i=1}^{N} ℓ_{i})^{2} and let d_{1}, …, d_{n} be the degrees of G. Macdonald’s conjecture [19] is the following equality,

∫_{R^{n}} Δ(x)^{s}e^{−(∑_{i} x_{i}^{2})/2}dx = (2π)^{n/2} ∏_{i=1}^{n} \frac{Γ(d_{i}s + 1)}{Γ(s + 1)},

when q(x) = ∑_{i} x_{i}^{2} and the ℓ_{i} are normalized in such a way that ∥ℓ_{i}∥ = \sqrt{2}. If we drop the condition that the ℓ_{i} are normalized, and if we replace x by \sqrt{2}x, we obtain

∫_{R^{n}} Δ(x)^{s}e^{− ∑_{i} x_{i}^{2}}dx = π^{n/2} (∏_{i=1}^{N} \frac{∥ℓ_{i}∥^{2}}{4} )^{s} ∏_{i=1}^{n} \frac{Γ(d_{i}s + 1)}{Γ(s + 1)}.

Hence for q an arbitrary positive definite quadratic form which is invariant under G, we obtain

(0.1) ∫_{R^{n}} Δ(x)^{s}e^{−q(x)}dx = π^{n/2}κ^{s} (∏_{i=1}^{n} \frac{Γ(d_{i}s + 1)}{Γ(s + 1)}) (discr q)^{− \frac{1}{2}} ,

with

κ = ∏_{i=1}^{N} \frac{q(ℓ_{i})}{4} , (0.2)

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors March 29, 1996 and, in revised form, January 24, 1997.

1991 Mathematics Subject Classification. Primary 11T24, 11L05; Secondary 33C80.

Key words and phrases. Character sums, finite fields, Coxeter groups, monodromy, ℓ-adic cohomology.

where we consider ℓ_{i} in q(ℓ_{i}) as a vector in R^{n}, identifying R^{n} with its dual, by means of the quadratic form q. Observe that

\sqrt{q(ℓ_{i})} = Max_{{x∈R^{n} \atop q(x)=1}} ℓ_{i}(x) . (0.3)

Formula (0.1) can be reformulated as

(0.4) ∫_{K} Δ(x)^{s}e^{−q(x)}\sqrt{Δ(x)}dx = π^{n/2}κ^{s+ \frac{1}{2}} (∏_{i=1}^{n} \frac{Γ(d_{i}(s + \frac{1}{2}))}{Γ(s + \frac{1}{2})}) (discr q)^{− \frac{1}{2}} ,

where K = R^{n}/G. Observe that \sqrt{Δ(x)}dx is a G-invariant differential form.

Over F, there is the notion of a Coxeter arrangement (1.4). It is a triple, A = (V, G, q), where V is a finite-dimensional F-vector space, G a finite subgroup of GL(V) generated by reflections, and q a G-invariant nondegenerate symmetric bilinear form on V. If p does not divide |G|, one may define the degrees of G, d_{1}, …, d_{n} (1.5). Let ℓ_{1}, …, ℓ_{N} be equations for the N different reflection hyperplanes. Put Δ(x) = (∏_{i=1}^{N} ℓ_{i})^{2}. Because p ≠ 2, we may define an element κ of F by (0.2). Fix a nontrivial additive character ψ : F → C. The analogue of the integral in (0.4) will be the character sum

S_{G}(χ) := ∑_{x∈(U/G)(F)} χ(Δ(x))ψ(q(x)),

where χ : F^{×} → C^{×} is a multiplicative character and U denotes the complement of the hypersurface Δ(x) = 0 in V. (We write q(x) for the quadratic form q(x, x) associated to the bilinear form q, and use the standard notation (U/G)(F) to denote the set of F-rational points on the quotient space U/G.) The analogue of the Gamma function will be the Gauss sum −g(χ), where g(χ) := − ∑_{x∈F^{×}} χ(x)ψ(x). Our main result is the following.

Main Theorem. Let A = (V, G, q) be a Coxeter arrangement over F. Assume that p does not divide |G|. Then κ ≠ 0 and, for every multiplicative character χ : F^{×} → C^{×},

S_{G}(χ) = (−1)^{n}ϕ(discr q)g(ϕ)^{n}ϕ(κ)χ(κ) ∏_{i=1}^{n} \frac{g((ϕχ)^{d_{i}})}{g(ϕχ)} ,

where ϕ denotes the unique multiplicative character of F of order 2.

Note the analogy with Macdonald’s formula (0.4), replacing \sqrt{π} = Γ(1/2) by −g(ϕ). (For some more remarks on this analogy, see (4.6) below.) Our proof of the Main Theorem depends on Macdonald’s formula (0.1) at one single point, namely for the proof of Theorem 3.3 we need to know that, with the notation of (0.1),

Max{Δ(x) | x ∈ R^{n}, q(x) = 1} = κN^{−N} ∏_{i=1}^{n} d_{i}^{d_{i}}, (0.5)

which follows directly from (0.1), by elementary calculus. Without relying on Macdonald’s formula we can nevertheless prove the Main Theorem for some κ in F\{0} which is independent of χ without the assertion that κ is given by (0.2).

In the special case where G is the symmetric group S_{n}, the Main Theorem has been proved for all p ≠ 2 by R. Evans [13], extending the results of G. Anderson on Selberg sums [1]. For the history and applications of such sums, we refer to [11] and [12]. For recent work on character sums related to relative invariants of prehomogeneous vector spaces, see [6].

Our proof of the Main Theorem is entirely based on the cohomological interpretation of character sums, using the Grothendieck-Lefschetz trace formula (see [26] Sommes trig. and [15]). In the present situation, the cohomology is concentrated in middle dimension (Proposition 2.1.2) and has rank 1 (Corollary 4.3.2 and formula (4.4.6)), so that we have only to calculate the determinant of the Frobenius action on the cohomology, which is done by Laumon’s product formula [16] 3.2.1.1.

The plan of the paper is the following. In the first section we study Coxeter arrangements over finite fields and their liftings to characteristic zero. In section 2 we study character sums associated to central arrangements that are invariant under a finite group. This will allow us to evaluate S_{G}(ϕ). In the next section we review results on monodromy and critical values for Coxeter arrangements from [10] and [7]. In the last section, using all the previous results, we prove the Main Theorem.

Most of the present work dates from 1992. It has been presented, under some additional hypotheses, in lectures by the first named author in Orsay (SAGA) in November 1992 and in Amsterdam in January 1993.

1. Liftable Coxeter arrangements over F

1.1. Let V be a finite dimensional vector space over a field. By a hyperplane arrangement in V we mean a finite set of affine hyperplanes in V. If all the hyperplanes contain 0, the arrangement is said to be central. We call an endomorphism of V a reflection if it has order 2 and fixes pointwise some hyperplane.

We define a classical Coxeter arrangement as a triple A = (V, G, q), where V is a finite dimensional vector space over R, G is a finite subgroup of GL(V) generated by reflections, and q is a G-invariant positive definite symmetric bilinear form on V. We define a Coxeter arrangement over C as a triple A = (V, G, q), where V is a finite dimensional vector space over C, G is a finite subgroup of GL(V) generated by reflections, and q is a G-invariant nondegenerate symmetric bilinear form on V, which arises by extension of scalars from a classical Coxeter arrangement. (In fact, by the argument given at the end of the proof of Proposition 1.6, the last condition in this definition is automatically verified.)

Similarly, when T is a subring of C with fraction field K, a Coxeter arrangement over T will be a triple A = (M, G, q), where M is a free T -module of finite rank, G a finite subgroup of GL(M) generated by reflections, and q a G-invariant nondegenerate symmetric bilinear form on M, which induces by extension of scalars a Coxeter arrangement over C.

The finite set of reflection hyperplanes in M ⊗_{T} K defines a central hyperplane arrangement in M ⊗_{T} K which we denote by A_{G}.

1.2. Let T be a discrete valuation ring with fraction field K. Let 𝔓 be the maximal ideal of T with residue field \overline{K} = T/𝔓. Let M be a free T -module of finite rank, and set V = M ⊗_{T} K, \overline{V} = M ⊗_{T} \overline{K}. We denote the reduction map M → \overline{V} by x ↦ \overline{x}. For any linear subspace W in V, we denote by \overline{W} the reduction modulo 𝔓 of W which is defined by

\overline{W} = \{\overline{x} | x ∈ W ∩ M\}.

Observe that dim_{K} W = dim\overline{K} \overline{W}.

Definition 1.2.1. Let A = (H_{i})_{i∈J} be a central arrangement in V. We say that A has good reduction mod 𝔓 if, for any I ⊂ J,

\overline{⋂_{i∈I} H_{i}} = ⋂_{i∈I} \overline{H_{i}}.

For A an arrangement in V consider the arrangement \overline{A} consisting of the hyperplanes \overline{H} with H in A. Denote the lattices of A and \overline{A} by L(A) and L(\overline{A}). There is a canonical inclusion preserving map

θ : \{ \begin{matrix}L(A) ⟶ L(\overline{A}) \\ X ↦ ⋂_{{H∈A \atop H⊃X}} \overline{H}. \end{matrix}

Proposition 1.2.2. If A has good reduction mod 𝔓, then θ is an isomorphism of lattices.

Proof. This is straightforward because θ(X) = \overline{X} when A has good reduction. Also the equality dim_{K} W = dim\overline{K} \overline{W} holds for any linear subspace W of V.

1.3. We assume the notation of (1.2). Suppose that T ⊂ C and let (M, G, q) be a Coxeter arrangement over T. Thus, in particular, G ⊂ GL(M). Let A_{G} be the central arrangement in V consisting of all the reflection hyperplanes of G. We denote the image of G in GL(\overline{V}) by \overline{G} and we choose linear forms ℓ_{i} over T which define the hyperplanes of A_{G} and which are not zero modulo 𝔓. Let κ be as in (0.2).

Proposition 1.3.1. Assume that 𝔓 does not contain |G|. Then A_{G} has good reduction mod 𝔓. Furthermore the canonical morphism G → \overline{G} is an isomorphism, and G acts freely on the complement in \overline{V} of the hyperplanes in \overline{A_{G}}.

Proof. For the first statement it suffices to show that for all I ⊂ J

dim ⋂_{i∈I} \overline{H_{i}} ≤ dim \overline{⋂_{i∈I} H_{i}}.

We may suppose that the H_{i}’s with i in I are linearly independent. Let s_{i} be the reflection with respect to H_{i} and set γ = ∏_{i∈I} s_{i}, Fix(γ) = {x ∈ V | γ(x) = x} and Fix(\overline{γ}) = \{\overline{x} ∈ \overline{V} | \overline{γ}(\overline{x}) = \overline{x}\}. From the proof of Theorem 6.27 (2) on p.225 of [22] we see that Fix(γ) = ⋂_{i∈I} H_{i}. Since \overline{H}_{i} ⊂ Fix(\overline{s}_{i}), dim ⋂_{i∈I} \overline{H_{i}} ≤ dim Fix(\overline{γ}). But this last dimension is equal to the number of eigenvalues of \overline{γ} which are equal to 1, i.e., the number of eigenvalues of γ which are congruent to 1 mod 𝔓 (counting multiplicities and enlarging T so that it contains all eigenvalues of γ). But if λ is an eigenvalue of γ which is congruent to 1 mod 𝔓, then λ = 1, because λ^{|G|} = 1, and 𝔓 does not divide |G|. But the number of eigenvalues of γ which are equal to 1 (counting multiplicities) is equal to dim Fix(γ). We now deduce the first statement from the equalities

dim Fix(γ) = dim ⋂_{i∈I} H_{i} = dim \overline{⋂_{i∈I} H_{i}}.

For the second statement we just have to observe that if an element γ of G is in the kernel of the morphism G → \overline{G}, all the eigenvalues of γ are congruent to 1 mod 𝔓, hence are equal to 1 by the above argument. The order of γ being finite, we conclude that γ is equal to 1. Finally the third assertion follows from the well known fact (cf. [4] Ch.V, § 3 Proposition 3) that G acts freely on the complement in V of the hyperplanes in A_{G}, and from the observation that Fix(\overline{α}) = \overline{Fix(α)}, for all α in G.

For each reflection s ∈ G we choose an eigenvector α_{s} ∈ M of s which is orthogonal (with respect to q) to the reflection hyperplane of s.

Set Φ = {α_{s} | s reflection in G}. Note that

s(x) = x − 2\frac{q(x, α_{s})}{q(α_{s}, α_{s})}α_{s} for all x ∈ V. (1)

If q(α, α) ∉ 𝔓 for each α ∈ Φ, then,

\overline{s}(\overline{x}) = \overline{x} − 2\frac{\overline{q}(\overline{x}, \overline{α}_{s})}{\overline{q}(\overline{α}_{s}, \overline{α}_{s})}\overline{α}_{s} for all \overline{x} ∈ \overline{V} . (2)

Denote by \overline{q} the symmetric bilinear form induced by q on \overline{V}.

Lemma 1.3.2. Assume that G is essential (meaning that V^{G} = {0}), that 𝔓 does not contain |G|, and that q(α, α) ∉ 𝔓 for all α ∈ Φ. Then the bilinear form \overline{q} is nondegenerate.

Proof. Assume that there exists \overline{x} ∈ \overline{V} with \overline{q}(\overline{x}, \overline{y}) = \overline{0} for all \overline{y} ∈ \overline{V}. Then (2) implies that \overline{s}(\overline{x}) = \overline{x} for every reflection s in G. Let H_{s} be the reflection hyperplane of the reflection s. Then \overline{x} ∈ \overline{H}_{s} because otherwise \overline{s} is the identity and all eigenvalues of s would be congruent to 1 mod 𝔓. Hence \overline{x} ∈ ⋂_{H∈A_{G}} \overline{H} = \overline{⋂_{H∈A_{G}} H} = \overline{V^{G}} = \overline{0} by Proposition 1.3.1.

Proposition 1.3.3. Assume that 𝔓 does not contain |G| and that the bilinear form \overline{q} is nondegenerate. Then κ ∈ T \ 𝔓, and, for any I ⊂ J, the restriction of \overline{q} to ⋂_{i∈I} \overline{H}_{i} is nondegenerate.

Proof. Observe first that the characteristic of \overline{K} is not 2 because 2 | |G|. For each reflection s ∈ G, choose an eigenvector α_{s} ∈ M of s which is orthogonal (with respect to q) to the reflection hyperplane of s such that \overline{α}_{s} ≠ 0. Since \overline{q} is nondegenerate there exists x ∈ M such that \overline{q}(\overline{x}, \overline{α}_{s}) ≠ 0. Thus q(x, α_{s}) ≢ 0 mod 𝔓. Because s(x) ∈ M, formula (1) yields q(α_{s}, α_{s}) ≢ 0 mod 𝔓. This implies that κ ∈ T \𝔓. Set U = ⋂_{i∈I} H_{i}, thus \overline{U} = ⋂_{i∈I} \overline{H}_{i}. We have to show that the restriction of \overline{q} to \overline{U} is nondegenerate. Because \overline{q} is nondegenerate, it suffices to prove that the restriction to (\overline{U})^{⊥} is nondegenerate. We have \overline{(U^{⊥})} = (\overline{U})^{⊥} because, clearly, \overline{(U^{⊥})} ⊂ (\overline{U})^{⊥} and both have the same dimension n − dim U. Thus it remains to prove that \overline{q} is nondegenerate on \overline{(U^{⊥})}. Let W be the subgroup of G which is generated by the reflections with respect to the reflection hyperplanes of G which contain U. Clearly, W is a finite Coxeter group in U^{⊥} and W ⊂ GL(U^{⊥} ∩M). It is easy to see that the hypothesis of Lemma 1.3.2 is satisfied if we replace V by U^{⊥}, M by U^{⊥} ∩ M and q by its restriction to U^{⊥} ∩ M. Indeed (U^{⊥})^{W} = {0} because V^{W} is contained in U, and thus (U^{⊥})^{W} ⊂ U^{⊥} ∩ U = {0} since q_{|U} arises from a positive definite form over R. Hence Lemma 1.3.2 shows that \overline{(q_{|U^{⊥}∩M})} is nondegenerate. Thus \overline{q}_{|\overline{(U^{⊥})}} is also nondegenerate.

1.4. A Coxeter arrangement over F is a triple A = (V, G, q), where V is a finite dimensional F-vector space, G a finite subgroup of GL(V) generated by reflections, and q a G-invariant nondegenerate symmetric bilinear form on V. By a reflection we mean an endomorphism of V of order 2, which fixes pointwise some hyperplane.

The finite set of reflection hyperplanes defines a central hyperplane arrangement which we denote by A_{G}. A Coxeter arrangement A = (V, G, q) over F will be called liftable if there exists an embedding T ↪ C of a discrete valuation ring T with residue field F and fraction field of finite degree over Q, and a Coxeter arrangement A_{T} = (M_{T} , G_{T} , q_{T}) over T such that V is isomorphic to M_{T} ⊗_{T} F and, after identification of M_{T} ⊗_{T} F with V, G = \overline{G}_{T} and q = \overline{q}_{T}. Here we denote the symmetric bilinear form induced by q_{T} on V by \overline{q}_{T}. Such a Coxeter arrangement A_{T} will be called a lifting of A over T.

The following statement is a direct consequence of Proposition 1.3.1.

Corollary 1.4. Let T be a discrete valuation ring with residue field F and fraction field of finite degree over Q. Fix an embedding T ↪ C. Let A = (M, G, q) be a Coxeter arrangement over T. Suppose that p does not divide |G| and that \overline{q}, the symmetric bilinear form induced by q on \overline{V}, is nondegenerate. Then \overline{A} := (\overline{V} , \overline{G}, \overline{q}) is a liftable Coxeter arrangement over F. Furthermore the arrangement A_{G} has good reduction and we have \overline{A_{G}} = A\overline{G}.

1.5. Let A = (V, G, q) be a Coxeter arrangement over F. Let us denote by S(V) the symmetric algebra of V and by R(V) the subalgebra of G-invariants. If p does not divide |G|, by Chevalley’s Theorem ([4] Ch.V, § 5 Théorème 4), R(V) is generated as an F-algebra by n := dim V algebraically independent homogeneous polynomials (together with 1). The degrees of these polynomials are called the degrees d_{1}, …, d_{n} of A (or the degrees of G). Their product being equal to |G|, they are also prime to p (loc. cit. p.115).

Proposition 1.5. Let A = (V, G, q) be a liftable Coxeter arrangement over F. Assume that p does not divide |G|. The degrees of A coincide with those of any lifting of A viewed as a complex Coxeter arrangement.

Proof. Let T ⊂ C be a discrete valuation ring having F as residue field and let A_{T} be a lifting of A over T. We have to prove that the degrees of A and A_{T} (viewed as a complex Coxeter arrangement) are the same. For that we may clearly suppose that T is complete. Let p_{1}, …, p_{n} be homogeneous generators of R(V). As p does not divide |G|, p_{1}, …, p_{n} can clearly be lifted, by averaging, to elements q_{1}, …, q_{n} of R(M_{T}), the subalgebra of G-invariants of the symmetric algebra of M_{T}. Let π be a uniformizing parameter of T. The polynomials q_{1}, …, q_{n} generate the algebra R(M_{T}) modulo π, hence, by induction, modulo π^{i} for any i. Passing to the limit, we see that q_{1}, …, q_{n} generate the algebra R(M_{T}), which implies the result.

1.6. By the following proposition, we have liftability of Coxeter arrangements over finite fields, as soon as the characteristic does not divide the order of the Coxeter group.

Proposition 1.6. Let A = (V, G, q) be a Coxeter arrangement over F. If p does not divide |G|, then A is liftable.

Proof. Since the group ring F[G] is semisimple, each simple component of the F[G]- module V is a direct summand of F[G]. Thus V is a projective F[G]-module, and can hence be lifted to an T [G]-module M, of finite rank over T, for some discrete valuation ring T ⊂ C, with residue field F and fraction field K of finite degree over Q. In fact, by [25] § 14.4 Proposition 42, we can lift V to an R[G]-module M, of finite rank over R, for some complete discrete valuation ring R ⊂ C, with residue field F, and it follows from a result of M. Greenberg [14] that we can work with a henselian discrete valuation ring instead of a complete one as in loc. cit. [As was suggested by the referee one could also observe that the group algebra Z[\frac{1}{|G|}][G] is an Azumaya algebra over its center which is unramified over Z[\frac{1}{|G|}], and take T to split them (cf. [20] Ch.IV § 1).] Since the reduction modulo p is injective on the roots of unity of order dividing |G|, we see that the reflections in G ⊂ GL(V) can be lifted to reflections in GL(M). By averaging, the form q can be lifted to a G-invariant nondegenerate symmetric bilinear form q_{T} on M. Moreover, the action of G on M ⊗_{T} C can be realized over R (see, e.g., [25] § 13.2 Théorème 31) in such a way that q_{T} corresponds to a positive definite symmetric bilinear form over R (indeed, this follows from the proof in loc. cit.). Thus (M, G, q_{T}) is a Coxeter arrangement over T.

2. Character sums associated to invariant central arrangements

2.0. The data. Let p be a prime number different from 2 and let F be a finite field of characteristic p. We denote by F̄ a fixed algebraic closure of F. Let f = ∏_{i∈I} ℓ_{i} be a product of (not necessarly distinct) linear forms ℓ_{i} on A_{F}^{n}. The locus of f = 0 is an arrangement A. We denote by U its complement in A_{F}^{n}. Let q be a quadratic form on A_{F}^{n} whose restriction to any stratum of A is nondegenerate. Choose a prime number ℓ ≠ p. Let χ be a multiplicative character F^{×} → Q̄_{ℓ}^{×} and let ψ be a nontrivial additive character F → Q̄_{ℓ}^{×}. (Here Q̄_{ℓ} denotes an algebraic closure of the field of ℓ-adic numbers.) We denote by L_{χ} and L_{ψ} the corresponding Kummer and Artin-Schreier sheaves (see [26] Sommes trig.). Let G be a finite group, leaving f and q invariant, and acting freely on U. We consider the exponential sum

S_{G}(χ) = ∑_{x∈(U/G)(F)} χ(f(x))ψ(q(x))
= Tr(F, (H_{c}^{·}(UF̄, f^{∗}L_{χ} ⊗ q^{∗}L_{ψ}))^{G}).

Here F denotes the geometric Frobenius automorphism and the second equality follows from Grothendieck’s trace formula, see loc. cit.

At this point we should perhaps recall how the action of G on the cohomology groups H_{c}^{·}(UF̄, f^{∗}L_{χ} ⊗q^{∗}L_{ψ}) is defined. It is induced by the canonical isomorphism

H_{c}^{·}(UF̄, f^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) ≃ H_{c}^{·}((U/G)F̄, π_{∗}Q̄_{ℓ} ⊗ f_{G}^{∗}L_{χ} ⊗ q_{G}^{∗}L_{ψ})

where π : U → U/G is the natural map, f_{G} and q_{G} : U/G → A^{1} are induced by f and q, with the natural action of G on π_{∗}Q̄_{ℓ} and the trivial action on f_{G}^{∗}L_{χ} and q_{G}^{∗}L_{ψ}.

2.1. Concentration of the cohomology in the middle dimension. We construct the following compactification q̃ of q. Let Γ be the closure in P^{n} × A^{1} of the graph of q in A^{n} × A^{1} and let q̃ be the projection Γ ⊂ P^{n} × A^{1} → A^{1}. Clearly q̃ is proper. We denote by j : U → Γ the canonical open immersion. Let F = j_{!}f^{∗}L_{χ}.

Lemma 2.1.1. Outside 0 ∈ A^{n} ⊂ Γ, the morphism q̃ is locally acyclic^{1} with respect to the sheaf F and Γ is smooth over F.

1For the notion of “locally acyclic”, see [26] Th. finitude 2.12.

Proof. We look locally at a ∈ Γ \ {0}. Case (i): a ∈ A^{n} ⊂ Γ. We may assume that the linear forms ℓ_{i} which vanish at a are x_{n}, x_{n−1}, …, x_{r} and linear combinations of these, with r > 1. The restriction of q to x_{n} = x_{n−1} = ⋯ = x_{r} = 0 is nondegenerate and a ≠ 0, thus the restriction of q is nonsingular at a. Hence x_{n}, x_{n−1}, …, x_{r}, q − q(a) are part of a system of local parameters for Γ at a. In the diagram

\xymatrix{Spec F[x_{r}, …, x_{n}] \ar[d] & Spec F[x_{r}, …, x_{n}, q] \ar[l] \ar[d] & A^{n} \ar[l]_{α=(x_{r},... ,x_{n},q)} \\ Spec F & Spec F[q] \ar[l]}

the square is cartesian, the map α is smooth at a and, locally at a for the etale topology, the sheaf F is the pull back of a sheaf on Spec F[x_{r}, …, x_{n}], thus q is locally acyclic for F at a. Case (ii): a ∉ A^{n} ⊂ Γ. The equation for Γ in P^{n} × A^{1} is yx_{0}^{2} = q(x_{1}, …, x_{n}), where x_{0}, x_{1}, …, x_{n} are projective coordinates on P^{n} and y is the coordinate on A^{1}. The map q̃ is given by (x, y) ↦ y. Let the coordinates of a be x_{i} = α_{i} and y = b. Since a ∉ A^{n}, we have α_{0} = 0. Hence q(α_{1}, …, α_{n}) = 0. We may assume that, e.g., α_{1} ≠ 0. We may assume also that the linear forms ℓ_{i} which vanish at (α_{1}, …, α_{n}) are x_{n}, x_{n−1}, …, x_{r} and linear combinations of these, with r > 1. Since the restriction of q to x_{n} = x_{n−1} = ⋯ = x_{r} = 0 is nondegenerate, (α_{1}, …, α_{n}) is a nonsingular projective point of the projective quadric q(x_{1}, …, x_{r−1}, 0, …, 0) = 0 and r > 2. Hence (1, \frac{α_{2}}{α_{1}} , \frac{α_{3}}{α_{1}} , …) is a nonsingular point of the affine quadric q(1, t_{2}, …, t_{r−1}, 0, …, 0) = 0. Moreover, Γ is defined in a neighbourhood of a by the affine equation yt_{0}^{2} = q(1, t_{2}, …, t_{n}). Here t_{0}, t_{2}, …, t_{n} are coordinates on A^{n} and y is the coordinate on A^{1}. We see that t_{0}, y − b, t_{r}, …, t_{n} are part of a system of local parameters for Γ at a. Since Γ \ U is a hypersurface of Γ given locally at a by an equation only involving t_{0}, t_{r}, …, t_{n}, we may conclude as in case (i) that q̃ is locally acyclic for F at a.

Proposition 2.1.2. Assume that the sheaves R^{i}q_{|U!}f^{∗}L_{χ} have tame ramification at ∞ for all integers i. Then

H_{c}^{i}(UF̄, f^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) = 0 if i ≠ n.

Proof. It is a direct consequence of Lemma 2.1.1 and of a straightforward adaptation of [8] Proposition 3.1. Indeed part 3.1.1 of that proposition remains valid when we replace the sheaf Q_{ℓ} by any constructible Q̄_{ℓ}-sheaf. Thus by Lemma 2.1.1 we get H_{c}^{i}(UF̄, f^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) = H_{c}^{i}(ΓF̄, F ⊗ q̃^{∗}L_{ψ}) = 0 when i > n provided that the sheaves R^{i}q_{|U!}f^{∗}L_{χ} have tame ramification at ∞ for all i. The case i < n follows by Poincaré duality and the affineness of U.

2.2. The case when χ is trivial. We assume throughout this paragraph that χ is trivial, thus

S_{G} = ∑_{x∈(U/G)(F)} ψ(q(x)).

We also consider

S = ∑_{x∈U(F)} ψ(q(x)) = Tr(F, H_{c}^{·}(UF̄, q^{∗}L_{ψ})).

We may assume that the ℓ_{i}’s define different hyperplanes.

Lemma 2.2.1. If A = ∅ and U = A^{n}, then

S = ϕ(discr q)(−g(ϕ))^{n} ,

where ϕ denotes the unique multiplicative character of F of order 2.

Proof. Reduce q to diagonal form and use ∑_{x∈A^{1}(F)} ψ(x^{2}) = −g(ϕ).

Lemma 2.2.2. The following formula holds,

H_{c}^{i}(UF̄, q^{∗}L_{ψ}) = 0 if i ≠ n.

Proof. For U = A^{n}, this is well known (reduce q to diagonal form). The general case is a direct consequence of Lemma 3.8.1 of [8], and of the fact that U is affine.

Lemma 2.2.3. The vector space H_{c}^{n}(An ; F̄, q^{∗}L_{ψ}) is one dimensional and the action of an element σ of G on this space is given by multiplication by its determinant (as an operator on A^{n}), det σ = ±1.

Proof. The first statement is well known. For the second statement, we may assume σ is a reflection since the orthogonal group is generated by reflections. By Künneth, the situation is reduced to the case where n = 1 and then it is easy.

Notation. We set

dim(U, G, q) := dim(F, H_{c}^{·}(UF̄, q^{∗}L_{ψ})^{G}) = dim(F, H_{c}^{n}(UF̄, q^{∗}L_{ψ})^{G})^{(−1)^{n}},

with n = dim U, and we set ν_{U,G} = 0 if there is a σ in G with dim σ ≠ 1, and ν_{U,G} = 1 otherwise, with dim σ denoting the determinant of the action of σ on the affine space of U. Let L(A), also denoted by L(U), be the lattice of the arrangement A. For X ∈ L(A) we set X^{∘} = X \⋃_{{Y ∈L(A) \atop X⊄Y}} Y and St_{G}(X) = \{σ ∈ G | σ(X) ⊂ X\}. We will use the above notation also when G does not act freely on U, for example when U is replaced by X^{∘} and G by St_{G}(X). From now on till the end of subsection (2.2), we do not assume that G acts freely on U.

Theorem 2.2.4. The following formula holds,

dim(U, G, q) = (ϕ(discr q)g(ϕ)^{n})^{(−1)^{n}ν_{U,G}} ∏_{X∈L(U)/G} dim(X^{∘}, St_{G}(X), q_{|X})^{−1}.

Proof. The theorem follows directly from Lemmas 2.2.1, 2.2.2 and 2.2.3 and the decomposition U = A^{n} \ ∐_{X̄∈L(U)/G} ∐_{X∈X̄} X^{∘}.

Notation. We denote by ch[A^{n}, X], for X ∈ L(A), the set of all chains in the lattice L(A) which connect A^{n} to X. By definition, such a chain is a sequence X⊂X_{1}⊂X_{2}⊂ ⋯ ⊂X_{m}⊂A^{n}, with X_{i} in L(A), the inclusions being strict. Note that St_{G}(X) acts on ch[A^{n}, X], so that we can consider the quotient chcl[X] := ch[A^{n}, X]/ St_{G}(X). For c in chcl[X] we denote by |c| the length of a chain representing c. For X in L(A), we let Repr_{X} be the following virtual representation of St_{G}(X)

Repr_{X} := (−1)^{n−dim X} ∑_{c∈chcl[X]}(−1)^{|c|−1} Ind_{St_{G}(c)}^{St_{G}(X)}(det_{|X}),

where Ind_{St_{G}(c)}^{St_{G}(X)}(det_{|X}) is the induced representation of the representation which to an element in St_{G}(c) assigns its determinant as an operator on X, and St_{G}(c) is the stabilizer of a chosen representative of c in ch[A^{n}, X].

Theorem 2.2.5. There is an isomorphism of virtual representations of G,

H_{c}^{n}(UF̄, q^{∗}L_{ψ}) ≃ ∑_{X∈L(A)/G} Ind_{St_{G}(X)}^{G}(Repr_{X}).

Moreover this isomorphism preserves the Frobenius action if we let it act by multiplication by (ϕ(discr q_{|X}))g(ϕ)^{dim X} on Repr_{X}.

Proof. By using the same decomposition of U as in the proof of Theorem 2.2.4, we deduce from Lemma 2.2.2 the following isomorphism of virtual representations of G,

H_{c}^{n}(UF̄, q^{∗}L_{ψ}) ≃ Repr_{A^{n}} −(−1)^{n} ∑_{{X∈L(A)/G \atop X≠A^{n}}} Ind_{St_{G}(X)}^{G}(−1)^{dim X}H_{c}^{dim X}(X∘ ; F̄, q^{∗}L_{ψ}),

which is compatible with the Frobenius action. The theorem follows now from Lemmas 2.2.1 and 2.2.3, by recursion and transitivity of induction.

The remaining results in this subsection (2.2) will not be used in the rest of the paper.

Notation. For X ∈ L(A), we set d(X) := dim (Repr_{X})^{St_{G}(X)}, thus

d(X) = (−1)^{n−dim X} ∑_{c∈chcl[X] dim(St_{G}(c)_{|X})⊂{1}}(−1)^{|c|−1},

where by dim(St_{G}(c)_{|X}) ⊂ {1} we mean that det(σ_{|X}) = 1 for all σ in St_{G}(c).

Corollary 2.2.6. The following equalities hold

(1) det(F, H_{c}^{n}(UF̄, q^{∗}L_{ψ})^{G}) = ∏_{X∈L(A)/G}(ϕ(discr(q_{|X}))g(ϕ)^{dim X})^{d(X)}

(2) dim (H_{c}^{n}(UF̄, q^{∗}L_{ψ})^{G})_{weight k} = ∑_{{X∈L(A)/G \atop dim X=k}} d(X).

Proposition 2.2.7. The virtual representation Repr_{X} is in fact a representation of St_{G}(X), and thus, in particular, d(X) ≥ 0, for any X in L(A).

Proof. Let A_{X} be the arrangement in A^{n} consisting of all hyperplanes of A which contain X. Clearly St_{G}(X) acts on A_{X}. Let U_{X} be the complement of A_{X} in A^{n}. From Theorem 2.2.5 applied to A_{X} we deduce that Repr_{X} is isomorphic, as a virtual representation of St_{G}(X), to H_{c}^{n}((U_{X})F̄, q^{∗}L_{ψ})_{weight dim X} which is an honest representation.

Corollary 2.2.8. Suppose dim H_{c}^{n}(UF̄, q^{∗}L_{ψ})^{G} = 1. Then there is exactly one X_{0} in L(A)/G such that d(X_{0}) ≠ 0. Moreover, d(X_{0}) = 1 and

det(F, H_{c}^{n}(UF̄, q^{∗}L_{ψ})^{G}) = ϕ(discr(q_{|X_{0}}))g(ϕ)^{dim X_{0}}.

Proof. The corollary is a direct consequence of Corollary 2.2.6 (2) and Proposition 2.2.7.

We will see in (4.4) that the hypothesis of Corollary 2.2.8 is satisfied for a Coxeter arrangement over F when p does not divide |G|.

2.3. The case when χ = ϕ. We keep the same notations as in (2.1) and (2.2).

Lemma 2.3.1. Suppose that f is the square of a polynomial over F on which G acts by multiplication by the determinant.

(1) There is a canonical isomorphism

H_{c}^{·}((U/G)F̄, f_{G}^{∗}L_{ϕ} ⊗ q_{G}^{∗}L_{ψ}) ≃ H_{c}^{·}(UF̄, q^{∗}L_{ψ})^{dim},

where the superscript det denotes the part on which G acts by multiplication by the determinant.

(2) For i ≠ n, H_{c}^{i}((U/G)F̄, f_{G}^{∗}L_{ϕ} ⊗ q_{G}^{∗}L_{ψ}) = 0.

(3) The vector space H_{c}^{n}((U/G)F̄, f_{G}^{∗}L_{ϕ} ⊗ q_{G}^{∗}L_{ψ})_{weight n} is one dimensional, and the Frobenius acts on it by multiplication by ϕ(discr q)g(ϕ)^{n}.

Proof. Let us prove (1). Let G^{+} = {σ ∈ G | dim σ = 1}. We may suppose G^{+} ≠ G, otherwise the assertion is trivial. We factor π : U → U/G into π_{1} : U → U/G^{+} and π_{2} : U/G^{+} → U/G. Note that π_{2} is the Kummer cover obtained by taking the square root of f, so we have canonical isomorphisms f_{G}^{∗}L_{ϕ} ≃ (π_{2∗}Q̄_{ℓ})^{−} ≃ (π_{2∗}Q̄_{ℓ})^{det}. From the canonical isomorphisms (π_{2∗}Q̄_{ℓ})^{det} ≃ (π_{2∗}(π_{1∗}Q̄_{ℓ})^{G^{+}})^{det} ≃ ((π_{2∗}π_{1∗}Q̄_{ℓ})^{G^{+}})^{det} ≃ (π_{∗}Q̄_{ℓ})^{det}, we obtain canonical isomorphisms

H_{c}^{·}((U/G)F̄, f_{G}^{∗}L_{ϕ} ⊗ q_{G}^{∗}L_{ψ}) ≃ H_{c}^{·}((U/G)F̄, (π_{∗}Q̄_{ℓ})^{det} ⊗ q_{G}^{∗}L_{ψ})
≃ H_{c}^{·}((U/G)F̄, (π_{∗}Q̄_{ℓ}) ⊗ q_{G}^{∗}L_{ψ})^{det}
≃ H_{c}^{·}(UF̄, q_{G}^{∗}L_{ψ})^{det}.

The assertion (2) is a direct consequence of (1) and Lemma 2.2.2, while (3) follows directly from (1) and Theorem 2.2.5.

Proposition 2.3.2. Suppose that the assumptions of Lemma 2.3.1 are valid and that H_{c}^{n}(UF̄, f^{∗}L_{ϕ} ⊗ q^{∗}L_{ψ})^{G} is of dimension 1. Then

S_{G}(ϕ) = (−1)^{n}ϕ(discr q)g(ϕ)^{n}.

Proof. It is a direct consequence of the preceding lemma.

3. Monodromy and critical values for Coxeter arrangements

3.1. Let (V, G, q) be a Coxeter arrangement over C. For each reflection hyperplane H, we choose a linear form ℓ_{H} : V → C defining H in V and we set δ = (∏_{H} ℓ_{H})^{2}. We denote by N the number of reflection hyperplanes and by Δ : V/G → C the map induced by δ. Thus Δ is the discriminant of G. We denote by F_{0} the Milnor fiber of Δ at 0 and by Z(T, G) the zeta function of local monodromy of Δ at 0, i.e.,

Z(T, G) = ∏_{i} det(1 − T M, H^{i}(F_{0}, C))^{(−1)^{i+1}},

where M denotes the monodromy automorphism (see, e.g., [3]).

Let d_{1}, …, d_{n} be the degrees of G. In [10] we proved the following result.

Theorem 3.1. For G a finite Coxeter group we have

∏_{E connected subgraph} Z(−T, G(E))^{(−1)^{|E|}} = ∏_{i=1}^{n} \frac{1 − T^{d_{i}}}{1 − T} ,

where the product on the left-hand side runs over all connected subgraphs E of the Coxeter diagram of G, G(E) denotes the Coxeter group with diagram E, and |E| the number of vertices of E.

The proof of Theorem 3.1 given in [10] depends on a case by case analysis. A more conceptual proof, but depending on Macdonald’s formula (0.1), is given in [7].

We use now terminology from [27] II Exp. XIII - XIV. Denote by η̄_{0} the generic geometric point of the henselization of A_{C}^{1} at 0 and by I_{0,C} its inertia group (i.e., the fundamental group of the complement of 0 in a small disk around 0). We denote by K_{I_{0,C}} the Grothendieck group of finite dimensional vector spaces with I_{0,C}-action. If L is an object in D_{c}^{b}(G_{m,C}, C), we denote by [Lη̄_{0}] the class of ∑(−1)^{i}[H^{i}(L)η̄_{0}] in K_{I_{0,C}} and we set [Lη̄_{∞}] = [inv^{∗}(L)η̄_{0}], where inv is the morphism x ↦ x^{−1}. If a finite group G acts on L, we denote by [LG ; η̄_{0}] the class of ∑(−1)^{i}[H^{i}(L)G ; η̄_{0}] and we define similarly [LG ; η̄_{∞}]. For any character χ : I_{0,C} → C^{×} we denote by V_{χ} the class in K_{I_{0,C}} of the rank one object with action given by χ and for any natural number m ≥ 1, we set V_{m} = [(π_{m∗}C)η̄_{0}], where π_{m} : G_{m} → G_{m} is given by x ↦ x^{m}. Of course, we have V_{m} = ∑_{χ^{m}=1} V_{χ}. We set M̄_{G} = (−1)^{n−1}[(Rψ_{Δ}(C))_{0}]. Here (Rψ_{Δ}(C))_{0} is the stalk at zero of the complex of nearby cycles with respect to Δ.

One can rephrase Theorem 3.1 as follows.

Theorem 3.1′. For G a finite Coxeter group, the following equality in K_{I_{0,C}} holds,

∑_{E connected subgraph} M̄_{G(E)} = V_{ϕ} ⊗ (∑_{i=1}^{n}(V_{d_{i}} − V_{1})),

where ϕ is the unique character of order 2.

3.2. Let U be the complement in V of δ = 0, let B (resp. B_{0}) be the intersection of U with the locus of q = 1 (resp. q = 0). The following proposition is proved in [7] by a detailed analysis of the geometry of a nice compactification of U. Here and in the following we denote by q_{|U}^{N} the restriction of the map defined by q on U raised to the N-th power.

Proposition 3.2.1. We have the following equalities in K_{I_{0,C}},

(1) [(Rδ_{|B_{0}!}C)G ; η̄_{∞}] = [(Rδ_{|B!}C)G ; η̄_{∞}].

(2) [(Rδ_{|B!}C)G ; η̄_{∞}] + [(Rq_{|U!}^{N}C)G ; η̄_{0}] = [(Rψ_{Δ}(C))_{0}].

(3) [(Rδ_{|B!}C)G ; η̄_{0}] = (−1)^{n} ∑_{E connected subgraph G(E)≠G} M̄_{G(E)}.

(4) There exist ā, b̄ in Z with ā+b̄ = (−1)^{n−1} such that [(Rq_{|U!}^{N}C)G ; η̄_{0}] = (ā−b̄)V_{N} + b̄V_{2N} and such that [(Rq_{|U!}δ^{∗}L_{χ})G ; η̄_{0}] = āV_{χ^{N}} + b̄V_{ϕχ^{N}}, for any character χ of I_{0,C}, where L_{χ} denotes the Kummer sheaf associated to χ.

Corollary 3.2.2. The equality

[(Rδ_{|B!}C)G ; η̄_{0}] − [(Rδ_{|B!}C)G ; η̄_{∞}] − [(Rq_{|U!}^{N}C)G ; η̄_{0}] = (−1)^{n}(V_{ϕ} ⊗ ∑_{i=1}^{n}(V_{d_{i}} − V_{1}))

holds in K_{I_{0,C}}.

Proof. The corollary follows directly from (2), (3) and Theorem 3.1′.

Actually, the values of ā and b̄ are determined by the following proposition, which is proved in [7], but not needed in the present paper.

Proposition 3.2.3. The following formula holds,

[(Rq_{|U!}C)G ; η̄_{0}] = (−1)^{n−1}V_{ϕ^{n+N}} = āV_{1} + b̄V_{ϕ}.

3.3. If f : X → G_{m,C} is a morphism of schemes of finite type over C, we set a(f) = ∏_{s∈G_{m,C}} s^{a_{s}(Rf_{!}C)}, where a_{s} is the drop of the rank of Rf_{!}(C) at s (i.e., the alternating sum of the drops of the rank of its cohomology sheaves, cf. [16] 3.1.5.2. We denote by Δ_{B} the function B/G → G_{m,C} induced by δ. We will use the following result which is proven in [7] as an easy consequence of Macdonald’s formula (0.1), by which one calculates the critical value (0.5) of Δ_{B}.

Theorem 3.3. The following formula holds,

a(Δ_{B}) = (κ\frac{∏_{i} d_{i}^{d_{i}}}{N^{N}} )^{(−1)^{n}},

where κ is as in (0.2).

4. Proof of the Main Theorem

4.1. Determinants and monodromy. Let U be a scheme of finite type over the finite field F. We assume that the characteristic p of F is not equal to 2 and we fix a prime ℓ distinct from p. We denote by F̄ an algebraic closure of F and we set UF̄ = U ⊗_{F} F̄. For any object F in D_{c}^{b}(U, Q̄_{ℓ}) (the derived category of “bounded complexes” of Q̄_{ℓ}-sheaves with constructible cohomology), we denote

ε_{0}(U/F, F) := det(−F, H_{c}^{·}(UF̄, F))^{−1}.

Let f : U → G_{m,F} be a morphism. We set a(f) = ∏_{s∈G_{m,F̄}} s^{a_{s}(Rf_{!}Q̄_{ℓ})} ∈ F, where a_{s} denotes the total drop at s of a complex of sheaves (cf. [16] 3.1.5.2). We also denote by η̄_{0} the generic geometric point of the henselization of A1 F̄ at 0 and by I_{0} (or I_{0,F̄}) the inertia group. We denote by K_{I_{0}} the Grothendieck group of finite dimensional Q̄_{ℓ}-vector spaces with continuous action of I_{0} defined on a finite extension of Q_{ℓ}. For L an object in D_{c}^{b}(G_{m,F}, Q̄_{ℓ}) we define as in (3.1) objects [Lη̄_{0}] and [Lη̄_{∞}] in K_{I_{0}}. Also if a finite group G acts on L, we define similarly [LG ; η̄_{0}] and [LG ; η̄_{∞}]. For χ : I_{0} → Q̄_{ℓ}^{×} a continuous character, we denote by V_{χ} the associated object in K_{I_{0}}. For N ≥ 1 an integer, we define V_{N} in K_{I_{0}} as in (3.1). We will denote by ϕ the character of order 2 of I_{0}. For any character χ : F^{×} → Q̄_{ℓ}^{×} we denote by L_{χ} the corresponding Kummer sheaf on G_{m,F}. We can also view χ as a continuous character I_{0} → Q̄_{ℓ}^{×} which we still denote by χ.

Suppose [(Rf_{!}Q̄_{ℓ})η̄_{0}] = ∑_{N∈N^{×}} α_{N}V_{N} and [(Rf_{!}Q̄_{ℓ})η̄_{∞}] = ∑_{N∈N^{×}} β_{N}V_{N} with α_{N} and β_{N} in Z. (Here N^{×} denotes N \ {0}.) We define the rational number b(f) as

b(f) := ( ∏_{N∈N^{×}} N^{Nα_{N}} )( ∏_{N∈N^{×}}(−N)^{−Nβ_{N}} ).

For any character χ : F^{×} → Q̄_{ℓ}^{×} we set

G(f, χ) = ∏_{N∈N^{×}}(g(χ^{N})^{α_{N}} g(χ^{−N})^{β_{N}} )

and

G̃(f, χ) = ∏_{N∈N^{×}}(g(χ^{N})^{α_{N} −β_{N}} ),

where g(χ) is the Gauss sum g(χ) := − ∑_{x∈F^{×}} χ(x)ψ(x) for ψ a fixed nontrivial additive character.

Theorem 4.1.1. Assume that α_{N} = β_{N} = 0 when p divides N and that the sheaves R^{i}f_{!}Q̄_{ℓ} are tame at 0 and ∞ for every i. Then, for every multiplicative character χ of F^{×},

ε_{0}(U/F, f^{∗}L_{χ}) = ε_{0}(U/F, Q̄_{ℓ}) χ(\frac{a(f)}{b(f)}) G(f, χ)

= ε_{0}(U/F, Q̄_{ℓ}) |F|^{∑_{χ^{N} ≠1} β_{N}} χ(\frac{a(f)}{|b(f)|}) G̃(f, χ).

Proof. The theorem follows directly from [9] Proposition 2.4.1, which is based on Laumon’s product formula [16] 3.2.1.1, using [16] 3.1.5.4 (iv), and g(χ^{−1}) = |F|χ(−1)g(χ)^{−1} if χ ≠ 1. For more details, see [6] Proposition 3.2.1. A much stronger result is contained in [23].

Remark. The advantage of the second equality upon the first is that |b(f)| and G̃(f, χ) only depend on [(Rf_{!}Q̄_{ℓ})η̄_{0}] − [(Rf_{!}Q̄_{ℓ})η̄_{∞}].

Let F be a constructible ℓ-adic sheaf on a scheme U (of finite type) over F and suppose a finite group G acts on U and F. We set

ε_{0}^{G}(U/F, F) := det(−F, H_{c}^{·}(UF̄, F)^{G})^{−1}.

Notation. For χ any multiplicative character of F^{×} and for F in K_{I_{0,C}} of the form F = ∑_{N∈N^{×}} γ_{N}V_{N}, with γ_{N} in Z, we set

G̃(F, χ) = ∏_{N} g(χ^{N})^{γ_{N}}

and

b(F) = ∏_{N} N^{Nγ_{N}} .

It is assumed here that a nontrivial additive character has been fixed. Note that G̃ and b are multiplicative in F.

Lemma 4.1.2. (1) If N ∈ N^{×} is even, V_{N} ⊗V_{ϕ} = V_{N}, G̃(V_{N} ⊗V_{ϕ}, χ) = G̃(V_{N}, χ)

and b(V_{N} ⊗ V_{ϕ}) = b(V_{N}).

(2) If N ∈ N^{×} is odd, V_{N} ⊗ V_{ϕ} = V_{2N} − V_{N}, G̃(V_{N} ⊗ V_{ϕ}, χ) = \frac{χ(4^{N})}{g(ϕ)} G̃(V_{N}, ϕχ) and b(V_{N} ⊗ V_{ϕ}) = 4^{N}b(V_{N}).

Proof. Everything is clear except perhaps the second relation in (2) which follows directly from the Hasse-Davenport formula (cf. [11]) and the first one.

4.2. Fix an integer d ≥ 1. Let R ⊂ C be a Dedekind domain containing the d-th roots of unity, with fraction field K of finite degree over Q. We denote by K̄ = Q̄ the algebraic closure of K in C. Let A = (V, G, q) be a Coxeter arrangement over R. For 𝔓 a maximal ideal of R we denote by (\overline{V}_{𝔓}, \overline{G}_{𝔓}, \overline{q}_{𝔓}) the corresponding data over k_{𝔓} := R/𝔓. When no confusion can arise, we will omit the index 𝔓. Choose linear forms ℓ_{H} over R which define the hyperplanes of A_{G}. Let 𝔓 be a maximal ideal of R which is relatively prime to d |G|, such that the linear forms ℓ_{H} are not zero modulo 𝔓 and such that \overline{q}_{𝔓} is a nondegenerate bilinear form. (In particular 𝔓 is relatively prime to 2 and to each degree d_{i} of G, because |G| is even and equal to the product of the d_{i}’s.) Thus the conclusions of Propositions 1.2.2, 1.3.1, and 1.3.3 hold. In this situation there is no ambiguity for the notations, δ_{𝔓}, U_{𝔓}, B_{𝔓}, B_{0𝔓}, etc, cf. the notation in 3.1 and 3.2, which refer to objects attached to the reduction of A mod 𝔓, which are also the reduction mod 𝔓 of the corresponding objects over R. We denote by I_{0,K̄} the inertia group at 0 of G_{m,K̄} and we fix a character χ : I_{0,K̄} → Q̄_{ℓ}^{×} of order d. The Kummer sheaf L_{χ} on G_{m,K̄} is obtained by base change from a sheaf, which we still denote by L_{χ}, on G_{m,R}. The character χ induces a character χ_{𝔓} : I_{0,k̄_{𝔓}} → Q̄_{ℓ}^{×} (here k̄_{𝔓} denotes an algebraic closure of k_{𝔓}) which is associated to a character, still denoted by the same symbol, χ_{𝔓} : k_{𝔓}^{×} → Q̄_{ℓ}^{×}. For s an element of K we write χ_{𝔓}(s) for the value of χ_{𝔓} on the residue class of s if s is a unit in R_{𝔓} and we set χ_{𝔓}(s) = 1 otherwise.

Proposition 4.2.1. Let 𝔓 be a maximal ideal of R satisfying the above conditions. Then a(Δ_{B})/b(F) and κ are units in R_{𝔓} and

\begin{matrix}χ_{𝔓}(\frac{a(Δ_{B})}{b(F)}) G̃(F, χ_{𝔓}) \\ = Cχ_{𝔓}(κ)^{(−1)^{n}} ∏_{1≤i≤n}[\frac{g((ϕχ_{𝔓})^{d_{i}})}{g(ϕχ_{𝔓})}]^{(−1)^{n}} \frac{g(χ_{𝔓}^{N})āg(ϕχ_{𝔓}^{N})b̄}{g(ϕ)b̄} \end{matrix} (1)

for F = [(Rδ_{|B!}C)G ; η̄_{0}] − [(Rδ_{|B!}C)G ; η̄_{∞}] in K_{I_{0,C}} (it follows from §3 and Lemma 4.1.2

that F has the required form), with C = [g(ϕ)^{(∑_{d_{i} odd} 1)−n}]^{(−1)^{n−1}}. Here, as usual, ϕ denotes the character of order 2.

Proof. It follows from Proposition 1.3.3 that κ is a unit in R_{𝔓}. By Corollary 3.2.2,

F − E = (−1)^{n}(V_{ϕ} ⊗ ∑_{i=1}^{n}(V_{d_{i}} − V_{1}))

with E = [(Rq_{|U!}^{N}C)G ; η̄_{0}], and by 3.2.1 (4), E = āV_{N} + b̄(V_{2N} − V_{N}). So by Lemma 4.1.2, we obtain that b(F − E) is a unit in R_{𝔓} and

\frac{G̃(F − E, χ_{𝔓})}{χ_{𝔓}(b(F − E))} = C ∏_{1≤i≤n}[\frac{g((ϕχ_{𝔓})^{d_{i}})}{g(ϕχ_{𝔓})}]^{(−1)^{n}}χ_{𝔓}( ∏_{1≤i≤n} d_{i}^{d_{i}})^{(−1)^{n−1}},

while b(E)N^{(−1)^{n}N} is a unit in R_{𝔓} and

\frac{G̃(E, χ_{𝔓})}{χ_{𝔓}(b(E)N^{(−1)^{n}N})} = \frac{g(χ_{𝔓}^{N})āg(ϕχ_{𝔓}^{N})b̄}{g(ϕ)b̄} .

(To verify this last equation when N is even, one has again to use the Hasse-Davenport formula as in the proof of 4.1.2.) Now the formula follows directly from the multiplicativity of G̃ and b and from Theorem 3.3.

Proposition 4.2.2. For almost all 𝔓 in Spec R (i.e., for all but a finite number),

ε_{0}^{G}(B_{𝔓}/k_{𝔓}, δ_{𝔓}^{∗}L_{χ_{𝔓}}) = ε_{0}^{G}(B_{𝔓}/k_{𝔓}, Q̄_{ℓ})|k_{𝔓}|^{∑_{χ^{j} ≠1} β_{j}}χ_{𝔓}(\frac{a(Δ_{B})}{b(F)})G̃(F, χ_{𝔓}) (1)

and

ε_{0}^{G}(B_{0𝔓}/k_{𝔓}, δ_{𝔓}^{∗}L_{χ_{𝔓}}) = ε_{0}^{G}(B_{0𝔓}/k_{𝔓}, Q̄_{ℓ})|k_{𝔓}|^{∑_{χ^{j} ≠1} β_{j}}, (2)

where the β_{j} are given by [(Rδ_{|B!}C)G ; η̄_{∞}] = ∑_{j} β_{j}V_{j}.

Proof. The first assertion is a direct consequence of Theorem 4.1.1, using standard constructibility, comparison and base change theorems. For the second assertion, note that because δ and q are homogeneous the sheaves R^{i}δ_{|B_{0}!}C are lisse on G_{m}. Thus |b(Δ_{|B_{0}/G})| = 1 (with b defined as in 4.1) and [(Rδ_{|B_{0}!}C)G ; η̄_{0}] = [(Rδ_{|B_{0}!}C)G ; η̄_{∞}]. Now the result follows similarly from Theorem 4.1.1 and Proposition 3.2.1 (1) (that

[(Rδ_{|B_{0}!}C)G ; η̄_{∞}] = [(Rδ_{|B!}C)G ; η̄_{∞}]).

4.3. Good reduction. Let A = (V, G, q) be a liftable Coxeter arrangement over F. We choose defining linear forms for each reflection hyperplane. Let d ≥ 1 be an integer and χ : F^{×} → Q̄_{ℓ}^{×} a character of order d. There exists a discrete valuation ring T ⊂ C, with residue field F and fraction field of finite degree over Q, containing the d-th roots of unity, and a lifting A_{T} of A over T. We denote by K the fraction field of T, by K̄ its algebraic closure in C and by L_{χ} the Kummer sheaf on G_{m,T} associated to χ. We assume that p does not divide |G| and we choose defining linear forms with coefficients in T for each reflection hyperplane, which reduce to those already chosen in F. We denote by δ_{T}, U_{T}, B_{T}, etc, the data associated to A_{T}.

Lemma 4.3.1. If p does not divide |G|, the specialization morphism

H_{c}^{i}(B ⊗ F̄, δ^{∗}L_{χ}) ⟶ H_{c}^{i}(B_{T} ⊗ K̄, δ_{T}^{∗} L_{χ})

is an isomorphism for all i.

Proof. The arrangement associated to A_{T} has good reduction by Proposition 1.3.1. Thus the T -scheme U_{T} admits a canonical smooth compactification Ū_{T} over T such that Ū_{T} \ U_{T} has normal crossings over T, and such that Ū_{T} is equipped with a morphism π : Ū_{T} → P_{T}^{n}, where P_{T}^{n} is the projective closure of V_{T}, extending the inclusion of U_{T} in V_{T}. The compactification Ū_{T} and the morphism π are obtained by blowing up successively the union of the strict transforms of the strata of dimension i of the projective arrangement associated to A_{T} for i increasing from 0 to n−2 (cf.

[17] § 7). We now define B̄_{T} to be the strict transform in Ū_{T} of the closure of B_{T} in P_{T}^{n}. It follows from Proposition 1.3.3 that B̄_{T} intersects the strata of Ū_{T} \ U_{T} transversally. Hence B̄_{T} is smooth and proper over T and B̄_{T} \ B_{T} has normal crossings over T. The assertion now follows from [26] Th. finitude Appendice 1.3.3 and 2.4.

Corollary 4.3.2. Let A = (V, G, q) be a liftable Coxeter arrangement over F. If p does not divide |G|, then

∑(−1)^{i} dim H_{c}^{i}(B ⊗ F̄, δ^{∗}L_{χ})^{G} = (−1)^{n−1}.

Proof. The isomorphism in Lemma 4.3.1 being G-equivariant, we only have to prove the analogous result for Coxeter arrangements over C. In this case

∑(−1)^{i} dim H_{c}^{i}(B, δ^{∗}L_{χ})^{G} = ∑(−1)^{i} dim H_{c}^{i}(B, C)^{G}
= |G|^{−1}χ(B, C).

But in [7] we proved that χ(B, C) = (−1)^{n−1}|G| for complex Coxeter arrangements. (The proof is by an easy induction in a more general setting, showing that (−1)^{n−1}χ(B, C) equals the number of Weyl chambers.)

Lemma 4.3.3. If p does not divide |G|, the sheaves R^{i}q_{T |U_{T}!}δ_{T}^{∗} L_{χ} are lisse on G_{m,T} for all i.

Proof. The polynomials associated to δ_{T} and q_{T} being homogeneous it is enough to prove that the specialization morphisms associated to the specialization of the generic point of t = 1 to the special point are isomorphims, which is precisely the content of Lemma 4.3.1.

Corollary 4.3.4. Let A = (V, G, q) be a liftable Coxeter arrangement over F. If p does not divide |G|, the sheaves R^{i}q_{|U!}δ^{∗}L_{χ} are lisse and tame on G_{m,F} for all i.

Proof. This is a direct consequence of Lemma 4.3.3, see, e.g., [8] (4.1).

Localizing the ring of algebraic integers in K by inverting a finite number of elements, we obtain a Dedekind ring R ⊂ T and a Coxeter arrangement A_{R} over R, such that A_{T} is obtained from A_{R} by extension of scalars. Let 𝔓_{0} ∈ Spec R be the intersection of the maximal ideal of T with R. Then 𝔓 = 𝔓_{0} satisfies the conditions stated at the beginning of 4.2.

Proposition 4.3.5. For 𝔓 = 𝔓_{0} or for 𝔓 any other maximal ideal of R satisfying the conditions stated at the beginning of 4.2, the following holds:

(1) The representation of Gal(K̄ | K) on det(H_{c}^{·}(B_{T} ⊗ K̄, δ^{∗}L_{χ})^{G})^{−1} is unramified at 𝔓 and the action of the geometric Frobenius relative to 𝔓 is given by −ε_{0}^{G}(B_{𝔓}/k_{𝔓}, δ_{𝔓}^{∗}L_{χ_{𝔓}}).

(2) The representation of Gal(K̄ | K) on det(H_{c}^{·}(B_{0T} ⊗ K̄, δ^{∗}L_{χ})^{G})^{−1} is unramified at 𝔓 and the action of the geometric Frobenius relative to 𝔓 is given by ε_{0}^{G}(B_{0𝔓}/k_{𝔓}, δ_{𝔓}^{∗}L_{χ_{𝔓}}). Moreover the Euler characteristic of RΓ_{c}(B_{0𝔓} ⊗ k̄_{𝔓}, δ_{𝔓}^{∗}L_{χ_{𝔓}}) is zero.

Proof. The first statement in (1) follows directly from Lemma 4.3.1 and smooth base change. The second statement in (1) follows now from Corollary 4.3.2. For (2), observe that by Lemma 4.3.3 and the Leray spectral sequence the analogous statement holds for the Galois module det(H_{c}^{·}(U_{T} \ B_{0T} ⊗ K̄, δ^{∗}L_{χ})^{G})^{−1}, by the same argument as at the end of the proof of Lemma 4.3.1, using the compactification P_{T}^{1} of G_{m,T}. So it is enough to prove the analogue statement for det(H_{c}^{·}(U_{T} ⊗ K̄, δ^{∗}L_{χ})^{G})^{−1}, which is clear if one considers the compactification used in the proof of Lemma 4.3.1. The assertion about the Euler characteristic follows in the same way.

4.4. Proof of the Main Theorem. Let A = (V, G, q) be a Coxeter arrangement over F. We assume that p does not divide |G|. Let d ≥ 1 be an integer and χ : F^{×} → Q̄_{ℓ}^{×} a character of order d. By Proposition 1.6, the Coxeter arrangement A is liftable. We use the notation and material of 4.3. It follows from Proposition 1.3.3 that κ ∈ T \ 𝔓_{0}. It is well known (see [4] Ch.V, § 6 Théorème 1) that ∑_{i=1}^{n}(d_{i} − 1) = N. This implies that the right-hand side of 4.2.1 (1) may be expressed in terms of a Jacobi sum Hecke character (cf. [26] Sommes trig. § 6). By Proposition 4.3.5, Proposition 4.2.2 and Cebotarev density, we deduce that relations (1) and (2) in Proposition 4.2.2 are also valid for 𝔓_{0}. Together with Proposition 4.2.1 this gives

\begin{matrix}ε_{0}^{G}(B/F, δ^{∗}L_{χ}) = ε_{0}^{G}(B/F, Q̄_{ℓ}) \\ · |F|^{∑_{χ^{j} ≠1} β_{j}}Cχ(κ)^{(−1)^{n}} ∏_{1≤i≤n}[\frac{g((ϕχ)^{d_{i}})}{g(ϕχ)}]^{(−1)^{n}} \frac{g(χ^{N})āg(ϕχ^{N})b̄}{g(ϕ)b̄} \end{matrix} (4.4.1)

and

ε_{0}^{G}(B_{0}/F, δ^{∗}L_{χ}) = ε_{0}^{G}(B_{0}/F, Q̄_{ℓ})|F|^{∑_{χ^{j} ≠1} β_{j}}. (4.4.2)

Here we have implicitly used Proposition 1.5 to identify the degrees of A and A_{T}.

By Lemma 4.3.3 the sheaves R^{i}q_{T |U_{T}!}δ_{T}^{∗} L_{χ} are lisse on G_{m,T}, hence we deduce from Proposition 3.2.1 (4) that [(Rq_{|U!}δ^{∗}L_{χ})G ; η̄_{0}] = āV_{χ^{N}} + b̄V_{ϕχ^{N}}. The sheaves (R^{i}q_{|U!}δ^{∗}L_{χ})^{G} being lisse and tame on G_{m,F} by Corollary 4.3.4, this implies, by the Leray spectral sequence, the relation

ε_{0}^{G}(U \ B_{0}/ F, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) = g(χ^{N})āg(ϕχ^{N})b̄ε_{0}^{G}(B/F, δ^{∗}L_{χ})^{−1}. (4.4.3)

(To verify this one uses the structure of tame irreducible lisse sheaves on G_{m,F} (see, e.g., [16] p.198), or, alternatively, one can use [6] 8.1.4 and 3.1.4,(4) and [16] 3.1.5.5.) Since we have

\begin{matrix}ε_{0}^{G}(U/F, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) \\ = ε_{0}^{G}(B_{0}/F, δ^{∗}L_{χ}) ε_{0}^{G}(U \ B_{0}/ F, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ}), \end{matrix} (4.4.4)

from equations (4.4.1) - (4.4.4), we obtain the relation

\begin{matrix}ε_{0}^{G}(U/F, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ}) = \frac{ε_{0}^{G}(B_{0}/F, Q̄_{ℓ})}{ε_{0}^{G}(B/F, Q̄_{ℓ})} \\ · C^{−1}χ(κ)^{(−1)^{n−1}}g(ϕ)b̄ ∏_{1≤i≤n}[\frac{g((ϕχ)^{d_{i}})}{g(ϕχ)}]^{(−1)^{n−1}} . \end{matrix} (4.4.5)

So,

dim(−F, H_{c}^{·}(UF̄, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ})^{G}) = A[χ(κ) ∏_{1≤i≤n} \frac{g((ϕχ)^{d_{i}})}{g(ϕχ)}]^{(−1)^{n}},

with A independent of the character χ.

Now note that by Proposition 2.1.2 and Corollary 4.3.4, H_{c}^{i}(UF̄, δ^{∗}L_{χ}⊗q^{∗}L_{ψ})^{G} = 0 for i ≠ n. On the other hand,

χ(RΓ_{c}(UF̄, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ})^{G}) = χ(RΓ_{c}(G_{m,F̄}, (Rq_{|U!}δ^{∗}L_{χ})^{G} ⊗ L_{ψ})),

because of Proposition 4.3.5 (2). Since the (R^{i}q_{|U!}δ^{∗}L_{χ})^{G} are lisse and tame on G_{m,F}, we obtain

χ(RΓ_{c}(G_{m,F̄}, (Rq_{|U!}δ^{∗}L_{χ})^{G} ⊗ L_{ψ})) = − ∑_{i}(−1)^{i} dim H_{c}^{i}(B ⊗ F̄, δ^{∗}L_{χ})^{G},

see, e.g., [15] 4.8.2. By Corollary 4.3.2, we deduce that

dim H_{c}^{n}(UF̄, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ})^{G} = 1. (4.4.6)

Hence we obtain by the Grothendieck trace formula,

\begin{aligned}S_{G}(χ) & = Tr(F, H_{c}^{·}(UF̄, δ^{∗}L_{χ} ⊗ q^{∗}L_{ψ})^{G}), \\ & = A′χ(κ) ∏_{1≤i≤n} \frac{g((ϕχ)^{d_{i}})}{g(ϕχ)}, \end{aligned} (4.4.7)

with A′ independent of the character χ. To find the value of A′ we take χ = ϕ, so A′ = S_{G}(ϕ)ϕ(κ) and by Proposition 2.3.2 we obtain

A′ = ϕ(κ)(−1)^{n}ϕ(discr q)g(ϕ)^{n}. (4.4.8)

The theorem now follows from (4.4.7) and (4.4.8).

4.5. Remark. A slightly different way of organizing the proof of the Main Theorem is by using material in [2] or [18] (or the results of [24]). This material provides an analogue of Theorem 4.1.1 for the determinant of period integrals. In this way Macdonald’s formula (0.4), which yields the period ∫_{B(R)} Δ^{s+1/2} \frac{dx}{dq}, directly implies an analogous expression for ε_{0}^{G}(B/F, δ^{∗}L_{χ}) ε_{0}^{G}(B/F, Q̄_{ℓ})^{−1}, when χ is generic and p ≫ 0, because the last expression and the period are given by similar formulas (compare Theorem 4.1.1 with loc. cit.). To obtain a full proof of the Main Theorem by this approach, one has to do similar calculations as in (4.4), but now one needs Proposition 3.2.1 (1) and Proposition 3.2.3.

4.6. Remark. That the analogy between our Main Theorem and Macdonald’s formula (0.4) is no coincidence, can be explained partially by the following conjecture of Deligne ([5] § 8.9): the period of a rank one motive over a number field is completely determined by the Frobenius action. Using this conjecture one directly obtains from Macdonald’s formula an expression for ε_{0}^{G}(B/F, δ^{∗}L_{χ}) when χ is generic and p ≫ 0. Note that to apply the conjecture one first has to get rid of the additive character. (For this reduction one needs again Proposition 3.2.1 (1) and Proposition 3.2.3.)

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Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium

   E-mail address: Jan.Denef@wis.kuleuven.ac.be

Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau (URA 169 du CNRS), and Institut de Mathématiques, Université P. et M. Curie, Case 82, 4 place Jussieu, F-75252 Paris Cedex 05 (UMR 9994 du CNRS)

   E-mail address: loeser@math.polytechnique.fr