Modules of differential operators of order 2 on Coxeter arrangements
aa r X i v : . [ m a t h . C O ] A p r Modules of differential operators of order 2 onCoxeter arrangements
Norihiro Nakashima
Abstract
The collection of reflection hyperplanes of a finite reflection group is calleda Coxeter arrangement. A Coxeter arrangement is known to be free. K. Saitohas constructed a basis consisting of invariant elements for the module ofderivations on a Coxeter arrangement. We study the module of A -differentialoperators as a generalization of the study of the module of A -derivations. Inthis article, we prove that the modules of differential operators of order 2 onCoxeter arrangements of types A, B and D are free, by exhibiting their bases.We also prove that the modules cannot have bases consisting of only invariantelements. Two keys for the proof of freeness are “Cauchy-Sylvester’s theoremon compound determinants” and “Saito-Holm’s criterion.” Key Words:
Coxeter arrangement, Cauchy-Sylvester’s compound determi-nants, Schur functions.
Primary 32S22, Secondary15A15.
Let V = R ℓ be a Euclidean space of dimension ℓ over R . Let { x , . . . , x ℓ } be a basisfor the dual space V ∗ , and let S := Sym( V ∗ ) ≃ R [ x , . . . , x ℓ ] be the polynomial ring.Put ∂ i := ∂∂x i for i = 1 , . . . , ℓ . Let D ( m ) ( S ) := L | α | = m S∂ α be the module of differen-tial operators (of order m ) of S , where | α | := α + · · · + α ℓ and ∂ α := ∂ α · · · ∂ α ℓ ℓ for amulti-index α = ( α , . . . , α ℓ ) ∈ N ℓ . A nonzero element θ = P | α | = m f α ∂ α ∈ D ( m ) ( S )is homogeneous of degree i if f α is zero or homogeneous of degree i for each α . When θ ∈ D ( m ) ( S ) is homogeneous of degree i , we write deg( θ ) = i . For a multi-index α ,1e put x α := ( x , . . . , x , x , . . . , x , . . . , x ℓ , . . . , x ℓ ) , (1.1) x α := ( x , . . . , x , x , . . . , x , . . . , x ℓ , . . . , x ℓ ) (1.2)where the number of x i (or x i ) is α i .Let A be a central (hyperplane) arrangement (i.e., every hyperplane contains theorigin) in V . Fix a linear form p H ∈ V ∗ such that ker( p H ) = H for each hyperplane H ∈ A , and put Q ( A ) := Q H ∈ A p H . We call Q ( A ) a defining polynomial of A .We define the module D ( m ) ( A ) of A -differential operators of order m by D ( m ) ( A ) := (cid:8) θ ∈ D ( m ) ( S ) | θ ( Q ( A ) S ) ⊆ Q ( A ) S (cid:9) . In the case m = 1, D (1) ( A ) is the module of A -derivations. We say A to be free if D (1) ( A ) is a free S -module. An excellent reference on arrangements is the book byOrlik and Terao [10].For a commutative R -algebra R , let D ( R ) denote the ring of differential oper-ators. Then D ( S ) is the Weyl algebra. For an ideal J of S , let D ( J ) denote thesubring of D ( S ) consisting of the operators preserving the ideal J . There is a ringisomorphism: D ( S/J ) ≃ D ( J ) /J D ( S )(see [7, Theorem 15.5.13]). Holm [3] showed that D ( Q ( A ) S ) decomposes into thedirect sum of D ( m ) ( A ). Thus we have an S -module isomorphism D ( S/Q ( A ) S ) ≃ L m ≥ D ( m ) ( A ) Q ( A ) D ( S ) . There has been a lot of research on finiteness properties of rings of differentialoperators. Systems of generators for D ( R ) are usefull to study finiteness properties.For example, it is known that D ( S/Q ( A ) S ) is a Noetherian ring when A is a 2-dimensional central arrangement, and an expression by a basis played a key role inthe proof of [8]. One of the aim to study freeness for the module D ( m ) ( A ) of A -differential operators is to give an S -basis (or S -generators) of the ring of differentialoperators D ( S/Q ( A ) S ). As the first step, we put the study of the module D (2) ( A )into practice when A is a Coxeter arrangement.Let W be a finite reflection group generated by reflections acting on V . Naturally W acts on S , and W acts on the tensor products D ( m ) ( S ) ≃ S ⊗ R P | α | = m R ∂ α .The collection of reflection hyperplanes of W is called a Coxeter arrangement (or2 reflection arrangement). Coxeter arrangements A ℓ − , B ℓ and D ℓ are respectivelydefined by A ℓ − := { H ij = { x i − x j = 0 } | ≤ i < j ≤ ℓ } , B ℓ := { H i = { x i = 0 } | i = 1 , . . . , ℓ } ∪ (cid:8) H ± ij = { x i ± x j = 0 } | ≤ i < j ≤ ℓ (cid:9) , D ℓ := (cid:8) H ± ij = { x i ± x j = 0 } | ≤ i < j ≤ ℓ (cid:9) . From now on, we assume that A is a Coxeter arrangement. The module of A -derivations is related to the invariant theory of the reflection group correspondingto A . K. Saito has proved that a Coxeter arrangement A is free, and the moduleof A -derivations is isomorphic to S ⊗ S W D (1) ( S ) W as an S -module, where S W and D (1) ( S ) W are the set of invariant elements of S and D (1) ( S ), respectively (see, forexample, Theorem 6.60 in [10]). In particular, there exists a basis for the moduleof A -derivations consisting of invariant elements. We can find an explicit basisfor D (1) ( A ) in [6] when A are Coxeter arrangements A ℓ − , B ℓ and D ℓ . However, D (2) ( A ) cannot have bases consisting of only invariant elements when A are Coxeterarrangements A ℓ − , B ℓ and D ℓ . We prove the assertion above in Section 6.In this paper we prove that the module of differential operators of order 2 onCoxeter arrangements A ℓ − , B ℓ and D ℓ are free by constructing bases in Section 4and 5. For this purpose, we introduce Cauchy-Sylvester’s theorem on compounddeterminants and Saito-Holm’s criterion. In Section 3, we give some applications ofthe Cauchy-Sylvester’s theorem on compound determinants.The results of this work (without proofs) have been submitted as an extendedabstract to FPSAC 2012 [9]. In this section, we explain Saito-Holm’s criterion. Put s m := (cid:0) ℓ + m − m (cid:1) and t m := (cid:0) ℓ + m − m − (cid:1) , and set { α (1) , . . . , α ( s m ) } = { α ∈ N ℓ | | α | = m } , where | α | = α + · · · + α ℓ for a multi-index α ∈ N ℓ . For operators θ , . . . , θ s m ∈ D ( m ) ( A ), define the coefficient matrix M m ( θ , . . . , θ s m ) of the operators θ , . . . , θ s m as follows: M m ( θ , . . . , θ s m ) := " θ i x α ( j ) α ( j ) ! ! ≤ i,j ≤ s m , α ! = α ! · · · α ℓ !. Thus the ( i, j )-entry of the coefficient matrix is the polyno-mial coefficient of ∂ α ( j ) in θ i .The following criterion was originally given by Saito [11] in the case m = 1, andwas generalized by Holm [2] into the case m general. Proposition 2.1 (Saito-Holm’s criterion) . Let θ , . . . , θ s m ∈ D ( m ) ( A ) be homoge-neous operators. Then the following two conditions are equivalent: (1) det M m ( θ , . . . , θ s m ) = cQ t m for some c ∈ R × . (2) θ , . . . , θ s m form a basis for D ( m ) ( A ) over S . When D ( m ) ( A ) is a free S -module, we define the exponents of D ( m ) ( A ) to bethe multi-set of degrees of a homogeneous basis { θ , . . . , θ s m } for D ( m ) ( A ), which isdenoted by exp D ( m ) ( A ):exp D ( m ) ( A ) = { deg( θ ) , . . . , deg( θ s m ) } . Throughout this paper, we assume ℓ ≥ m . In this section, we will follow thenotation of the paper by Ito and Okada [5] as far as possible. We denote by ≻ thelexicographic order on Z m . That is, for µ = ( µ , . . . , µ m ) and ν = ( ν , . . . , ν m ) ∈ Z m ,we write µ ≻ ν if there exist an index k such that µ = ν , . . . , µ k − = ν k − , and µ k > ν k . Set Z := { µ = ( µ , . . . , µ m ) ∈ Z m | ≤ µ < µ < · · · < µ m ≤ ℓ } . Then Z is a totally ordered subset of Z m . Put x µ := ( x µ , . . . , x µ m ) ∈ S m .Let A = ( a i,j ) ≤ i,j ≤ ℓ be a square matrix of order ℓ . For µ, ν ∈ Z put A µ,ν := (cid:0) a µ i ,ν j (cid:1) ≤ i,j ≤ m . We define the m -th compound matrix A ( m ) by A ( m ) := (det A µ,ν ) µ,ν ∈ Z , Z .The following was obtained by Cauchy and Sylvester (see, for example, [5, Propo-sition 3.1]). Proposition 3.1 (Cauchy-Sylvester) . Let A = ( a i,j ) ≤ i,j ≤ ℓ be a square matrix. Thenthe determinant of the m -th compound matrix A ( m ) is given by det A ( m ) = (det A )( ℓ − m − ) . (3.1)Put Λ := { λ = ( λ , . . . , λ m ) ∈ Z m | ℓ − m ≥ λ ≥ λ ≥ · · · ≥ λ m ≥ } . We regard Λ as a totally ordered subset of Z m by the order ≻ . Then the map Z ∋ ( µ , . . . , µ m ) ( ℓ − m + 1 − µ , ℓ − m + 2 − µ , . . . , ℓ − µ m ) ∈ Λis a bijection between Λ and Z , and this bijection reverses the ordering on Λ and Z .For λ ∈ Λ, we define the following symmetric polynomials and a Laurent poly-nomial: s A λ := det( t λ j + m − ji ) ≤ i,j ≤ m det( t m − ji ) ≤ i,j ≤ m ∈ S [ t , . . . , t m ] , (3.2) s B λ := det( t λ j + m − j )+1 i ) ≤ i,j ≤ m det( t m − j ) i ) ≤ i,j ≤ m ∈ S [ t , . . . , t m ] , (3.3) s D λ := det( t λ j + m − j ) − i ) ≤ i,j ≤ m det( t m − j ) i ) ≤ i,j ≤ m ∈ S [ t ± , . . . , t ± m ] . (3.4)The polynomial s A λ is the Schur polynomial corresponding to the partition λ . TheLaurent polynomials s B λ and s D λ may be expressed by s A λ as follows: s B λ = t · · · t m · det(( t i ) λ j + m − j ) ≤ i,j ≤ m det(( t i ) m − j ) ≤ i,j ≤ m = t · · · t m s A λ ( t , . . . , t m ) (3.5) s D λ = 1 t · · · t m · det(( t i ) λ j + m − j ) ≤ i,j ≤ m det(( t i ) m − j ) ≤ i,j ≤ m = 1 t · · · t m s A λ ( t , . . . , t m ) (3.6)We remark that s D λ is a symmetric polynomial if λ m ≥
1. Now the degrees of theseLaurent polynomials are following:deg s A λ = | λ | , deg s B λ = 2 | λ | + m, deg s D λ = 2 | λ | − m, (3.7)where | λ | := λ + · · · + λ m . 5 roposition 3.2. We have the following determinant identities: det (cid:0) s A λ ( x µ ) (cid:1) λ ∈ Λ µ ∈ Z = " Y ≤ i Apply the formula (3.1) to the matrices A = ( x ℓ − ji ) ≤ i,j ≤ ℓ , A = ( x ℓ − j )+1 i ) ≤ i,j ≤ ℓ and A = ( x ℓ − j ) − i ) ≤ i,j ≤ ℓ .We will use these determinant identities for proving that D (2) ( A ) are free when A are reflection arrangements types A, B and D in Section 4 and 5. Example 3.3. We assume that ℓ = 3 , m = 2 . Then Λ = { ( λ , λ ) | ≥ λ ≥ λ ≥ } = { (1 , , (1 , , (0 , } . The Schur polynomials are following: s A (1 , ( t , t ) = t t − t t t − t = t t ,s A (1 , ( t , t ) = t − t t − t = t + t ,s A (0 , ( t , t ) = t − t t − t = 1 . Let A = x x x x x x . Then the determinant of A is Vandermonde’s determinant, and is equal to the directproduct ( x − x )( x − x )( x − x ) . et us consider the second compound matrix A (2) : A (2) = x x − x x x − x x − x x x − x x x − x x − x x x − x x x − x x − x = ( x − x ) s A (1 , ( x , x ) ( x − x ) s A (1 , ( x , x ) ( x − x ) s A (0 , ( x , x )( x − x ) s A (1 , ( x , x ) ( x − x ) s A (1 , ( x , x ) ( x − x ) s A (0 , ( x , x )( x − x ) s A (1 , ( x , x ) ( x − x ) s A (1 , ( x , x ) ( x − x ) s A (0 , ( x , x ) . By the identity (3.1), we have the determinant identity (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s A (1 , ( x , x ) s A (1 , ( x , x ) s A (0 , ( x , x ) s A (1 , ( x , x ) s A (1 , ( x , x ) s A (0 , ( x , x ) s A (1 , ( x , x ) s A (1 , ( x , x ) s A (0 , ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( x − x )( x − x )( x − x ) . Example 3.4. Let ℓ = 3 , m = 2 . Then s B (1 , ( t , t ) = t t − t t t − t , s B (1 , ( t , t ) = t t − t t t − t , s B (0 , ( t , t ) = t t − t t t − t . Let B = x x x x x x x x x . Then the determinant det B is equal to x x x ( x − x )( x − x )( x − x ) . Since det B (2) = ( x x x ( x − x )( x − x )( x − x )) , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s B (1 , ( x , x ) s B (1 , ( x , x ) s B (0 , ( x , x ) s B (1 , ( x , x ) s B (1 , ( x , x ) s B (0 , ( x , x ) s B (1 , ( x , x ) s B (1 , ( x , x ) s B (0 , ( x , x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = x x x ( x − x )( x − x )( x − x ) . A and B Let A be an arbitrary arrangement. By [3, Proposition 2.3] and [3, Theorem 2.4],we have D ( m ) ( A ) = \ H ∈ A D ( m ) ( p H S ) , (4.1)7here D ( m ) ( p H S ) = (cid:8) θ ∈ D ( m ) ( S ) | θ ( p H x α ) ∈ p H S for any | α | = m − (cid:9) for H ∈ A . Recall that the defining polynomials of Coxeter arrangements A ℓ − and B ℓ oftypes A and B are Q ( A ℓ − ) = Y ≤ i For k = 1 , . . . , ℓ , we have that η A k ∈ D ( m ) ( A ℓ − ) and η B k ∈ D ( m ) ( B ℓ ) .Proof. For any 1 ≤ i < j ≤ ℓ and a multi-index β with | β | = m − m ! ∂ mk (cid:0) ( x i ± x j ) x β (cid:1) = i = k and β i + 1 = m, ± j = k and β j + 1 = m, . If i = k and β i + 1 = m or j = k and β i + 1 = m , then η A k (( x i − x j ) · x β ) . = h A k ∈ ( x i − x j ) S . Therefore we obtain η A k ∈ D ( m ) ( A ℓ − ) from (4.1).Similarly we have η B k ∈ T ≤ i 1, we have η B k (cid:0) x i · x β (cid:1) = ( h B k if i = k and β i + 1 = m, . This leads to that η B k ∈ T ℓi =1 D ( m ) ( x i S ). Therefore we obtain η B k ∈ D ( m ) ( B ℓ ).8or a Laurent polynomial f ( t , . . . , t m ) ∈ S [ t ± , . . . , t ± m ] satisfying f ( x α ) ∈ S forany α with | α | = m , we define an operator θ f := X | α | = m f ( x α ) 1 α ! ∂ α . We say a Laurent polynomial f ( t , . . . , t m ) is symmetric if f ( t , . . . , t i , . . . , t j , . . . , t m ) = f ( t , . . . , t j , . . . , t i , . . . , t m )for all pairs ( i, j ). Lemma 4.2. Assume that f ( t , . . . , t m ) is a symmetric Laurent polynomial. Thenwe have that θ f ∈ D ( m ) ( A ℓ − ) .Proof. Since f ( t , . . . , t m ) is symmetric, we have θ f (cid:0) ( x i − x j ) · x β (cid:1) | x i = x j = (cid:0) f ( x β + e i ) − f ( x β + e j ) (cid:1) | x i = x j = 0for any 1 ≤ i < j ≤ ℓ and a multi-index β with | β | = m − 1. We obtain θ f (cid:0) ( x i − x j ) · x β (cid:1) ∈ ( x i − x j ) S . Hence it follows from (4.1) that θ f ∈ D ( m ) ( A ℓ − ).For λ ∈ Λ, define operators θ A λ := X | α | = m s A λ ( x α ) 1 α ! ∂ α , θ B λ := X | α | = m s B λ ( x α ) 1 α ! ∂ α . Then deg θ A λ = | λ | , deg θ B λ = 2 | λ | + m by the formula (3.7). Proposition 4.3. For λ ∈ Λ , we have θ A λ ∈ D ( m ) ( A ℓ − ) and θ B λ ∈ D ( m ) ( B ℓ ) .Proof. Since Laurent polynomials s A λ and s B λ are symmetric, we obtain θ A λ , θ B λ ∈ D ( m ) ( A ℓ − ) by Lemma 4.2.By (4.1), we can write D ( m ) ( B ℓ ) = D ( m ) ( A ℓ − ) ∩ ℓ \ i =1 D ( m ) ( x i S ) ! ∩ \ ≤ i 1, we have θ B λ ( x i · x β ) = s B λ ( x β + e i ) = x i · x β s A λ ( x β + e i ) ∈ x i S. This implies T ℓi =1 θ B λ ∈ D ( m ) ( x i S ).For any 1 ≤ i < j ≤ ℓ and a multi-index β with | β | = m − θ B λ (( x i + x j ) · x β ) = s B λ ( x β + e i ) + s B λ ( x β + e j ) = x β (cid:16) x i s A λ ( x β + e i ) + x j s A λ ( x β + e j ) (cid:17) Then we have θ B λ (( x i + x j ) · x β ) | x i = − x j = 0, and this implies θ B λ (cid:0) ( x i + x j ) · x β (cid:1) ∈ ( x i + x j ) S . Hence we obtain θ B λ ∈ D ( m ) ( B ℓ ). Theorem 4.4. Let m = 2 . (1) The set C A := (cid:8) η A i | i = 1 , . . . ℓ (cid:9) ∪ (cid:8) θ A λ | λ ∈ Λ (cid:9) forms an S -basis for D (2) ( A ℓ − ) . Hence exp D (2) ( A ℓ − ) = { ℓ − , . . . , ℓ − } ∪ {| λ | | λ ∈ Λ } . (2) The set C B := (cid:8) η B i | i = 1 , . . . ℓ (cid:9) ∪ (cid:8) θ B λ | λ ∈ Λ (cid:9) forms an S -basis for D (2) ( B ℓ ) . Hence exp D (2) ( B ℓ ) = { ℓ − , . . . , ℓ − } ∪ { | λ | + 2 | λ ∈ Λ } . Proof. (1) All operators in C A belong to D (2) ( A ℓ − ) by Proposition 4.1 and Propo-sition 4.3.By Proposition 2.1, we only need to prove that the determinant of the coefficientmatrix M m ( C A ) of the operators of C A is equal to Q ( A ℓ − ) ℓ up to a nonzero constant.By Proposition 3.2, we obtain det (cid:0) s A λ ( x α ) (cid:1) λ ∈ Λ , α ∈ Z = Q ( A ) ℓ − . Hence we havedet M m ( C A ) . = Q ( A ℓ − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ℓ ∗ (cid:0) s A λ ( x α ) (cid:1) λ ∈ Λ α ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Q ( A ℓ − ) · Q ( A ℓ − ) ℓ − = Q ( A ℓ − ) ℓ . (2) We have an identitydet M m ( C B ) . = x · · · x ℓ Y ≤ i 10 0 0 x x x + x 10 0 0 x x x + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − ( x − x ) ( x − x ) ( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x + x x x x + x x x x + x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . = Q ( A ) . D In this section, we assume m = 2, and we construct a basis for D (2) ( D ℓ ). Recall thedefining polynomial Q ( D ℓ ) = Q ≤ i For λ ∈ Λ , we have θ D λ ∈ D (2) ( D ℓ ) .Proof. By Lemma 4.2, we have θ D λ ∈ D (2) ( A ℓ − ) for any λ ∈ Λ.12ince X | α | =2 s D λ ( x α ) 1 α ! ∂ α (( x i + x j ) · x k )= s D λ ( x i , x k ) + s D λ ( x j , x k )= 1 x i x k s A λ ( x i , x k ) + 1 x j x k s A λ ( x j , x k )= 1 x i x j x k (cid:0) x j s A λ ( x i , x k ) + x i s A λ ( x j , x k ) (cid:1) , we obtain θ D λ (( x i + x j ) x k ) | x i = − x j = 0 for 1 ≤ i < j ≤ ℓ , k = 1 , . . . , ℓ and λ ∈ Λ.Hence we have θ D λ ∈ D (2) ( D ) for any λ ∈ Λ.We introduce other operators h D k of D (2) ( D ℓ ). For k = 1 , . . . , ℓ put h D k := ( x k − x ) · · · ( x k − x k − )( x k − x k +1 ) · · · ( x k − x ℓ ), and define η D k := h D k x k ∂ k − ( − ℓ − x k θ D λ (0) . The coefficient of ∂ k in η D k is h D k x k − ( − ℓ − ( x · · · x ℓ ) x k · x k = h D k − ( − ℓ − ( x · · · x k − x k +1 · · · x ℓ ) x k ∈ S. Hence we obtain η D k ∈ D (2) ( S ), and deg η D k = 2 ℓ − Proposition 5.2. For k = 1 , . . . , ℓ , we have that η D k ∈ D (2) ( D ℓ ) .Proof. Let k = 1 , . . . , ℓ . It is clear that h D k ∂ k ∈ D (2) ( D ℓ ) , and we have θ D λ (0) ∈ D (2) ( D ℓ ) by Proposition 5.1. Thus we have h D k ∂ k − ( − ℓ − θ D λ (0) ∈ D (2) ( D ℓ ). Thisleads to η D k ∈ D (2) ( D ℓ ). Theorem 5.3. Assume m = 2 . The set C D := (cid:8) η D i | i = 1 , . . . ℓ (cid:9) ∪ (cid:8) θ D λ | λ ∈ Λ (cid:9) forms an S -basis for D (2) ( D ℓ ) . Hence exp D (2) ( D ℓ ) = { ℓ − , . . . , ℓ − } ∪ { λ + 2 λ − | ℓ − ≥ λ ≥ λ ≥ }∪ { λ − ℓ | ℓ − ≥ λ ≥ } ∪ { ℓ − } . roof. By Proposition 5.1 and Proposition 5.2, we have C D ⊆ D (2) ( D ℓ ). Let M ( C D )be the coefficient matrix of the operators in C D . We shall show that det M ( C D ) . = Q ( D ℓ ) ℓ .Put θ λ := P | α | =2 s D λ ( x α ) α ! ∂ α for λ ∈ Λ. Thendet M ( C D ) = det M ( η D i , θ D λ | i = 1 , . . . , ℓ, λ ∈ Λ)= det M ( η D i + ( − ℓ − x i θ D λ (0) , θ D λ | i = 1 , . . . , ℓ, λ ∈ Λ)= ( x · · · x ℓ ) ℓ det M ( η D i + ( − ℓ − x i θ D λ (0) , θ λ | i = 1 , . . . , ℓ, λ ∈ Λ) . = (cid:18) h D x · · · h D ℓ x ℓ (cid:19) ( x · · · x ℓ ) ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I ℓ ∗ (cid:0) s D λ ( x α ) (cid:1) λ ∈ Λ α ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = Q ( D ℓ ) ( x · · · x ℓ ) ℓ − Q ( D ℓ ) ℓ − ( x · · · x ℓ ) ℓ − = Q ( D ℓ ) ℓ by Proposition 3.2. Hence we conclude that the set C D forms an S -basis for D (2) ( D ℓ )by Proposition 2.1. Let W be a finite reflection group generated by reflections acting on V . Then W acts on S by ( w · f )( v ) = f ( w − · v ) for f ∈ S , w ∈ W and v ∈ V . The action of W on D ( m ) ( S ) is defined by ( w · θ )( f ) = w · ( θ ( w − · f )) for w ∈ W , θ ∈ D ( m ) ( S ) and f ∈ S .Let S ℓ be the symmetric group acting on V by permuting the coordinates. Let Z / Z = { , − } . An abelian group ( Z / Z ) ℓ acts on V by change of signs. Let( Z / Z ) ℓ − be the subgroup of ( Z / Z ) ℓ defined by( Z / Z ) ℓ − = (cid:8) ( a , . . . , a ℓ ) ∈ ( Z / Z ) ℓ | a · · · a ℓ = 1 (cid:9) . The group S ℓ acts on ( Z / Z ) ℓ and ( Z / Z ) ℓ − by permuting the coodinates.The finite irreducible reflection groups of types A, B and D are defined by W A := S ℓ ,W B := S ℓ ⋉ ( Z / Z ) ℓ ,W D := S ℓ ⋉ ( Z / Z ) ℓ − . ℓ ≥ W A , W B and W D act on V . Hence the groups W A , W B and W D act on S and D ( m ) ( S ). Proposition 6.1. Let W be a finite reflection group, and A the reflection arrange-ment consisting of all reflection hyperplanes of W . Then the submodule D ( m ) ( A ) of D ( m ) ( S ) is closed under the action of W .Proof. For w ∈ W and θ ∈ D ( m ) ( A ), we prove that w · θ ∈ D ( m ) ( A ).For f ∈ S , we have w · θ ( Qf ) = w · ( θ ( w − · ( Qf )))= w · ( θ (det( w − ) Q ( w − · f ))) . Since det( w − ) Q ( w − · f ) ∈ QS , we have θ (det( w − ) Q ( w − · f )) ∈ QS . Then thereexists g ∈ S such that θ (det( w − ) Q ( w − · f )) = Qg . Hence w · θ ( Qf ) = w · ( Qg ) = (det( w ) Q )( w · g ) ∈ QS. By Proposition 6.1, the groups W A , W B and W D act on D ( m ) ( A ), D ( m ) ( B ) and D ( m ) ( D ), respectively.In case m = 1, the modules D (1) ( A ), D (1) ( B ) and D (1) ( D ) have bases consistingof only invariant elements [10, Theorem 6.60]. In this section, we prove that D (2) ( A ), D (2) ( B ) and D (2) ( D ) cannot have bases consisting of only invariant elements, when m = 2.The actions of a transposition σ i,j := ( i j ) ∈ S ℓ on { x , . . . x ℓ } and { ∂ , . . . , ∂ ℓ } are as follows: σ i,j · x k = x σ i,j · k , σ i,j · ∂ k = ∂ σ i,j · k ( k = 1 , . . . , ℓ ) . The group S ℓ acts on the set of multi-indices by permuting the coordinates: σ i,j · ( α , . . . , α i , . . . , α j , . . . , α ℓ ) = ( α , . . . , α j , . . . , α i , . . . , α ℓ )The action of S ℓ preserves the norm of a multi-index. Then σ i,j · x α = x σ i,j · α for amulti-index α ∈ N ℓ .Let τ i ∈ ( Z / Z ) ℓ be the element of change of signs of the i -th coordinate: τ i · x k = a i,k x k , τ i · ∂ k = a i,k ∂ k ( k = 1 , . . . , ℓ ) (6.1)where a i,i = − a i,k = 1 for k = i . 15 emma 6.2. Let λ ∈ Λ . (1) The operator θ A λ is W A -invariant. (2) The operator θ B λ is W B -invariant. (3) The operator θ D λ is W D -invariant when m = 2 .Proof. (1) Since W A is generated by transpositions σ , , . . . , σ ℓ − ,ℓ (see, for example,[1]), it is enough to prove that θ A λ = X | α | = m s A λ ( x α ) 1 α ! ∂ α is invariant under the actions of transpositions σ , , . . . , σ ℓ − ,ℓ .Clearly, we have that ( σ i,i +1 · α )! = α ! and | σ i,i +1 · α | = | α | for a multi-index α with | α | = m . Then a transposition σ i,i +1 is a bijection between the set { α ∈ N ℓ || α | = m } and itself. Therefore we have that, for i = 1 , . . . , ℓ − σ i,i +1 · θ A λ = X | σ i,i +1 · α | = m σ i,i +1 · (cid:0) s A λ ( x σ i,i +1 · α ) (cid:1) σ i,i +1 · α )! σ i,i +1 · ( ∂ σ i,i +1 · α )= X | σ i,i +1 · α | = m s A λ ( x σ i,i +1 · σ i,i +1 · α ) 1( σ i,i +1 · α )! ∂ σ i,i +1 · σ i,i +1 · α = X | α | = m s A λ ( x α ) 1 α ! · ∂ α = θ A λ . (2) The group W B is generated by τ ℓ and transpositions σ , , . . . , σ ℓ − ,ℓ (see [1]).It is enough to prove that θ B λ is invariant under the actions of the generators.By the formulas (1.2) and (3.5), we have s B λ ( x α ) = x α · · · x α ℓ ℓ s A λ ( x α ) (6.2)for a multi-index α with | α | = m . Then σ i,i +1 · s B λ ( x α ) = x α σ i,i +1 · · · · x α ℓ σ i,i +1 · ℓ s A λ ( x σ i,i +1 · α )= x α σi,i +1 · · · · x α σi,i +1 · ℓ ℓ s A λ ( x σ i,i +1 · α ) = s B λ ( x σ i,i +1 · α ) . Hence we have that σ i,i +1 · θ B λ = X | α | = m s B λ (cid:0) x σ i,i +1 · α (cid:1) σ i,i +1 · α )! · ∂ σ i,i +1 · α = θ B λ . 16t remains to prove that θ B λ is τ ℓ -invariant.By the formulas (6.1) and (6.2), we have τ ℓ · ∂ α = ( − α ℓ ∂ α and τ ℓ · s B λ ( x α ) =( − α ℓ s B λ ( x α ). Then we have τ ℓ · θ B λ = X | α | = m ( − α ℓ s B λ ( x α ) 1 α ! ( − α ℓ ∂ τ ℓ · α = θ B λ . Hence the operator w · θ B λ coincides with θ B λ for w ∈ W B .(3) The proof of (3) goes similarly to (2). The set { σ , , . . . , σ ℓ − ,ℓ , σ ℓ − ,ℓ τ ℓ − τ ℓ } is a system of generators of W D (see [1]).The formulas (1.2) and (3.6) imply that s D λ ( x α ) = 1 x α · · · x α ℓ ℓ s A λ ( x σ i,i +1 · α ) . (6.3)Then the formula (6.3) leads to σ i,i +1 · s D λ ( x α ) = 1 σ i,i +1 · ( x α · · · x α ℓ ℓ ) s A λ ( x σ i,i +1 · α ) = s D λ ( x σ i,i +1 · α ) . We obtain σ i,i +1 · θ D λ = θ D λ by a straightforward calculation using the formulas (5.1),(5.2) and (5.3).In order to verify σ ℓ − ,ℓ τ ℓ − τ ℓ · θ D λ = θ D λ , we compute the actions of σ ,ℓ − τ ℓ − τ ℓ on polynomials and differential operators: σ ℓ − ,ℓ τ ℓ − τ ℓ · s D λ ( x α ) = ( − α ℓ − + α ℓ σ ℓ − ,ℓ · ( x α · · · x α ℓ ℓ ) s A λ ( x σ ℓ − ,ℓ · α )= ( − α ℓ − + α ℓ s D λ ( x α ) ,σ ℓ − ,ℓ τ ℓ − τ ℓ · ∂ α = ( − α ℓ − + α ℓ ∂ σ ℓ − ,ℓ · α ,σ ℓ − ,ℓ τ ℓ − τ ℓ · x · · · x ℓ = 1 x · · · x ℓ ,σ ℓ − ,ℓ τ ℓ − τ ℓ · x · · · x ℓ ) = 1( x · · · x ℓ ) . By the case-by-case checking, we can verify that σ ℓ − ,ℓ τ ℓ − τ ℓ · θ D λ coincides with θ D λ for λ ∈ Λ. Therefore the operator θ D λ is W D -invariant.17 emma 6.3. (1) The vector space spanned by (cid:8) η A k | k = 1 , . . . ℓ (cid:9) is closed underthe action of W A . Moreover, the vector space spanned by (cid:8) η A k | k = 1 , . . . ℓ (cid:9) isisomorphic to the Euclidean space V as W A -modules. (2) The vector space spanned by (cid:8) η B k | k = 1 , . . . ℓ (cid:9) is closed under the action of W B . Moreover, the vector space spanned by (cid:8) η B k | k = 1 , . . . ℓ (cid:9) is isomorphicto the Euclidean space V as W B -modules. (3) Let m = 2 . The vector space spanned by (cid:8) η D k | k = 1 , . . . ℓ (cid:9) is closed underthe action of W D . Moreover, the vector space spanned by (cid:8) η D k | k = 1 , . . . ℓ (cid:9) is isomorphic to the Euclidean space V as W D -modules.Proof. (1) Let h η A , . . . , η A ℓ i R be the vector space spanned by (cid:8) η D k | k = 1 , . . . ℓ (cid:9) .Define a linear isomorphism φ A : h η A , . . . , η A ℓ i R −→ V by φ A ( η A k ) = e k for k = 1 , . . . , ℓ .A transposition σ i,i +1 acts on the standard basis e , . . . , e ℓ of V by σ i,i +1 · e k = e σ i,i +1 · k for k = 1 , . . . , ℓ . To prove the assertion, we verify σ i,i +1 · η A k = η A σ i,i +1 · k for k = 1 , . . . , ℓ .Recall that the definitions h A k = ( x k − x ) · · · ( x k − x k − )( x k − x k +1 ) · · · ( x k − x ℓ )and η A k = h A k m ! ∂ mk . For k = i, i + 1, we have σ i,i +1 · h A k = h A k . Also we have σ i,i +1 · h A i = ( x i +1 − x ) · · · ( x i +1 − x i − )( x i +1 − x i )( x i +1 − x i +2 ) · · · ( x k − x ℓ ) = h A i +1 , and similarly σ i,i +1 · h A i +1 = h A i . Then we obtain σ i,i +1 · η A k = η A k ( k = i, i + 1) , σ i,i +1 · η A i = η A i +1 , σ i,i +1 · η A i +1 = η A i . Hence we conclude that the map φ A is a W A -isomorphism.(2) A system of generators σ , , . . . , σ ℓ − ,ℓ and τ ℓ of W B acts on the standardbasis e , . . . , e ℓ of V by σ i,i +1 · e k = e σ i,i +1 · k for k = 1 , . . . , ℓ and τ ℓ · e ℓ = − e ℓ , τ ℓ · e k = e k ( k = 1 , . . . , ℓ − . Define a linear isomorphism φ B : h η B , . . . , η B ℓ i R −→ V by φ B ( η B k ) = e k for k = 1 , . . . , ℓ . We prove that φ B is a W B -isomorphism by checkingall actions of the generators are the same.18ecall that the definitions h B k = x k ( x k − x ) · · · ( x k − x k − )( x k − x k +1 ) · · · ( x k − x ℓ )and η B k = h B k m ! ∂ mk . We have the following: σ i,i +1 · h B k = h B k ( k = i, i + 1) ,σ i,i +1 · h B i = h B i +1 ,σ i,i +1 · h B i +1 = h B i ,τ ℓ · h B k = h B k ( k = ℓ ) ,τ ℓ · h B ℓ = − h B ℓ . Then σ i,i +1 · η B k = η B k ( k = i, i + 1) , σ i,i +1 · η B i = η B i +1 , σ i,i +1 · η B i +1 = η B i ,τ ℓ · η B k = η B k ( k = ℓ ) , τ ℓ · η B ℓ = η B ℓ . The actions of the generators on { η B k | k = 1 , . . . , ℓ } coincide with the actions of thegenerators on { e k | k = 1 , . . . , ℓ } . Hence the map φ B is a W B -isomorphism.(3) The group W D is generated by elements σ , , . . . , σ ℓ − ,ℓ and σ ℓ − ,ℓ τ ℓ − τ ℓ . Thegenerators acts on the standard basis e , . . . , e ℓ of V by σ i,i +1 · e k = e σ i,i +1 · k for k = 1 , . . . , ℓ and σ ℓ − ,ℓ τ ℓ − τ ℓ · e ℓ = − e ℓ − , σ ℓ − ,ℓ τ ℓ − τ ℓ · e ℓ − = − e ℓ ,σ ℓ − ,ℓ τ ℓ − τ ℓ · e k = e k ( k = 1 , . . . , ℓ − . We prove that a linear isomorphism φ D : h η D , . . . , η D ℓ i R −→ V defined by φ D ( η D k ) = e k for k = 1 , . . . , ℓ is a W D -isomorphism.Recall that the definitions h D k = ( x k − x ) · · · ( x k − x k − )( x k − x k +1 ) · · · ( x k − x ℓ )and η D k = h D k x k ∂ k − ( − ℓ − x k θ D λ (0) . Clearly we have σ i,i +1 · h D i = h D i +1 , σ i,i +1 · h D i +1 = h D i , σ i,i +1 · h D k = h D k ( k = i, i + 1) , and then σ i,i +1 · η D i = η D i +1 , σ i,i +1 · η D i +1 = η D i , σ i,i +1 · η D k = η D k ( k = i, i + 1)by Lemma 6.2. The actions of σ ℓ − ,ℓ τ ℓ − τ ℓ on { h D k | k = 1 , . . . , ℓ } are σ ℓ − ,ℓ τ ℓ − τ ℓ · h D ℓ = h D ℓ − , σ ℓ − ,ℓ τ ℓ − τ ℓ · h D ℓ − = h D ℓ ,σ ℓ − ,ℓ τ ℓ − τ ℓ · h D k = h D k ( k = 1 , . . . , ℓ − . σ ℓ − ,ℓ τ ℓ − τ ℓ · η D ℓ = − h D ℓ − x ℓ − ∂ ℓ − + ( − ℓ − x ℓ − θ D λ (0) = − η D ℓ − ,σ ℓ − ,ℓ τ ℓ − τ ℓ · η D ℓ − = − h D ℓ x ℓ ∂ ℓ + ( − ℓ − x ℓ θ D λ (0) = − η D ℓ ,σ ℓ − ,ℓ τ ℓ − τ ℓ · η D k = η D k ( k = 1 , . . . , ℓ − . Thus the actions of the generators on { η D k | k = 1 , . . . , ℓ } coincide with the actions ofthe generators on { e k | k = 1 , . . . , ℓ } . Hence the map φ D is a W D -isomorphism. Corollary 6.4. Assume that m = 2 . Let W be a finite reflection group of typeA, B or D. Let A be the reflection arrangement corresponding to W . Then thevector space X generated by the basis C A in Theorem 4.4 or Theorem 5.3 is a W -module, and X is isomorphic as a representeation to X ♯ Λ0 ⊕ V where X is the trivialrepresentation. In the case of type A ℓ − , the reflection group W A stabilizes the subspace R ( e + · · · + e ℓ ) of V pointwisely. The orthogonal complement V ′ of R ( e + · · · + e ℓ ) is closedunder the action of the reflection group W A , and V ′ is essensial (see [4]). Moreover V ′ is an irreducible representation. In the case of type B or D, a representation V is irreducible (see Bourbaki [1, Chap. 5, Sect. 3, Proposition 5]).Let D ( m ) ( S ) W be the set of invariant elements of D ( m ) ( S ). Theorem 6.5. Assume that m = 2 . Let W be a finite reflection group of type A,B or D, and let A be the reflection arrangement corresponding to W . Then Themodule D (2) ( A ) cannot have bases consisting of only invariant elements.Proof. Since proofs of type B and D are almost the same with the proof of type A,we prove the assertion only in the case of type A.We assume that A = A ℓ − and W = W A . Let θ ′ , . . . , θ ′ s be an S -basisfor D (2) ( A ). Since the degrees do not depend on a choice of a basis, we have { deg θ ′ , . . . , deg θ ′ s } = exp D (2) ( A ℓ − ) by Theorem 4.4. Assume deg θ ′ ≤ · · · ≤ deg θ ′ s . We show that there exists θ ′ j with deg θ ′ j = ℓ − ℓ − ℓ − ℓ − θ ′ j / ∈ D (2) ( S ) W .Suppose that θ ′ j ∈ D (2) ( S ) W for any j . Since C A is a basis for D (2) ( A ) byTheorem 4.4, we may write θ ′ j = X λ f λ θ A λ + ℓ X k =1 a k η A k f λ ∈ S and a k ∈ R . For any w ∈ W , we have w · θ ′ j = X λ ( w · f λ ) θ A λ + w · ℓ X k =1 a k η A k by Lemma 6.2. Since w · θ ′ j = θ ′ j and C A is linearly independent over S , we havethat f λ is W -invariant for λ ∈ Λ. Then ℓ X k =1 a k η A k ∈ D (2) ( S ) W . By Lemma 6.3, the vector space h η A , . . . , η A ℓ i R / R ( η A + · · · + η A ℓ ) is a non-trivial irreducible representation. Thus we have a = · · · = a ℓ (replace it by a = 0 , . . . , a ℓ = 0 in the case of type B and D). This leads that θ ′ , . . . , θ ′ s islinearly dependent over S . 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