Séminaire Lotharingien de Combinatoire 50 (2005), Article B50j

Some new Applications of Orbit Harmonics

by

A.M. Garsia and N.R. Wallach

Abstract

We prove a new result in the Theory of Orbit Harmonics and derive from it a new proof of the Cohen–Macauliness of the ring QI_{m}(G) of m-Quasi-Invariants of a Coxeter Group G. Using the non-degeneracy of the fundamental bilinear form on QI_{m}(G), this approach yields also a direct and simple proof that the quotient of QI_{m}(G) by the ideal generated by the homogeneous G-invariants affords a graded version of the left regular representation of G. Originally all of these results were obtained as a combination of some deep work of Etingof– Ginzburg [3], Feigin–Veselov [6] and Felder–Veselov [5]. The arguments here are quite elementary and self contained, except those using the non-degeneracy of the fundamental bilinear form.

Introduction

Throughout this paper we let G be a finite reflection group of n×n matrices, Σ(G) will denote its class of reflections and for each s ∈ Σ(G) we choose once and for all a vector α_{s} perpendicular to the reflecting hyperplane of s. In this manner the linear form giving the equation of this reflecting hyperplane is given by the scalar product (x, α_{s}). This given, a polynomial P (x) = P (x_{1}, x_{2}, …, x_{n}) is said to be G-m-Quasi-Invariant if and only if for all s ∈ Σ(G) the polynomial (1−s)P (x) is divisible by (x, α_{s})^{2m+1}. It easily shown that G-m-Quasi-Invariants form a finitely generated G-invariant graded subalgebra of the polynomial ring Q[X_{n}], where X_{n} is short for x_{1}, x_{2}, …, x_{n}. Denoting this algebra by QI_{m}[G], we have the proper inclusions

QI_{1}[G] ⊃ QI_{2}[G] ⊃ ⋯ ⊃ QI_{m}[G] ⊃ · · · .

Clearly, QI_{0}[G] = Q[X_{n}], and QI_{∞}[G] may be viewed as the algebra Λ_{G} of G-invariant polynomials. Our goal here is to derive new proofs of some of the basic results on m-Quasi-Invariants by means of the Theory of Orbit Harmonics. More generally, using this theory, we will prove the following basic result. Theorem I.1

Let A be a degree-graded G-invariant subalgebra of Q[X_{n}], and suppose that (i) A ⊇ Λ_{G}, (ii) for some non-trivial homogeneous G-invariant B(x) we have B(x)Q[X_{n}] ⊆

A, (iii) A has a G-invariant, non-degenerate, symmetric bilinear form ⟨ , ⟩_{A} , graded

by degree†, (iv) the orthogonal complement H_{G}(A), with respect to ⟨ , ⟩_{A}, of the ideal J_{G}(A)

generated in A by the homogeneous G-invariants has dimension bounded by the order of G.

Then both H_{G}(A) and A/J_{G}(A) afford the regular representation of G and A is free over Λ_{G}.

To state a further application of this Theorem we need to introduce further notation and make some preliminary observations. To begin with, we should note that within G-m-Quasi-Invariants we find m-analogues of all the ingredients that occur in the relationship between the polynomial ring Q[X_{n}] and the ring of invariants Λ_{G}. For instance, let us recall that the space H_{G} of “G-Harmonics” is defined as the orthogonal complement of the ideal J_{G} generated by the homogeneous Ginvariants. Now, it is well known that, for a Coxeter group G of n × n matrices, Λ_{G} is a free polynomial ring on n homogeneous generators q_{1}(x), q_{2}(x), … q_{n}(x). It follows from this that we have

H_{G} = {P ∈ Q[X_{n}] : q_{k}(∂_{x})P (x) = 0 for all k = 1, 2, …, n},  (I.1)

where for a polynomial P (x) we set P (∂_{x}) = P (∂_{x_{1}}, ∂_{x_{2}}, …, ∂_{x_{n}}). It is also shown in [12] that H_{G} is the linear span of all the partial derivatives of the discriminant

Π_{G}(x) = ∏_{s∈Σ(G)}(x, α_{s}) .  (I.2)

† Homogeneous elements of A of different degrees are orthogonal with respect to ⟨ , ⟩_{A}.

In symbols,

H_{G} = {Q(∂_{x})Π_{G}(x) Q ∈ Q[X_{n}] }.  (I.3)

We have m-analogues of both (I.1) and (I.3). To describe them we need to recall that in [2] Chalykh and Veselov show that to each homogeneous m-Quasi-Invariant Q(x) of degree d there corresponds a unique homogeneous differential operator, acting on QI_{m}[G], of the form

γ_{Q}(x, ∂_{x}) = Q(∂_{x}) + ∑_{|q|<d} c_{q}(x)∂_{x}^{q},  (I.4)

where ∂_{x}^{q} = ∂_{x_{1}}^{q_{1}}∂_{x_{2}}^{q_{2}} ⋯ ∂_{x_{n}}^{q_{n}} and |q| = q_{1} + q_{2} + ⋯ + q_{n}, with c_{q}(x) a rational function in x_{1}, x_{2}, …, x_{n} with a denominator which factors into a product of the linear forms (x, α_{s}). In fact, there is even an explicit formula for γ_{q}(x, ∂_{x}) which is due to Berest [1]. This is

γ_{Q}(x, ∂_{x}) = \frac{1}{2^{d}d!} ∑_{k=0}^{d} \binom{d}{k}L_{m}(G)^{d−k}\underline{Q}L_{m}(G)^{k}  (I.5)

where \underline{Q} denotes the operator “multiplication by Q(x)”,

L_{m}(G) = Δ_{2} − 2m ∑_{s∈Σ(G)} \frac{1}{(x, α_{s})}∂_{α_{s}}  (I.6)

with Δ_{2} the ordinary Laplacian and ∂_{α_{s}} the directional derivative corresponding to α_{s}. In fact, it develops that the linear extension of the map Q → γ_{Q}(x, ∂_{x}) defined by (I.5) yields an algebra isomorphism of QI_{m}[G] onto the algebra of operators of the form (I.4) that commute with L_{m}(G). In particular for all P, Q ∈ QI_{m}[G] we have

γ_{P Q}(x, ∂_{x}) = γ_{P} (x, ∂_{x})γ_{Q}(x, ∂_{x}).  (I.7)

This given, a deep result of Opdam [11] implies that the bilinear form defined by setting, for P, Q ∈ QI_{m}[G]

⟨P , Q⟩_{m} = γ_{P} (s, ∂_{x})Q(x)|_{x=0}  (I.8)

is non-degenerate on QI_{m}[G] × QI_{m}[G]. Now, the space H_{G}(m) of m-Harmonics is simply defined as the orthogonal complement, with respect to ⟨ , ⟩_{m}, of the ideal J_{G}(m) generated in QI_{m}[G] by the homogeneous G invariants. This given, the m-analogue of (I.1) is simply

H_{G}(m) = {P ∈ Q[X_{n}] : γ_{q_{k}}(x, ∂_{x})P (x) = 0 for all k = 1, 2, …, n} (I.9)

It should be mentioned that it follows from this that H_{G}(m) ⊆ QI_{m}[G]. This is an immediate consequence of the remarkable property of the operator L_{m}(G) to the effect that for any two polynomials P, Q we have L_{m}(G)P = Q with Q ∈ QI_{m}[G] if and only if P ∈ QI_{m}[G]. In particular, any polynomial in the kernel of L_{m}(G) is necessarily in QI_{m}[G].

Now, it develops that a beautiful m-analogue of (I.3) was conjectured by Feigin and Veselov in [6] and proved by Etingov and Ginsburg in [3]. In the present notation, this result may be stated as follows. Theorem I.2 (Theorem 6.20 of [3])

H_{G}(m) = {γ_{Q}(x, ∂_{x})Π_{G}^{2m+1}(x) : Q ∈ QI_{m}[G] }.  (I.10)

In fact, if B ⊂ QI_{m}[G] is any basis for the quotient of QI_{m}[G]/J_{G}(m), then the collection

F = {γ_{b}(x, ∂_{x}) Π_{G}^{2m+1}(x) : b ∈ B}  (I.11)

is a basis for H_{G}(m).

We shall show here that, by combining the theory of orbit harmonics with a Hilbert series result of Felder–Veselov, we can also obtain a rather nice new proof of this result.

This paper consists of 6 sections. In the first four sections we establish Theorem I.1 by a sequence of five steps as follows. In the first step we introduce, for a given G-regular vector a = (a_{1}, a_{2}, …, a_{n}), the ring A_{[a]} of the G-orbit of a and show that it affords the regular representation of G. In the second step we introduce its graded version gr A_{[a]} and show that it carries a graded version of the regular representation of G. In the third step, we introduce the space H_{[a]}(A) of orbit A-harmonics and use the non-degeneracy of the form ⟨ , ⟩_{A} to show that H_{[a]}(A) and gr A_{[a]} are equivalent as graded G-modules. In the fourth step we use the dimension bound in (iv) to show that H_{A} and H_{[a]}(A) are one and the same. In the fifth step we use again the non-degeneracy of the form to show that H_{A} and the quotient ring A/I_{G}(A) are equivalent as graded G-modules. In the sixth and final step we combine property (ii) with a simple Hilbert series argument and derive that A is free over Λ_{G}.

In the fifth section we use some basic facts about Coxeter groups and their corresponding m-Quasi-Invariants to show that the hypotheses of Theorem I.1 are satisfied when A = QI_{m}[G] and thereby derive that QI_{m}[G] is a Cohen–Macaulay module over the ring of G-invariants. We believe that the resulting proof is simpler, more elementary and more revealing than the previous proofs.

In the sixth section we prove Theorem I.2. This last section has a substantially different character than the previous ones. It makes crucial use of deep results such as the symmetry of the Hilbert series of m-Harmonics and properties of the Baker–Akhiezer function of QI_{m}[G] whose proofs, to this date, are far from being elementary.

Finally, we should mention that some of the methods of the Theory of Orbit Harmonics we use here were developed and successfully used in [7] in the study of the Garsia–Haiman modules. This may not be an accident since, in a sense, m-Quasi-Invariants may be viewed as the Jack polynomial case of the so-called n!- conjecture. In particular, since m-Quasi-Invariants arose in the study of operators which commute with L_{m}(G), by analogy, there must be spaces arising from a study of operators that commute with the Macdonald operator. We believe that we have only seen here the tip of a mathematical iceberg gravid with combinatorial implications. To dispell any doubts we may have on this score, we urge the reader to view the surprising facts that emerged in [9] in the study of the simplest possible cases. Namely when the underlying reflection group reduces to the symmetric group S_{2} or the dihedral group D_{2}.

1. The orbit Ring In this paper we adopt the convention that an n×n matrix A = ∥a_{ij}∥_{1≤i,j≤n}

acts on a point x = (x_{1}, x_{2}, …, x_{n}) by right multiplication. In this manner the action of A on a polynomial P (x) = P (x_{1}, x_{2}, …, x_{n}) is simply expressed in the form

T_{A}P (x) = P (xA).

We will work with a fixed finite Coxeter group G of n × n matrices and a general algebra A which satisfies the hypotheses of Theorem I.1. The specialization A = QI_{m}[G] will only take place in Section 5. In the first four sections we will make use of the invariant B(x) which satisfies hypothesis in (ii). This given, for our developments it is necessary that we choose once and for all a point a = (a_{1}, a_{2}, …, a_{n}) which satisfies the following two conditions

a) B(a) ≠ 0 ,  and  b) Π_{G}(a) = ∏_{s∈Σ(G)}(a, α_{s}) ≠ 0.  (1.1)

Throughout the paper we denote by [a]_{G} the G-orbit of a. In symbols,

[a]_{G} = {aA : A ∈ G} .  (1.2)

Note that because of (1.1) b) the point a cannot lie in any of the reflecting hyperplanes of G. Since G is a Coxeter group, it follows that the stabilizer of a, namely the subgroup

G_{a} = {A ∈ G : aA = a}

reduces to the identity. This assures that [a]_{G} consists of |G| distinct points. Next we need a polynomial ϕ_{a}(x) such that

ϕ_{a}(x) = \{ \begin{matrix}1 & if x = a, \\ 0 & if x = b with b ∈ [a]_{G} and b ≠ a. \end{matrix}  (1.3)

The construction of this polynomial can be carried out in many ways. For the moment it is immaterial how we pick ϕ_{a}(x). However, to get the best results in Section 5, it will be necessary that ϕ_{a}(x) is constructed to have the smallest possible degree. It turns out that we can never do better than the degree of Π_{G}. Indeed, suppose (1.3) holds true and set

F (x) = ∑_{B∈G} det(B)ϕ_{a}(xB) .

Now note that from (1.3) it follows that F (a) = 1 ≠ 0. Moreover we also have

F (xA^{−1}) = det(A) F (x) (for all A ∈ G)

and this implies that F (x) is a G-invariant multiple of Π_{G}(x). Thus degree(F (x)) ≥ degree(Π_{G}(x)) and this can only happen when degree(ϕ_{a}(x))) ≥ degree(Π_{G}(x)).

To show that this minimum can be achieved, we use the well known fact that if B = {h_{1}, h_{2}, …, h_{|G|}} is a basis for the G-Harmonics then every polynomial P (x) has an expansion of the form

P (x) = ∑_{i=1}^{|G|} h_{i}(x)A_{i}(x)  (with A_{i}(x) ∈ Λ_{G}).  (1.4)

This given, we start by constructing in any manner we please an initial polynomial P_{a}(x) satisfying

P_{a}(x) = \{ \begin{matrix}1 & if x = a, \\ 0 & if x = b with b ∈ [a]_{G} and b ≠ a, \end{matrix}

then use (1.4) and obtain the expansion

P_{a}(x) = ∑_{i=1}^{|G|} h_{i}(x)A_{i,a}(x) (with A_{i,a}(x) ∈ Λ_{G}) .

This done, we claim that we can take

ϕ_{a}(x) = ∑_{i=1}^{|G|} h_{i}(x)A_{i,a}(a) .  (1.5)

In fact, note that the G-invariance of the coefficients A_{i,a}(x) gives

P_{a}(b) − ϕ_{a}(b) = ∑_{i=1}^{|G|} h_{i}(b)(A_{i,a}(b) − A_{i,a}(a)) = 0 for all b ∈ [a]_{G}.

Thus this choice of ϕ_{a}(x) will also satisfy (1.3). Note next that since (1.5) defines ϕ_{a}(x) to be a G-harmonic polynomial, it follows from (I.3) that

degree(ϕ_{a}(x)) ≤ degree(Π_{G}(x)) .

Since we already have the reverse inequality, equality must hold true for this choice of ϕ_{a}.

Now for each b ∈ [a]_{G} with b = aA we set
ϕ_{b}(x) = ϕ_{a}(xA^{−1}) and ϵ(b) = det(A) .

Note that from (1.3) it immediately follows that

ϕ_{b}(x) = \{ \begin{matrix}1 & if x = b, \\ 0 & if x = b′ with b′ ∈ [a]_{G} and b′ ≠ b. \end{matrix}  (1.6)

Next, we let J_{[a]_{G}}(A) denote the ideal generated in A by the elements of A that vanish on [a]_{G}. In symbols,

J_{[a]_{G}}(A) = {P (x) ∈ A : P (b) = 0 for all b ∈ [a]_{G}}.  (1.7)

This given, it follows that Proposition 1.1

The quotient ring

A_{[a]_{G}} = A/J_{[a]_{G}}(A)

has dimension |G| and affords the left regular representation of G. Proof

In view of (1.1), we can set

ψ_{b}(x) = ϕ_{b}(x)\frac{B(x)}{B(a)} .  (1.8)

Note that, since B(x) is G-invariant, we derive from (1.6) that

ψ_{b}(x) = \{ \begin{matrix}1 & if x = b, \\ 0 & if x = b′ with b′ ∈ [a]_{G} and b′ ≠ b. \end{matrix}  (1.9)

Moreover, from (ii) we also derive that

ψ_{b}(x) ∈ A for all b ∈ [a]_{G} .

Now note that, for any P ∈ A, (1.9) gives

P (x) − ∑_{b∈[a]_{G}} P (b) ψ_{b}(x) ∈ J_{[a]_{G}}(A) .  (1.10)

It will be convenient to express this relation by writing

P (x) ≅_{[a]_{G}} ∑_{b∈[a]_{G}} P (b) ψ_{b}(x) .  (1.11)

Thus the collection

{ψ_{b}(x)}_{b∈[a]_{G}}  (1.12)

is a basis for the quotient ring A_{[a]_{G}}.

Since the ideal J_{[a]_{G}}(A) is G-invariant it immediately follows that G acts on A_{[a]_{G}}. Thus to complete our proof we only need to compute the character of this action. To this end, note that, for all σ and β in G, we derive from (1.11) that

T_{σ}ψ_{aβ}(x) = ψ_{a}(xσβ^{−1}) ≅_{[a]_{G}} ∑_{α∈G} ψ_{a}(aασβ^{−1})ψ_{aα}(x)

≅_{[a]_{G}} ∑_{α∈G} χ(ασβ^{−1} = id)ψ_{aα}(x)

≅_{[a]_{G}} ∑_{α∈G} χ(ασ = β)ψ_{aα}(x),

where χ(A) = 1 if A is true and χ(A) = 0 otherwise. Thus the action of G on the basis {ψ_{b}(x)}_{b∈[a]_{G}} is given by the matrix A(σ) = ∥χ(ασ = β)∥_{α,β∈G}. It follows that the character of the G action on A_{[a]_{G}} is given by

trace A(σ) = ∑_{α∈G} χ(ασ = α) = \{ \begin{matrix}|G| & if σ = id, \\ 0 & if σ ≠ id, \end{matrix}

and this is the character of the left regular representation of G, precisely as asserted.

Remark 1.1 The ring A_{[a]_{G}} is not graded but it has a filtration given by the subspaces

H_{≤k}(A_{[a]_{G}}) = L_{[a]_{G}}[P : P ∈ H_{≤k}(A)]  (1.13)

where “L_{[a]_{G}}” denotes “Linear Span” modulo J_{[a]_{G}}(A) and H_{≤k}(A) is the subspace of A spanned by its elements of degree ≤ k. Now note that, since by construction we have

degree ψ_{b}(x) = degree(B(x)) × degree(ϕ_{a}),  (1.14)

it immediately follows from the expansion in (1.11) that

A_{[a]_{G}} = H_{≤d_{A}}(A_{[a]_{G}})  (1.15)

where for convenience we have set

d_{A} = degree(B(x)) × degree(ϕ_{a}).  (1.16)

It will be convenient here and after to adopt the convention that if V is a graded vector space then H_{=k}(V) denotes the subspace spanned by homogeneous elements of degree k. Likewise H_{≤k}(V) denotes the subspace spanned by the homogeneous elements of degree ≤ k.

2. The graded version of A_{[a]_{G}} For each polynomial P we shall here and after denote by h(P) the homo-

geneous component of highest degree in P. This given, we let gr J_{[a]_{G}}(A) be the ideal in A generated by the highest degree components of elements of J_{[a]_{G}}(A). In symbols,

gr J_{[a]_{G}}(A) = (h(P) : P ∈ J_{[a]_{G}}(A) )_{A} .  (2.1)

This brings us to define the “graded version” of A_{[a]_{G}} as the quotient

gr A_{[a]_{G}} = A/gr J_{[a]_{G}}(A) .  (2.2)

Since gr J_{[a]_{G}}(A) is generated by homogeneous polynomials, gr A_{[a]_{G}} is necessarily a graded ring. Note further that if P is any homogeneous polynomial of degree > d_{A} then the fact that

P (x) − ∑_{b∈[a]_{G}} P (b) ψ_{b}(x) ∈ J_{[a]_{G}}(A)

immediately implies that P ∈ gr J_{[a]_{G}}(A). Thus it follows that gr A_{[a]_{G}} has the direct sum decomposition

gr A_{[a]_{G}} = ⨁_{k=0}^{d_{A}} H_{=k}(gr A_{[a]_{G}}).  (2.3)

It will be convenient to choose once and for all, for each 1 ≤ k ≤ d_{A}, a collection B^{=k} ⊂ H_{=k}(A) yielding a basis for the subspace H_{=k}(gr A_{[a]_{G}}). This given, we have the following useful fact. Proposition 2.1

The collection

B^{≤k} = B^{=0} ⊎ B^{=1} ⊎ ⋯ ⊎ B^{=k}

is also a basis for the subspace H_{≤k}(A_{[a]_{G}}). In particular, B^{≤d_{A}} is a basis of A_{[a]_{G}}. Proof

Clearly the result holds true for k = 0 since B_{=0} reduces to the single constant 1. So we may proceed by induction on k. Let us then assume that each P ∈ H_{≤k−1}(A_{[a]_{G}}) has an expansion in terms of B^{≤k−1}. This given, note that any P ∈ H_{≤k}(A), viewed as a representative of an element of gr A_{[a]_{G}}, may be expanded in terms of the basis B^{≤k}. In other words, we will have constants c_{ϕ} such that

P = ∑_{ϕ∈B^{≤k}} c_{ϕ} ϕ + R ,  (2.4)

for a suitable R ∈ gr J_{[a]_{G}}(A). Now the definition of gr J_{[a]_{G}}(A) yields that there must be elements f_{i} ∈ J_{[a]_{G}}(A) and A_{i} ∈ A giving

R = ∑_{i} A_{i} h(f_{i}) .  (2.5)

Since R is necessarily of degree ≤ k we see that there is no loss in assuming that we have

degree A_{i} ≤ k − degree h(f_{i}) .  (2.6)

Using (2.5) we may rewrite (2.4) in the form

P = ∑_{ϕ∈B^{≤k}} c_{ϕ} ϕ + ∑_{i} A_{i} f_{i} − ∑_{i} A_{i} (f_{i} − h(f_{i})) ,  (2.7)

and this implies that

P − ∑_{ϕ∈B^{≤k}} c_{ϕ} ϕ ≅_{[a]_{G}} − ∑_{i} A_{i} (f_{i} − h(f_{i})) .  (2.8)

But now (2.6) and the definition of h(f_{i}) yield that

degree ∑_{i} A_{i} (f_{i} − h(f_{i})) ≤ k − 1 .  (2.9)

Thus our inductive hypothesis yields that we have constants d_{ϕ} giving

∑_{i} A_{i} (f_{i} − h(f_{i})) ≅_{[a]_{G}} ∑_{ϕ∈B^{≤k−1}} d_{ϕ} ϕ ,

and this combined with (2.8) completes the induction. This proves the first assertion. The second assertion follows from the identity in (1.15).

Proposition 2.1 has the following remarkable corollary.

Theorem 2.1

The ring gr A_{[a]_{G}} yields a graded version of the regular representation. In fact, for every 0 ≤ k ≤ d_{A}, we have the character relation

char H_{=k}(gr A_{[a]_{G}}) = char H_{≤k}(A_{[a]_{G}}) − char H_{≤k−1}(A_{[a]_{G}}).  (2.10)

In particular the characters of the subspaces H_{≤k}(A_{[a]_{G}}) are related to the graded character of gr A_{[a]_{G}} by the following identity

∑_{k=0}^{d_{A}} q^{k}char H_{=k}(gr A_{[a]_{G}}) = q^{d_{A}}char A_{[a]_{G}} + (1 − q) ∑_{k=0}^{d_{A}−1} q^{k} char H_{≤k}(A_{[a]_{G}}).

(2.11)

Proof

Let B^{≤k} = {ϕ_{i}}_{i=1}^{m_{k}} so that B^{=k} = {ϕ_{i}}_{m_{k−1}<i≤m_{k}}, and let
A(σ) = ∥a_{i,j}(σ)∥_{i,j=1}^{|G|}

be the matrix expressing the action of G on the basis B^{≤d_{A}} as elements of A_{[a]_{G}}. Since H_{≤k}(A_{[a]_{G}}) is G-invariant it follows that for any j ≤ m_{k} we have the expansion

T_{σ}ϕ_{j} = ∑_{i=1}^{m_{k}} ϕ_{i} a_{i,j}(σ) + R_{σ,j}  (with R_{σ,j} ∈ J_{[a]_{G}}(A) of degree ≤ k). (2.12)

This means that the action of G on the subspace H_{≤k}(A_{[a]_{G}}) induces a representation given by the matrix

A^{≤k}(σ) = ∥a_{i,j}(σ)∥_{1≤i,j≤m_{k}} .  (2.13)

Note further that when m_{k−1} < j ≤ m_{k}, then ϕ_{j} is homogeneous of degree k and equating homogeneous terms of degree k in (2.12) we get

T_{σ}ϕ_{j} = ∑_{m_{k−1}<i≤m_{k}} ϕ_{i} a_{i,j}(σ) + \{ \begin{matrix}h(R_{σ,j}) & if degree R = k, \\ 0 & if degree R_{σ,j} < k, \end{matrix}

(with R_{σ,j} ∈ J_{[a]_{G}}(A)).

In either case we derive that

T_{σ}ϕ_{j} ≅ ∑_{m_{k−1}<i≤m_{k}} ϕ_{i} a_{i,j}(σ)  (modulo gr J_{[a]_{G}}(A)).

This shows that the action of G on the subspace H_{=k}(gr A_{[a]_{G}}) induces a representation given by the matrix

A^{=k}(σ) = ∥a_{i,j}(σ)∥_{m_{k−1}<i,j≤m_{k}} .

Thus

char H_{=k}(gr A_{[a]_{G}}) = trace A^{=k} = trace A^{≤k} − trace A^{≤k−1} .

This proves (2.10). This given, multiplying (2.10) by q^{k} and summing for 0 ≤ k ≤ d_{A}, the identity in (1.15) yields (2.11).

3. The orbit A-Harmonics Recall that by the hypothesis in (iii) of Theorem I.1, the algebra A has a

non-degenerate bilinear form ⟨ , ⟩_{A}. Using this form we define the A-harmonics of the orbit [a]_{G} to constitute the orthogonal complement of the ideal grJ_{[a]_{G}}(A). In symbols

H_{[a]_{G}}(A) = {P ∈ A : ⟨Q , P ⟩_{A} = 0 for all Q ∈ gr J_{[a]_{G}}(A)} .  (3.1)

Note that, since the bilinear form ⟨ , ⟩_{A} is also graded, it immediately follows from (3.1) that all the homogeneous components of any P ∈ H_{[a]}(A) must be also in H_{[a]_{G}}(A). Thus H_{[a]_{G}}(A) is a graded vector space. However we can establish a considerably stronger result. Theorem 3.1

H_{[a]_{G}}(A) is a graded G-module equivalent to gr A_{[a]_{G}}. In particular it also carries the regular representation and thus it has dimension the order of G. Proof

Note that the G-invariance of the bilinear form ⟨ , ⟩_{A} may be also expressed in the form

⟨T_{σ}P , Q⟩_{A} = ⟨P , T_{σ}^{−1}Q⟩_{A}  (for all σ ∈ G),  (3.2)

and, since the ideal grJ_{[a]_{G}}(A) is G-invariant, it follows from (3.2) and the definition in (3.1) that also H_{[a]_{G}}(A) is G-invariant. Thus to prove the assertion we need only compute its character. To this end, for a given 1 ≤ k ≤ d_{A} let {ψ_{1}, ψ_{2}, …, ψ_{s}} ⊆ A be a basis for H_{=k}(gr J_{[a]_{G}}(A)), and let {ϕ_{1}, ϕ_{2}, …, ϕ_{r}} ⊆ A be constructed so that, together with {ψ_{1}, ψ_{2}, …, ψ_{s}} they yield a basis for H_{=k}(A). In particular, {ϕ_{1}, ϕ_{2}, …, ϕ_{r}} ⊆ A are independent modulo gr J_{[a]_{G}}(A), and therefore the elements of gr A_{[a]_{G}} which they represent are a basis for H_{=k}(gr A_{[a]_{G}}).

Now note that the non-degeneracy of the form ⟨ , ⟩_{A} on the subspace H_{=k}(A) assures the non-singularity of the block matrix

M = [ \begin{matrix}∥⟨ϕ_{i}, ϕ_{j}⟩_{A}∥_{1≤i,j≤r} & ∥⟨ϕ_{i}, ψ_{j}⟩_{A}∥\begin{matrix}1≤i≤r \\ 1≤j≤s \end{matrix} \\ ∥⟨ψ_{j}, ϕ_{i}⟩_{A}∥\begin{matrix}1≤i≤r \\ 1≤j≤s \end{matrix} & ∥⟨ψ_{i}, ψ_{j}⟩_{A}∥_{1≤i,j≤s} \end{matrix} ] .  (3.3)

It then follows that the matrix product

⟨ϕ_{1}, ϕ_{2}, …, ϕ_{r}, ψ_{1}, ψ_{2}, …, ψ_{s}⟩M^{−1}

yields a basis for H_{=k}(A) that is “dual” to ⟨ϕ_{1}, ϕ_{2}, …, ϕ_{r}, ψ_{1}, ψ_{2}, …, ψ_{s}⟩ with respect to the bilinear form < , >_{A}. That means that if this new basis is

⟨η_{1}, η_{2}, …, η_{r}, γ_{1}, γ_{2}, …, γ_{s}⟩

then we shall have

a) ⟨ϕ_{i}, η_{j}⟩_{A} = \{ \begin{matrix}1 & if i = j \\ 0 & if i ≠ j \end{matrix} , b) ⟨ψ_{i}, γ_{j}⟩_{A} = \{ \begin{matrix}1 & if i = j \\ 0 & if i ≠ j \end{matrix}  (3.4)

and

a) ⟨ϕ_{i}, γ_{j}⟩_{A} = 0 for all i, j , b) ⟨ψ_{i}, η_{j}⟩_{A} = 0 for all i, j  (3.5)

Note that, since ψ_{1}, ψ_{2}, …, ψ_{s} span H_{=k}(gr J_{[a]_{G}}(A)), the relations in (3.5) b) imply that η_{1}, η_{2}, …, η_{r} lie in H_{=k}(H_{[a]_{G}}(A)). We claim that ⟨η_{1}, η_{2}, …, η_{r}⟩ are in fact a basis of H_{=k}(H_{[a]_{G}}(A)). To show this, note that the duality relations in (3.4) and (3.5) yield that for any polynomial P ∈ H_{=k}(A) we have the expansions

P = ∑_{i=1}^{r} ⟨P, η_{i}⟩_{A}ϕ_{i} + ∑_{j=1}^{s} ⟨P, γ_{j}⟩_{A}ψ_{i} ,  (3.6)

P = ∑_{i=1}^{r} ⟨P, ϕ_{i}⟩_{A}η_{i} + ∑_{j=1}^{s} ⟨P, ψ_{j}⟩_{A}γ_{i} .  (3.7)

In particular, if P ∈ H_{=k}(H_{[a]_{G}}(A)), the latter expansion reduces to

P = ∑_{i=1}^{r} ⟨P, ϕ_{i}⟩_{A}η_{i} .  (3.8)

This shows that ⟨η_{1}, η_{2}, …, η_{r}⟩ span H_{=k}(H_{[a]_{G}}(A)). Since they are independent by construction, it follows that they are a basis as asserted. This given, let A(σ) = ∥a_{i,j}(σ)∥_{i,j=1}^{r} be the matrix that expresses the action of G on ⟨η_{1}, η_{2}, …, η_{r}⟩. That is, for all σ ∈ G we have

T_{σ} η_{i} = ∑_{u=1}^{r} η_{u} a_{u,i}(σ)  for 1 ≤ i ≤ r.  (3.9)

Using the expansion in (3.6) for P = T_{σ}ϕ_{v}, we get

T_{σ}ϕ_{v} = ∑_{i=1}^{r} ⟨T_{σ}ϕ_{v}, η_{i}⟩_{A}ϕ_{i} + ∑_{j=1}^{s} ⟨T_{σ}ϕ_{v}, γ_{j}⟩_{A}ψ_{j}
(by (3.2)) = ∑_{i=1}^{r} ⟨ϕ_{v}, T_{σ}^{−1}η_{i}⟩_{A}ϕ_{i} + ∑_{j=1}^{s} ⟨T_{σ}ϕ_{v}, γ_{j}⟩_{A}ψ_{j}

≅ ∑_{i=1}^{r} ⟨ϕ_{v}, T_{σ}^{−1}η_{i}⟩_{A}ϕ_{i} (modulo gr J_{[a]_{G}}(A)).

(3.10)

But, using (3.9), we derive (from (3.4) a)):

⟨ϕ_{v}, T_{σ}^{−1}η_{i}⟩_{A} = ∑_{u=1}^{r} ⟨ϕ_{v}, η_{u}⟩_{A}a_{u,i}(σ^{−1}) = a_{v,i}(σ^{−1}) ,

and thus (3.10) reduces to

T_{σ}ϕ_{v} ≅ ∑_{i=1}^{r} a_{v,i}(σ^{−1})ϕ_{i} (modulo gr J_{[a]_{G}}(A)).

In other words, the action of G on the basis ⟨ϕ_{1}, ϕ_{2}, …, ϕ_{r}⟩ of H_{=k}(gr A_{[a]_{G}}) is given by the matrix

A^{⊤}(σ^{−1}) .

Since

trace A(σ) = trace A^{⊤}(σ^{−1})

we can thereby conclude that

char H_{=k}(H_{[a]_{G}}(A)) = char H_{=k}(gr A_{[a]_{G}})) .

This proves that H_{[a]_{G}}(A) and gr A_{[a]_{G}} are equivalent as graded G-modules and completes the proof of the theorem.

Remark 3.1

We should note that if V is a finite dimensional vector space with a nondegenerate, symmetric, bilinear form ⟨ , ⟩_{V}, and U ⊂ V is a proper subspace, and if we set

U^{⊥} = {P ∈ V : ⟨P , Q⟩_{V} = 0 for all Q ∈ U } ,

then

U^{⊥}^{⊥} = U .  (3.11)

Note that, since the form ⟨ , ⟩_{A} is graded we can apply this result with V = H_{=k}(A) and U = H_{=k}(grJ_{[a]_{G}}(A)) and deduce the relations

H_{=k}(H_{[a]_{G}}(A)) = (H_{=k}(grJ_{[a]_{G}}(A)))^{⊥}

and

H_{=k}(grJ_{[a]_{G}}(A)) = (H_{=k}(H_{[a]_{G}}(A)))^{⊥} .

Similarly we get

H_{[a]_{G}}(A) = (grJ_{[a]_{G}}(A))^{⊥} and grJ_{[a]_{G}}(A) = (H_{[a]_{G}}(A))^{⊥} .  (3.12)

4. The A-Harmonics of G In the classical case the space H_{G} of “Harmonics” of G is defined as the

orthogonal complement of the ideal generated by the homogeneous G-invariants. For A-harmonics the definition is entirely analogous. We simply let J_{G}(A) be the ideal generated in A by the homogeneous G-invariants and set, (using the notation in Remark 3.1):

H_{G}(A) = (J_{G}(A))^{⊥}  (4.1)

Note that, since the bilinear form ⟨ , ⟩_{A} is graded, H_{G}(A) will necessarily be a graded vector space, as was the case for H_{[a]_{G}}(A) itself. However, the hypothesis in (iv) immediately yields the following remarkable result. Theorem 4.1

The two subspaces H_{[a]_{G}}(A) and H_{G}(A) are identical. In particular, H_{G}(A) carries a graded version of the left regular representation of G. Moreover we also have

J_{G}(A) = grJ_{[a]_{G}}(A)  (4.2)

Proof

Note that if Q is a G-invariant polynomial then the polynomial Q(x) − Q(a) vanishes at all points of the orbit [a]_{G}. That is Q(x) − Q(a) ∈ J_{[a]_{G}}(A). Thus if Q is also homogeneous of positive degree it necessarily follows that

Q(x) ∈ gr J_{[a]_{G}}(A) .

This proves the containment

J_{G}(A) ⊆ gr J_{[a]_{G}}(A) .

Therefore we must also have the reverse containment

(J_{G}(A))^{⊥} ⊇ (gr J_{[a]_{G}}(A))^{⊥} ,

and this can be written as

H_{G}(A) ⊇ H_{[a]_{G}}(A) .  (4.3)

But now Theorem 3.1 and the hypothesis in (iv) give

|G| = dim H_{G}(A) ≥ dim H_{[a]_{G}}(A) = |G| ,

and (4.3) forces the desired equality

H_{G}(A) = H_{[a]_{G}}(A) .

Then it follows that we must also have

H_{G}(A)^{⊥} = H_{[a]_{G}}(A)^{⊥} ,

and the equality in (4.2) follows immediately from Remark 3.1.

We should note that from (4.2) we can derive the following result. Theorem 4.2

If P ∈ A is homogeneous then

degree(P) > d_{A} ⟹ P ∈ J_{G}(A)  (4.4)

Proof

We have seen that, in terms of the basis elements defined in (1.8), every element P ∈ A satisfies the identity in (1.10), that is

P (x) − ∑_{b∈[a]_{G}} P (b) ψ_{b}(x) ∈ J_{[a]_{G}}(A) .  (4.5)

Since all the terms in the sum have degree ≤ d_{A}, if P is homogeneous of degree > d_{A} then (4.5) forces

P (x) ∈ grJ_{[a]_{G}}(A) ,

and the equality in (4.2) proves (4.4).

But the most important consequence of (4.2) is given by the following truly remarkable result.

Theorem 4.3

The quotient ring

A/J_{G}(A)

carries the regular representation of G and therefore has dimension the order of G. Proof

The equality J_{G}(A) = grJ_{[a]_{G}}(A) forces the equality

A/J_{G}(A) = A/grJ_{[a]_{G}}(A) ,

and therefore the assertion follows from Theorem 2.1.

Before we can complete the proof of Theorem I.1 we need to recall some basic facts about Cohen–Macaulay algebras. To begin with, let us recall that the Hilbert series of a finitely generated, graded algebra A is given by the formal sum

F_{A}(t) = ∑_{k≥0} t^{k} dim H_{k}(A) ,  (4.6)

where H_{k}(A) denotes the subspace spanned by the elements of A that are homogeneous of degree k. It is well known that F_{A}(t) is a rational function of the form

F_{A}(t) = \frac{P (t)}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} ,

with d_{1}, d_{2}, …, d_{n} positve integers and P (t) a polynomial that does not vanish at t = 1. The minimum n for which this is possible characterizes the growth of dim H_{k}(A) as k→∞. This integer is customarily called the “Krull dimension” of A and is denoted “dim_{K} A”. It is easily shown that we can always find in A homogeneous elements θ_{1}, θ_{2}, …, θ_{n} such that the quotient of A by the ideal generated by θ_{1}, θ_{2}, …, θ_{n} is a finite dimensional vector space. In symbols

dim A/(θ_{1}, θ_{2}, …, θ_{n})_{A} < ∞  (4.7)

It is also a fact that dim_{K} A is also equal to the minimum n for which this is possible. When (4.7) holds true and n = dim_{K} A then {θ_{1}, θ_{2}, …, θ_{n}} is called a “homogeneous system of parameters”, HSOP in brief.

It follows from (4.7) that if η_{1}, η_{2}, …, η_{N} ∈ A give a basis for the quotient in (4.7) then every element of A has an expansion of the form

P = ∑_{i=1}^{N} η_{i}P_{i}(θ_{1}, θ_{2}, …, θ_{n}) ,  (4.8)

with coefficients P_{i}(θ_{1}, θ_{2}, …, θ_{n}) polynomials in their arguments. The algebra A is said to be Cohen–Macaulay, when the coefficients P_{i}(θ_{1}, θ_{2}, …, θ_{n}) are uniquely determined by P. This amounts to the requirement that the collection

{η_{i} θ_{1}^{p_{1}}θ_{2}^{p_{2}} ⋯ θ_{n}^{p_{n}}}_{i,p}  (4.9)

is a basis for A as a vector space and therefore A is a free module over Q[θ_{1}, θ_{2}, …, θ_{n}]. Note that when this happens and θ_{1}, θ_{2}, …, θ_{n}; η_{1}, η_{2}, …, η_{N} are homogeneous of degrees d_{1}, d_{2}, …, d_{n}; r_{1}, r_{2}, …, r_{N} then we must necessarily have

F_{A}(t) = \frac{∑_{i=1}^{N} t^{r_{i}}}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} ,  (4.10)

from which it follows that dim_{K} A = n.

This brings us to a useful criterion for assuring the Cohen–Macauliness of a finitely generated graded algebra.

Proposition 4.1

Let θ_{1}, θ_{2}, …, θ_{n} be an HSOP, and let d_{i} = degree(θ_{i}), Then A is free over Q[θ_{1}, θ_{2}, …, θ_{n}] and therefore Cohen–Macaulay if and only if

lim_{t→1^{−}}(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})F_{A}(t) = dim A/(θ_{1}, θ_{2}, …, θ_{n})_{A}  (4.11)

Proof†

Note first that the necessity of the condition follows immediately from (4.10). To prove the sufficiency, let η_{1}, η_{2}, …, η_{N} be a homogeneous basis for the quotient A/(θ_{1}, θ_{2}, …, θ_{n})_{A} and set r_{i} = degree(η_{i}). Next let M_{i} denote the subspace spanned by the collection

{η_{j}θ_{1}^{p_{1}}θ_{2}^{p_{2}} ⋯ θ_{n}^{p_{n}} : 1 ≤ j ≤ i; p_{j} ≥ 0} .  (4.12)

† This is a known result but we include a proof for the sake of completeness.

It is easily seen that if H_{m}(M_{i}) and H_{m}(M_{i}/M_{i−1}) denote the subspaces of M_{i} and M_{i}/M_{i−1} spanned by their homogeneous elements of degree m then we must have

dim H_{m}(M_{i}) = dim H_{m}(M_{i}/M_{i−1}) + dim H_{m}(M_{i−1}) .

Multiplying by t^{m} and summing, we derive the Hilbert series identities

F_{M_{i}}(t) = F_{M_{i}/M_{i−1}}(t) + F_{M_{i−1}}(t) (for 1 ≤ i ≤ N with M_{0} = {0}) .

This implies that

F_{A}(t) = F_{M_{1}}(t) + F_{M_{2}/M_{1}}(t) + ⋯ + F_{M_{N} /M_{N−1}}(t) .  (4.13)

Now, for a given 1 ≤ i ≤ N, let ϕ be the map from the polynomial ring Q[x_{1}^{d_{1}}, x_{2}^{d_{2}}, …, x_{n}^{d_{n}}] onto M_{i}/M_{i−1} defined by setting for every polynomial P (x_{1}^{d_{1}}, x_{2}^{d_{2}}, …, x_{n}^{d_{n}})

ϕP = η_{i} P (θ_{1}, θ_{2}, …, θ_{n}) .

Note that, a priori the kernel J of ϕ will be an ideal of Q[x_{1}^{d_{1}}, x_{2}^{d_{2}}, …, x_{n}^{d_{n}}], and since ϕ preserves degrees, we will have

F_{M_{i}/M_{i−1}}(t) = t^{r_{i}}F_{Q[x_{1}^{d_{1}} ,x_{2}^{d_{2}} ,...,x_{n}^{dn}]/J} (t) .  (4.14)

Thus, if J happens to be trivial, it follows that

F_{M_{i}/M_{i−1}}(t) = \frac{t^{r_{i}}}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} .  (4.15)

On the other hand, if J contains a non trivial homogeneous element P of degree d, from (4.14) we derive the coefficient-wise inequality of Hilbert series

F_{M_{i}/M_{i−1}}(t) << t^{r_{i}}F_{Q[x_{1}^{d_{1}} ,x_{2}^{d_{2}} ,...,x_{n}^{dn}]/(P)}(t) ,  (4.16)

where the symbol “<<” is to indicate that the inequality is coefficient-wise. Since the ring Q[x_{1}^{d_{1}}, x_{2}^{d_{2}}, …, x_{n}^{d_{n}}] has no zero divisors, we will have

F_{Q[x_{1}^{d_{1}} ,x_{2}^{d_{2}} ,...,x_{n}^{dn}]/(P)}(t) = \frac{1 − t^{d}}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} ,

and from (4.16) we derive that

F_{M_{i}/M_{i−1}}(t) << t^{r_{i}} \frac{1 − t^{d}}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} .  (4.17)

In conclusion, we see that we will have

lim_{t→1^{−}}(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})F_{M_{i}/M_{i−1}}(t) = ϵ_{i} ,  (4.18)

with

ϵ_{i} = \{ \begin{matrix}1 & if (4.15) holds true, \\ 0 & if (4.17) holds true. \end{matrix}

Thus, passing to the limit as t→1^{−} in (4.13) and using (4.18) together with the hypothesis in (4.11), we finally obtain

ϵ_{1} + ϵ_{2} + ⋯ + ϵ_{N} = dim A/(θ_{1}, θ_{2}, …, θ_{n})_{A} = N .  (4.19)

This forces all the ϵ_{i} to be equal to one. However, this can only hold true when the collection

{η_{j}θ_{1}^{p_{1}}θ_{2}^{p_{2}} ⋯ θ_{n}^{p_{k}}}_{j,p}

forms an independent set. Indeed any relation of the form

∑_{j=1}^{N} η_{j}P_{j}(θ_{1}, θ_{2}, … θ_{n}) = 0 .

would force one of the ϵ_{i} to vanish and contradict (4.19). This shows that

F_{A}(t) = \frac{F_{A/(θ_{1},θ_{2},...,θ_{n})_{A}}(t)}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})}  (4.20)

and proves the Cohen–Macauliness of A.

We are now ready to complete the proof of Theorem I.1.

Theorem 4.4

The algebra A is a free Λ_{G}-module and therefore Cohen–Macaulay. Proof

Let q_{1}(x), q_{2}(x), … q_{n}(x) be a fundamental set of homogeneous generators of Λ_{G} and suppose that d_{1}, d_{2}, …, d_{n} are their respective degrees. It is well known (see [10]) that we must have the equality

d_{1}d_{2} ⋯ d_{n} = |G|.  (4.21)

From the hypothesis (ii) it follows that we have the containments

B(x)Q[X_{n}] ⊂ A ⊂ Q[X_{n}] ,  (4.22)

and since A is degree graded, and B is homogeneous, it follows from (4.6) that the Hilbert series F_{A}(q) of A will satisfy the inequalities

\frac{t^{degree(B)}}{(1 − t)^{n}} << F_{A}(t) << \frac{1}{(1 − t)^{n}} .  (4.23)

In particular, this shows that the Krull dimension of A is n.

This given, multiplying both sides of (4.23) by (1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}}) and passing to the limit as t→1^{−}, the equality in (4.21) implies that

lim_{t→1^{−}}(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})F_{A}(t) = |G| .  (4.24)

Moreover, since q_{1}(x), q_{2}(x), … q_{n}(x) are also generators of the ideal J_{G}(A) we see that Theorem 4.3 implies that

dim A/(q_{1}, q_{2}, … q_{n})_{A} = |G| ,

and Proposition 4.1 then yields that A is Cohen–Macaulay over Λ_{G}. Remark 4.1

We should note that (4.20) yields the Hilbert series identity

F_{A}(t) = \frac{F_{A/(q_{1},q_{2},...q_{n})_{A}}(t)}{(1 − t^{d_{1}})(1 − t^{d_{2}}) ⋯ (1 − t^{d_{n}})} .

In particular it follows that if B is any basis for the quotient

A/(q_{1}, q_{2}, … q_{n})_{A}

then the collection

{b q_{1}^{p_{1}}q_{2}^{p_{2}} ⋯ q_{n}^{p_{n}}}_{b∈B , p_{i}≥0}

is a basis for A.

5. QI_{m}(G) is a Cohen–Macaulay algebra.

In this section we show that QI_{m}(G) satisfies the hypotheses (i)-(iv) of Theorem I.1.

Now, (i) is no problem since we have seen that the definition gives QI_{m}(G) ⊃ QI_{∞}(G) = Λ_{G} for all m ≥ 0. For property (ii), we can take

B(x) = Π_{G}(x)^{2m}.  (5.1)

To see this, we need only observe that for any polynomial P (x) ∈ Q[X_{n}] and any s ∈ Σ(G), the G-invariance of Π_{G}(x)^{2m} gives

(1 − s)Π_{G}(x)^{2m}P (x) = Π_{G}(x)^{2m}(1 − s)P (x).

Clearly, Π_{G}(x)^{2m} yields the factor (x, α_{s})^{2m} and another factor (x, α_{s}) comes from (1 − s)P (x). This proves that

Π_{G}(x)^{2m}Q[X_{n}] ⊂ QI_{m}(G) ,  (5.2)

as desired. For (iii) we need to make sure that the bilinear form

⟨P , Q⟩_{G} = γ_{P} (x, ∂_{x})Q(x) |_{x=0}  (5.3)

mentioned in the introduction has the required properties. To begin with, in the S_{n} case a reasonably elementary proof of non-degeneracy can be found in [8], but in the general case we will have to rely on Opdam’s proof [11]. The G-invariance as expressed in (3.2) immediately follows from the identity

T_{σ}γ_{Q}(x, ∂_{x})T_{σ}^{−1} = γ_{T_{σ}Q}(x, ∂_{x}) ( for all σ ∈ G)

satisfied by all the operators γ_{Q}. The latter in turn follows from the Berest formula (I.5) and the identity

T_{σ}L_{m}(G)T_{σ}^{−1} = L_{m}(G) ( for all σ ∈ G),

which can be easily verified from the definition in (I.6). The symmetry of ⟨ , ⟩_{m} is a consequence of the symmetry of the Baker–Akhiezer function (see Section 6 for more information on that function) of the algebra QI_{m}(G). A reasonably accessible proof of this result can be found in [8]. The fact that ⟨ , ⟩_{m} is graded is an immediate consequence of the fact that for Q homogeneous of degree d the differential operator γ_{Q}(x, ∂_{x}) is also homogeneous of order d. Again this can be easily seen from the Berest formula.

We are thus left with the verification of property (iv). Before we do this, we need to fix some notation and establish a few auxiliary facts that are of interest by themselves. To begin with, we recall that the point a = (a_{1}, a_{2}, …, a_{n}) has to be chosen to have a trivial G-stabilizer. In this case we can satisfy all the conditions we need (including (1.1)) by requiring that

Π_{G}(a) ≠ 0  (5.4)

Here, as in Section 1, ϕ_{b}(x) denotes the polynomial defined by (1.6). But it will be helpful to assume that the polynomial ϕ_{a}(x) satifying (1.3) is chosen so as to have the minimal degree |Σ(G)|. Thus, since degree(Π(x_{G})) = |Σ(G)|, the definition in (1.8) and (5.1) yield that the polynomial ψ_{b}(x) satisfying (1.9) has degree

d_{m}(G) = (2m + 1)|Σ(G)| .  (5.5)

Before we proceed with our arguments, it will be convenient to adopt a notation that is more adherent to the present choice A = QI_{m}(G). To this end, for the rest of this paper we shall assume that (1) The ideal of G-m-Quasi-Invariants that vanish in [a]_{G} is denoted

J_{[a]_{G}}(m).

(2) The quotient of QI_{m}(G) by J_{[a]_{G}}(m) is denoted R_{[a]_{G}}[m].

(3) The graded version of J_{[a]_{G}}(m)] is denoted gr J_{[a]_{G}}(m).

(4) The graded version of R_{[a]_{G}}[m] is denoted gr R_{[a]_{G}}[m]

(5) The orthogonal complement of gr J_{[a]_{G}}(m) with respect to the bilinear form in (5.3) is denoted H_{[a]_{G}}(m) and its elements are called “orbit m-Harmonics”.

(6) The ideal generated in QI_{m}(G) by the G-invariants is denoted J_{G}(m).

(7) The orthogonal complement of J_{G}(m) with respect to the bilinear form in (5.3) is denoted H_{G}(m), and its elements are called “G-m-Harmonics”.

(5.6)

This given, since the only place we have used property (iv) is in the proof of Theorem 4.1, all the results obtained in Sections 1, 2 and 3 hold true with A = QI_{m}(G). Therefore we can state Theorem 5.1

(a) R_{[a]_{G}}[m] is of dimension |G| and affords the regular representation of G.

(b) R_{[a]_{G}}[m] and H_{[a]_{G}}(m) are equivalent G-modules affording a graded version of the regular representation of G.

However, here we have the following two additional results. Theorem 5.2

H_{[a]_{G}}(m) = {Q ∈ QI_{m}(G) : γ_{P} Q = 0 for all P ∈ gr J_{[a]_{G}}(m)} .  (5.7)

Proof

In view of the definition in (5.3), we clearly see that the condition γ_{P} Q = 0 is stronger than ⟨P , Q⟩_{m} = 0. Thus we need only establish the containment

H_{[a]_{G}}(m) ⊆ {P ∈ QI_{m}(G) : γ_{P} Q = 0 for all P ∈ gr J_{[a]_{G}}(m)} .  (5.8)

To this end, note that since gr J_{[a]_{G}}(m) is an ideal of QI_{m}(G), the defining condition

H_{[a]_{G}}(m) = {Q ∈ QI_{m}(G) : ⟨P , Q⟩_{m} = 0 for all P ∈ gr J_{[a]_{G}}(m)}

can be also written in the form

H_{[a]_{G}}(m) = {Q ∈ QI_{m}(G) : ⟨RP , Q⟩_{m} = 0
for all P ∈ gr J_{[a]_{G}}(m) and R ∈ QI_{m}(G) }.

On the other hand from (I.7) and the definition in (5.3) of the bilinear form ⟨ , ⟩_{m} we derive that

⟨RQ , P ⟩_{m} = ⟨R , γ_{Q}P ⟩_{m}

Thus

P ∈ H_{[a]_{G}}(m) implies ⟨R , γ_{Q}P ⟩_{m} = 0
for all Q ∈ gr J_{[a]_{G}}(m) and R ∈ QI_{m}(G) .

But then the non-degeneracy of ⟨ , ⟩_{m} yields that

P ∈ H_{[a]_{G}}(m) ⟹ γ_{Q}P = 0 for all Q ∈ gr J_{[a]_{G}}(m) .

This proves (5.8) and completes the proof of the theorem.

The next result proves the identity in (I.9). Theorem 5.3

H_{G}(m) = {P ∈ Q[X_{n}] : γ_{q_{k}}(x, ∂_{x})P (x) = 0 for all k = 1, 2, …, n} . (5.9)

Proof By definition,

H_{G}(m) = {P ∈ Q[X_{n}] : ⟨Q , P ⟩_{m} = 0 for all Q ∈ J_{G}(m)} ,

and, since the fundamental invariants q_{1}, q_{2}, …, q_{n} generate the ideal J_{G}(m), this is equivalent to

H_{G}(m) = {P ∈ Q[X_{n}] : ⟨Rq_{k} , P ⟩_{m} = 0
for all R ∈ QI_{m}(G) and k = 1, 2, …, n}

(5.10)

Now, from (5.3) and (I.7) we derive that

⟨Rq_{k} , P ⟩_{m} = ⟨R , γ_{q_{k}}P ⟩_{m} .  (5.11)

But the non-degeneracy of ⟨ , ⟩_{m} yields that

⟨R , γ_{q_{k}}P ⟩_{m} = 0 for all R ∈ QI_{m}(G) implies γ_{q_{k}}P = 0.  (5.12)

Combining (5.10), (5.11) and (5.12) proves (5.9).

We are now finally in a position to establish property (iv) for A = QI_{m}(G). Theorem 5.4 We have

dim H_{G}(m) ≤ |G|.  (5.13)

Proof

Notice to the reader. The following argument was provided to us by E-mail by Feigin and Veselov. We feel compelled to reproduce it here since it is not available in the present literature. Although this result is stated in many places (see [3], [4], [6]), proofs (when not omitted) give no indication that it could be established in such a simple and elementary manner.

The idea is to show that each m-Harmonic Q ∈ H_{G}(m) is completely determined by |G| of its derivatives at the point a. To do this, we fix once and for all a fundamental set q_{1}, q_{2}, …, q_{n} of G-invariants and a monomial basis {x^{ϵ_{1}}, x^{ϵ_{2}}, …, x^{ϵ_{N}} } for the quotient Q[X_{n}]/(q_{1}, q_{2}, …, q_{n})_{Q[X_{n}]}. Since it is well known that dim Q[X_{n}]/(q_{1}, q_{2}, …, q_{n})_{Q[X_{n}]} = |G|, we must necessarily have

N = |G|.

This given, we need only show that every Q(x) ∈ H_{G}(m) is completely determined by the |G| derivatives

∂_{x}^{ϵ_{1}}Q(x) |_{x=a} , ∂_{x}^{ϵ_{2}}Q(x)|_{x=a} , …, ∂_{x}^{ϵ_{N}} Q(x) |_{x=a} .

That is we must show that

∂_{x}^{ϵ_{1}}Q(x) |_{x=a} = 0 , ∂_{x}^{ϵ_{2}}Q(x)|_{x=a} = 0 , …, ∂_{x}^{ϵ_{N}} Q(x) |_{x=a} = 0  (5.14)

forces

Q ≡ 0.  (5.15)

To do this, the point of departure is the well known fact that every polynomial P ∈ Q[X_{n}] may be given an expansion of the form

P = ∑_{r=1}^{|G|} x^{ϵ_{r}}A_{P,r}(q_{1}, q_{2}, …, q_{n}) ,  (5.16)

with the coefficients A_{P,r} polynomials in their arguments. Specializing P to the monomial x^{p} (5.16) gives

x^{p} = ∑_{r=1}^{|G|} x^{ϵ_{r}} a_{p,r}(x)  (5.17)

with

a_{p,r}(x) = A_{x^{p},r}(q_{1}, q_{2}, …, q_{n})  (5.18)

Two important facts should be kept in mind here:

(1) We can assume that

degree(a_{p,r}(x)) = |p| − |ϵ_{r}| .  (5.19)

(2) From (5.18), (5.9) and (I.7) it follows that

γ_{a_{p,r}}(x, ∂_{x})Q(x) = A_{x^{p},r}(0, 0, …, 0) Q(x) for all Q ∈ H_{G}(m) .

(5.20)

Note first that, since it is well known that we must have

∑_{r=1}^{N} q^{ϵ_{r}} = ∏_{i=1}^{n}(1 + q + ⋯ + q^{d_{i}−1})

with d_{i} = degree(q_{i}), it follows that one of the exponents ϵ_{r} vanishes. Thus one of the conditions in (5.14) reduces to

Q(a) = 0 .  (5.21)

Our next task is to show that the remaining conditions in (5.14) force

∂_{x}^{p}Q(x) |_{x=a} = 0  (5.22)

for all

p = (p_{1}, p_{2}, …, p_{n}) .

In view of (5.21), we can proceed by induction on |p| = p_{1} + p_{2} + ⋯ + p_{n}. We assume (5.22) to be true for |p| < d and show that it holds true for |p| = d. To this end we use (5.17) and write for |p| = d,

∂_{x}^{p}Q(x) = ∑_{|ϵ_{r}|=d} a_{p,r} ∂_{x}^{ϵ_{r}}Q(x) + ∑_{|ϵ_{r}|<d} ∂_{x}^{ϵ_{r}}a_{p,r}(∂_{x})Q(x)

Note that (5.19) says that a_{p,r} in the first sum reduces to a scalar and in the second sum it must be a homogeneous polynomial of degree d − |ϵ_{r}| > 0. In particular the conditions in (5.14) immediately give us that

∂_{x}^{p}Q(x) |_{x=a} = ∑_{|ϵ_{r}|<d} ∂_{x}^{ϵ_{r}}a_{p,r}(∂_{x})Q(x)|_{x=a} .  (5.23)

Now, from (I.4) and the fact that degree(a_{p,r}) = d − |ϵ_{r}|, we obtain an expansion of the form

γ_{a_{p,r}}(x, ∂_{x}) = a_{p,r}(∂_{x}) + ∑_{|q|<d−|ϵ_{r}|} c_{q,p,r}(x)∂_{x}^{q} .

Using this in (5.23), we derive that

\begin{matrix}∂_{x}^{p}Q(x) |_{x=a} = ∑_{|ϵ_{r}|<d} ∂_{x}^{ϵ_{r}}γ_{a_{p,r}}(x, ∂_{x})Q(x)|_{x=a} \\ − ∑_{|ϵ_{r}|<d} ∑_{|q|<d−|ϵ_{r}|} ∂_{x}^{ϵ_{r}}c_{q,p,r}(x)∂_{x}^{q}Q(x)|_{x=a} . \end{matrix}

But (5.20) and the inductive hypothesis reduces this to

∂_{x}^{p}Q(x) |_{x=a} = − ∑_{|ϵ_{r}|<d} ∑_{|q|<d−|ϵ_{r}|} ∂_{x}^{ϵ_{r}}c_{q,p,r}(x)∂_{x}^{q}Q(x)|_{x=a} .  (5.24)

Note that this makes perfectly good sense since our assumption in 5.4 assures that the denominators that will be produced by the term

∂_{x}^{ϵ_{r}}c_{q,p,r}(x)∂_{x}^{q}Q(x)

will not vanish at x = a. However, (5.24) completes the induction since the derivatives of Q that will be produced by these terms will necessarily be of order < d and by the inductive hypothesis they will all vanish at x = a..

This completes the proof of the dimension bound in (5.13).

Theorem 5.3 has a number of immediate corollaries that are worth stating explicitly.

Theorem 5.5 The following remarkable equalities hold true for every integer m ≥ 0:

H_{[a]_{G}}(m) = H_{G}(m) and J_{G}(m) = gr J_{[a]_{G}}(m) .  (5.25)

Thus the m-Harmonics H_{G}(m) and the quotient ring

QI_{m}(G)/(q_{1}, q_{2}, …, q_{n})_{QI_{m}(G)}  (5.26)

have both dimension |G| and afford the same graded regular representation of |G| as the the space of orbit harmonics H_{[a]_{G}}(m).

Theorem 5.6

For every integer m ≥ 0 the algebra of G-m-Quasi-Invariants is a free module over the ring of invariants Λ_{G}. Theorem 5.7

Every homogeneous G-m-Quasi-Invariant Q ∈ QI_{m}(G) of degree greater than (2m + 1)|Σ(G)| lies in the ideal J_{G}(m).

Since we have verified that the algebra A = QI_{m}(G) satisfies conditions (i), (ii), (iii), (iv) of Theorem I.1, all of these results are simply specializations to A = QI_{m}(G) of the corresponding results established in Section 4.

6. More on the G-m-Harmonics The goal of this section is to establish Theorem I.2. The basic tool in this

task is a space Γ_{[a]_{G}}(m) of formal power series in x_{1}, x_{2}, …, x_{n} which may be viewed as the orthogonal complement of the ideal J_{[a]_{G}}(m). More precisely we set

Γ_{[a]_{G}}(m) = {Φ(x) : γ_{Q}(x, ∂_{x})Φ(x) = 0 for all Q ∈ J_{[a]_{G}}(m)}  (6.1)

Perhaps a few words are necessary here to assure that this is a well defined space. To begin with, we shall view each formal power series Φ(x) as the formal sum

Φ = Φ^{(0)} + Φ^{(1)} + Φ^{(2)} + ⋯ + Φ^{(k)} + ⋯

with Φ^{(k)} a polynomial in x_{1}, x_{2}, …, x_{n} homogeneous of degree k. Moreover if Q ∈ J_{[a]_{G}}(m) has the decomposition

Q = Q_{0} + Q_{1} + ⋯ + Q_{h}

with Q_{r} homogeneous of degree r, then the equation

γ_{Q}(x, ∂_{x})Φ(x) = 0  (6.2)

simply means that we must have

∑_{r=0}^{h} γ_{Q_{r}}(x, ∂_{x})Φ^{(r+k)}(x) = 0  ( for all k ≥ 0).  (6.3)

Thus no infinite sums are involved in checking containment in Γ_{[a]_{G}}(m).

To deal with this space, we need an important ingredient which has been in the background up to this moment but which nevertheless is the most significant tool in the Theory of m-Quasi-Invariants. This is a formal power series Ψ_{m}(x, y) referred to as the “Baker–Akhiezer function” of QI_{m}(G). To use Ψ_{m}(x, y), we shall have to state a number of facts whose original proofs are in a series of papers scattered over several years. However, a reasonably self contained account of this material with detailed proofs of everything we use here (except for the non-degeneracy of the bilinear form) can be found in [8]. This is a monograph we have put together for the benefit of future researchers in this area.

To begin with, it will be good to see how Ψ_{m}(x, y) is defined. Indeed, although this definition will play no role here, the novice in this area has great difficulty locating it in the literature. Remarkably, Ψ_{m}(x, y) may be given an explicit (though quite forbidding) construction based on a truly remarkable family SD_{n} of “Shift-Differential” operators that act on polynomials by a combination of the ordinary G-action followed by differentiation. These operators are of the form

A = ∑_{σ∈G} a_{σ}(x, ∂_{x}) T_{σ} ,  (6.4)

where

a_{σ}(x, ∂_{x}) = ∑_{p} a_{p}(x)∂_{x}^{p}  (∂_{x}^{p} = ∂_{x_{1}}^{p_{1}}∂_{x_{2}}^{p_{2}} ⋯ ∂_{x_{n}}^{p_{n}}),  (6.5)

with each a_{p}(x) a special rational function in the ring SR_{n}(x) generated by the variables x_{i} together with the fractions 1/(x, α_{s}) for s ∈ Σ(G). It it is easily shown that SD_{n} is in fact an algebra since the algebra of differental operators of the form given in (6.5) is invariant under conjugation by elements of G. The building blocks of SD_{n} are the m-Dunkl operators, which can be written in the form

∇_{i}(m) = ∂_{x_{i}} − m ∑_{s∈Σ(G)}(α_{s}, e_{i})\frac{1}{(x, α_{s})} (1 − s)  (for i = 1, 2, …, n), (6.6)

with e_{i} the i^{th} coordinate unit vector. For fixed m, the operators {∇_{i}(m)}_{i=1}^{n} are a commuting set, and thus it makes perfectly good sense to evaluate a polynomial at ∇_{1}(m), ∇_{2}(m), …, ∇_{n}(m). We adopt the notation

Q[∇(m)] = Q(∇_{1}(m), ∇_{2}(m), …, ∇_{n}(m)) (for all Q ∈ Q[X_{n}]).

Clearly, all these operators belong to the family SD_{n}. But once they are written in the form given in (6.4), we can simply “forget” the G action and set

ΓA = ∑_{σ∈G} a_{σ}(x, ∂_{x}) .

It develops that this seemingly innocent operation can achieve miracles. For instance one obtains that

Γ ∑_{i=1}^{n} ∇_{i}(m)^{2} = L_{m}(G).  (6.7)

More generally, it can be shown that for each G-invariant Q we have the beautiful identity

γ_{Q}(x, ∂_{x}) = ΓQ[∇(m)].  (6.8)

This given, let us set

O_{m} = Γ Π_{G}(∇(m))Π_{G}(\underline{x}) ,

where “Π_{G}(\underline{x})” denotes the operator multiplication by Π_{G}(x). Remarkably, it can be shown that we have the Opdam commutation relation

L_{m}(G) O_{m} = O_{m}L_{m−1}(G) .  (6.9)

This given, the Baker–Akhiezer function Ψ_{m}(x, y) is simply defined by setting

Ψ_{m}(x, y) = O_{m}O_{m−1} ⋯ O_{1}e^{(x,y)} .  (6.10)

Since the definition in (I.6) gives that L_{0}(G) = Δ_{2}, we see that it follows from (6.9) that

L_{m}(G)Ψ_{m}(x, y) = (y, y)Ψ_{m}(x, y) .  (6.11)

The properties of Ψ_{m}(x, y) we will use here may be stated as follows:

(a) Symmetry: Ψ_{m}(x, y) = Ψ_{m}(y, x).

(b) γ_{Q}(x, ∂_{x})Ψ_{m}(x, y) = Q(y)Ψ_{m}(x, y) for all Q ∈ QI_{m}(G). (6.12)

(c) For all σ ∈ G we have Ψ_{m}(xσ, y) = Ψ_{m}(x, yσ^{−1}).

(d) We have the decomposition Ψ_{m}(x, y) = c_{m}(G) + ∑_{k≥1} Ψ_{m}^{(k)}(x, y) with (i) Ψ_{m}(0, 0) = Ψ_{m}(0, y) = Ψ_{m}(x, 0) = c_{m}(G) ≠ 0, (ii) Ψ_{m}^{(k)}(x, y) a polynomial homogeneous of degree k in the x′s and y′s

separately, (iii) Ψ_{m}^{(k)}(x, y) in QI_{m}(G) in x_{1}, x_{2}, …, x_{n} and y_{1}, y_{2}, …, y_{n}.

We should note that (6.12) (d) (ii) immediately follows from the definition in (6.10) and the fact that the operator Ω_{m} = O_{m}O_{m−1} ⋯ O_{1} does not change degrees. In fact, from the expansion of the exponential e^{(x,y)} we derive that

Ψ_{m}^{(k)}(x, y) = ∑_{|p|=k} \frac{y^{p}}{p!} Ω_{m}x^{p} .

Moreover, property (6.12) (d) (iii) is an immediate consequence of the following remarkable fact which will also be needed in the sequel.

We have the relation L_{m}(G)P = Q with Q ∈ QI_{m}(G) if and only if P ∈ QI_{m}(G). (6.13)

We now have all the tools we need to proceed with our developments. We begin with some basic properties of Γ_{[a]_{G}}(m). Proposition 6.1

(1) All the homogeneous components of every Φ ∈ Γ_{[a]_{G}}(m) are in QI_{m}(G).

(2) The collection {Ψ_{m}(x, b)}_{b∈[a]_{G}} is a basis for Γ_{[a]_{G}}(m).

(3) Γ_{[a]_{G}}(m) is a G-module affording the regular representation of G.

(4) Every element Φ ∈ Γ_{[a]_{G}} may be written in the form

a) Φ(x) = γ_{Q}Θ_{[a]_{G}}(x) (for some Q ∈ QI_{m}(G)),

where

b) Θ_{[a]_{G}}(x) = ∑_{b∈[a]_{G}} ϵ_{b}Ψ_{m}(x, b) = cΠ(x)^{2m+1} + · · · .

Proof

We have seen in (6.7) that L_{m}(G) = γ_{p_{2}}(x, ∂_{x}) with p_{2}(x) = (x, x). Since (x, x) is G-invariant, it follows that the difference (x, x)−(a, a) belongs to Γ_{[a]_{G}}(m). Thus, from the definition in (6.1) it follows that

L_{m}(G)Φ = (a, a)Φ  ( for all Φ ∈ Γ_{[a]_{G}}(m) ).  (6.14)

But if Φ^{(i)} is the homogeneous component of degree i in Φ, then (6.14) yields that

L_{m}(G)Φ^{(0)} = 0 , L_{m}(G)Φ^{(1)} = 0 and L_{m}(G)Φ^{(i)} = (a, a)Φ^{(i−2)} for all i ≥ 2 .

Thus property (1) follows from (6.13). To show (2) we note first that property (b) of the Baker–Akhiezer function implies that for all Q ∈ J_{[a]_{G}}(m) we have γ_{Q}(x, ∂_{x})Ψ_{m}(x, b) = 0 for all b ∈ [a]_{G}. Thus

{Ψ_{m}(x, b)}_{b∈[a]_{G}} ⊂ Γ_{[a]_{G}}(m) .

To show that this collection spans Γ_{[a]_{G}}(m), we will use the polynomials {ψ_{b}(x)}_{[a]_{G}} defined in (1.8) with the specialization A = QI_{m}(G) and B(x) = Π_{G}(x)^{2m}. Since (1.9) immediately gives the identity

1 ≡ ∑_{b∈[a]_{G}} ψ_{b}(x) (mod J_{[a]_{G}}(m) )

we derive from the definition in (6.1) that, for all Φ ∈ Γ_{[a]_{G}}(m) we have the decomposition

Φ(x) = ∑_{b∈[a]_{G}} Φ_{b}(x) ,  (6.15)

with

Φ_{b}(x) = γ_{ψ_{b}}(x, ∂_{x})Φ(x) .  (6.16)

We claim that the latter is none other than a scalar multiple of Ψ_{m}(x, b). To prove this, note first that, for all Q ∈ QI_{m}(G) we have the relation

Q(x)ψ_{b}(x) − Q(b)ψ_{b}(x) ≡ 0  (mod J_{[a]_{G}}(m) )  (6.17)

Thus

γ_{Q}(x, ∂_{x})Φ_{b}(x) = γ_{Q}(x, ∂_{x})γ_{ψ_{b}}(x, ∂_{x})Φ(x)
(by (I.7)) = γ_{Qψ_{b}}(x, ∂_{x})Φ(x)
(by (6.17)) = Q(b)γ_{ψ_{b}}(x, ∂_{x})Φ(x) = Q(b)Φ_{b}(x).

(6.18)

Using this relation with Q(x) = Ψ_{m}^{(k)}(x, y), we get

γ_{Ψ_{m}^{(k)}}Φ_{b} |_{x=0} = Ψ_{m}^{(k)}(b, y)Φ_{b}(0).  (6.19)

On the other hand, since γ_{Ψ_{m}^{(k)}} decreases degrees by k, denoting by Φ_{b}^{(k)} the k^{th} homogeneous component of Φ_{b}, it follows that

γ_{Ψ_{m}^{(k)}}Φ_{b}(x) |_{x=0} = ⟨Ψ_{m}^{(k)} , Φ_{b}^{(k)}⟩_{m}
= ⟨Φ_{b}^{(k)} , Ψ_{m}^{(k)}⟩_{m}
= γ_{Φ_{b}^{(k)}}Ψ_{m}^{(k)}(x, y) |_{x=0}
= γ_{Φ_{b}^{(k)}}Ψ_{m}(x, y) |_{x=0}
(by (b) of (6.12) ) = Φ_{b}^{(k)}(y)Ψ_{m}(x, y) |_{x=0}
(by (d) (i) of (6.12)) = Φ_{b}^{(k)}(y)c_{m}(G)

(6.20)

Combining (6.19) and (6.20) we get that

Ψ_{m}^{(k)}(b, y)Φ_{b}(0) = Φ_{b}^{(k)}(y)c_{m}(G) .

This holding true for all k yields that

Ψ_{m}(b, y)Φ_{b}(0) = Φ_{b}(y)c_{m}(G) .

Solving for Φ_{b}(y) and using the symmetry of Ψ_{m}(x, y), we now obtain

Φ_{b}(y) = \frac{Φ_{b}(0)}{c_{m}(G)} Ψ_{m}(y, b) ,  (6.21)

as desired. Combining (6.15) with (6.21) proves that {Ψ_{m}(x, b)}_{b∈[a]_{G}} spans Γ_{[a]_{G}}(m).

To complete the proof of (2), we need to show independence. To this end, suppose that for some constants c_{b} we have

Φ = ∑_{b∈[a]_{G}} c_{b}Ψ_{m}(x, b) .  (6.22)

Then the relations in (1.9) and (b) of (6.12) give, for b′ ∈ [a]_{G},

⟨ψ_{b′} , Φ⟩_{m} = γ_{ψ_{b′}} Φ|_{x=0} = ∑_{b∈[a]_{G}} c_{b} ψ_{b′}(b)Ψ_{m}(x, b)|_{x=0} = c_{b′} c_{m}(G) . (6.23)

Thus

Φ = 0 ⟹ c_{b} = 0 for all b ∈ [a]_{G} ,

proving independence. Incidentally, (6.23) yields that the expansion in (6.22) may be written in the form

Φ = ∑_{b∈[a]_{G}} ^{\frac{⟨ψ_{b} , Φ⟩_{m}}{c_{m}(G)}} Ψ_{m}(x, b) .

Finally, note that property (c) of Ψ_{m}(x, y) gives that

T_{σ}Ψ_{m}(x, b) = Ψ_{m}(x, bσ^{−1}) ( for all σ ∈ G).

Thus the character χ of the action of G on the basis {Ψ_{m}(x, b)}_{b∈[a]_{G}} has the expansion

χ(σ) = ∑_{b∈[a]_{G}} Ψ_{m}(x, bσ^{−1})|_{Ψ_{m}(x,b)} = ∑_{b∈[a]_{G}} χ(bσ^{−1} = b) = \{ \begin{matrix}|G| & if σ = id, \\ 0 & if σ ≠ id. \end{matrix}

This gives (3).

Finally, note that if

Φ(x) = ∑_{be∈[a]_{G}} c_{b}Ψ_{m}(x, b)  (6.24)

and we set

Q = ∑_{b′∈[a]_{G}} \frac{c_{b′}}{ϵ_{b′}} ψ_{b′}(x)

then clearly Q ∈ QI_{m}(G), and we also have from (6.24)

γ_{Q}Θ_{[a]_{G}} = ∑_{b∈[a]_{G}} ϵ_{b} ∑_{b′∈[a]_{G}} \frac{c_{b}}{ϵ_{b′}} γ_{ψ_{b′}} Ψ_{m}(x, b)
(by (6.12) (b)) = ∑_{b∈[a]_{G}} ϵ_{b} ∑_{b′∈[a]_{G}} \frac{c_{b}}{ϵ_{b′}} ψ_{b′}(b)Ψ_{m}(x, b)
(by (1.9)) = ∑_{b∈[a]_{G}} ϵ_{b} \frac{c_{b}}{ϵ_{b}} Ψ_{m}(x, b) = Φ(x).

This proves (4) (a).

To prove (4) (b), note that from the definition of Θ_{[a]_{G}}(x) it follows that for any σ ∈ G we have

T_{σ}Θ_{[a]_{G}} = det(σ) Θ_{[a]_{G}}.

This forces all the homogeneous components of Θ_{[a]_{G}} to be G-invariant multiples of Π_{G}(x)^{2m+1}. Thus we can write

Θ_{[a]_{G}}(x) = A(x)Π_{G}(x)^{2m+1} + ⋯ (for some A(x) ∈ Λ_{G}).

Now, from the definition of Θ_{[a]_{G}} and (6.12) (b) we derive that

γ_{Π^{2m+1}}Θ_{[a]_{G}}(x) = ∑_{b∈[a]_{Sn}} ϵ(b)Π_{G}(b)^{2m+1}Ψ_{m}(x, b) = Π_{G}(a)^{2m+1} ∑_{b∈[a]_{Sn}} Ψ_{m}(x, b) .

(6.25)

Since γ_{Π^{2m+1}} decreases degrees by (2m + 1)|Σ(G)|, we have

degree(γ_{Π^{2m+1}}A(x)Π_{G}(x)^{2m+1}) = degree(A(x)) .

Thus,

degree (μ(γ_{Π^{2m+1}}Θ_{[a]_{G}}(x))) ≥ degree(A(x)) ,  (6.26)

where, for each formal power series Φ, we let μ(Φ) denote the homogeneous component of least degree in Φ. However, the right hand side of (6.25) (using (6.12) (d) (i)) yields that

μ(γ_{Π^{2m+1}}Θ_{[a]_{G}}(x)) = Π_{G}(a)^{2m+1}|G|c_{m}(G) ≠ 0 .  (6.27)

Combining (6.26) with (6.27) forces A(x) to be a scalar.

Note that this argument also shows that for some constant C we must have

γ_{Π^{2m+1}}Π(x)^{2m+1} = C ≠ 0  (6.28)

To introduce our next construct, we need further notation. To begin with, we shall choose once and for all a degree lexicographic order of monomials. For instance, the one which corresponds to the total order x_{1} > x_{2} > … > x_{n}. It will be convenient to denote this order by “<_{dl}”. Recalling that μ(Φ) denotes the homogeneous component of least degree in Φ, we will let l(Φ) denote the d-lex least monomial in μ(Φ). We also set l(f) = l(Φ) when f = μ(Φ). Remark 6.1

It will be convenient in our further developments to use the symbol “Φ |_{=k}” to denote the the homogeneous components of Φ of degree k. In the same vein, we let “Φ|_{<k}” and “Φ|_{≤k}” the sum of all the homogeneous components of Φ of degree < k and ≤ k respectively. In our previous notation

Φ|_{=k} = Φ^{(k)} , Φ|_{<k} = ∑_{i=0}^{k−1} Φ^{(i)} and Φ|_{≤k} = ∑_{i=0}^{k} Φ^{(i)} .  (6.29)

Thus we have

μ(Φ) = Φ|_{=k} if and only if Φ|_{<k} = 0 and Φ|_{=k} ≠ 0 .

Thus we clearly see that the map Φ ↦ μ(Φ) is not linear. Nevertheless, if μ(Φ_{1}) and μ(Φ_{2}) have the same degree then

μ(Φ_{1}) + μ(Φ_{2}) ≠ 0 implies μ(Φ_{1} + Φ_{2}) = μ(Φ_{1}) + μ(Φ_{2}).  (6.30)

Keeping this in mind, we let μ(Γ_{[a]_{G}}(m)) denote the linear span of the homogeneous components of least degree of elements of Γ_{[a]_{G}}(m). In symbols

μ(Γ_{[a]_{G}}(m)) = L[ μ(Φ) : Φ ∈ Γ_{[a]_{G}}(m) ] .  (6.31)

The following result shows the intimate relation between Γ_{[a]_{G}}(m) and μ(Γ_{[a]_{G}}(m)). Proposition 6.2

We can find in Γ_{[a]_{G}}(m) a collection of formal power series

Φ_{1} , Φ_{2} , …, Φ_{|G|}  (6.32)

with the property that

l(Φ_{1}) <_{dl} l(Φ_{2}) <_{dl} ⋯ <_{dl} l(Φ_{|G|})  (6.33)

and such that each f ∈ μ(Γ_{[a]_{G}}(m)) has a unique expansion of the form

f = ∑_{l(Φ_{i}) ≥_{dl} l(f)} c_{i} μ(Φ_{i}) .  (6.34)

In particular, the collection

μ(Φ_{1}) , μ(Φ_{2}) , …, μ(Φ_{|G|})  (6.35)

is a basis of μ(Γ_{[a]_{G}}(m)) and we have

dim μ(Γ_{[a]_{G}}(m)) = |G| .  (6.36)

Proof

The collection in (6.32) satisfying (6.33) can be constructed by starting with the basis

{Ψ_{μ}(x, b)}_{b∈[a]_{G}} ,

then progressively reducing it to echelon form with respect to the degree-lexico-graphic order of least monomials. This done, (6.33) yields that their minimum components in (6.35) are linearly independent. In fact, more than that is true. Note first that no element Φ ∈ Γ_{[a]_{G}}(m) can have a least monomial that is different from each of the monomials in (6.33). Indeed, such an element would necessarily be independent of the Φ_{i}’s, and then Γ_{[a]_{G}}(m) would have dimension greater than |G|. The existence of constants c_{i} giving (6.34) may be established by descent induction on the degree-lexicographic order of least monomials. To see this let

f = μ(Φ).

Now the result is immediate if l(f) = l(Φ_{|G|}). In fact, let c be chosen so that l(Φ_{|G|}) does not occur in f − c μ(Φ_{|G|}). Then the difference Φ − cΦ_{|G|} must be identically zero, for otherwise its degree-lexicographically least monomial would necessarily be larger than l(Φ_{|G|}) and this, as we have seen, is not possible. This gives

f = c μ(Φ_{|G|}),

and we are done in this case. So assume by induction that the expansion in (6.34) exists when l(f) >_{dl} l(Φ_{i_{o}}). Let l(f) = l(Φ_{i_{o}}). Again chose c so that l(Φ_{i_{o}}) does not occur in f − cΦ_{i_{o}}. Here there are two cases. If f = c μ(Φ_{i_{o}}) we are done. If, on the other hand, this difference does not vanish identically, then, since l(f) = l(Φ_{i_{o}}) implies that degree(μ(Φ)) = degree(μ(Φ_{i_{o}})), we can use (6.30) with Φ_{1} = Φ and Φ_{2} = −c Φ_{i_{o}} and conclude that

f = μ(Φ) = c μ(Φ_{i_{o}}) + μ(Φ − c Φ_{i_{o}}) .

Since now l(Φ − c Φ_{i_{o}}) >_{dl} l(Φ_{i_{o}}), we can use the induction hypothesis and deduce that, for some suitable constants c_{i}, we must have

f = c μ(Φ_{i_{o}}) + ∑_{i=i_{o}+1}^{|G|} c_{i} μ(Φ_{i}) .

This completes the induction and establishes (4.34). The remaining assertions are immediate consequences of (6.34).

The following remarkable fact provides us with a new tool for studying m-Harmonics.

Theorem 6.1 For all finite reflection groups G and all m ≥ 0 we have

μ(Γ_{[a]_{G}}(m)) = H_{G}(m).  (6.37)

Proof

In view of the fact that from Theorem 5.5 and Proposition 6.2 it follows that these two spaces have the same dimension, to show the equality in (6.37) it is sufficient to derive the containment

μ(Γ_{[a]_{G}}(m)) ⊆ H_{G}(m) .  (6.38)

This in turn immediately follows from Theorem 5.3 once we verify that

P ∈ μ(Γ_{[a]_{G}}(m)) implies γ_{q_{k}}P (x) = 0 (for k = 1, 2, …, n) .  (6.39)

This given, let Q(x) be a homogeneous G-invariant and note that since the difference Q(x) − Q(a) vanishes throughout [a]_{G}, it follows from the definition in (6.1) that we have

γ_{Q}(x, ∂_{x})Φ(x) = Q(a)Φ(x) ( for all Φ ∈ Γ_{[a]_{G}}(m) ).  (6.40)

Thus, if Q is of degree d, equating homogeneous components of both sides of (6.40), we derive that

γ_{Q}(x, ∂_{x})Φ^{(k)}(x) = \{ \begin{matrix}0 & if k < d, \\ Q(a)Φ^{(k−d)}(x) & if k ≥ d. \end{matrix}  (6.41)

Thus, if μ(Φ(x)) has degree d_{o}, then it follows from this that

γ_{Q}(x, ∂_{x})Φ^{(k)}(x) = 0  for all k − d < d_{o} .  (6.42)

In particular, we must have

γ_{Q}(x, ∂_{x})μ(Φ(x)) = 0 ,

and (6.39) then immediately follows from the definition in (6.31). This establishes (6.38) and completes our proof.

We now have all the tools we need for the proof of Theorem I.2. The argument is based on the following two basic observations

Proposition 6.3 Let d_{mn} = maxdegree(H_{G}(m)). Then Ob_{1} : To show that an element Φ ∈ Γ_{[a]_{G}}(m) vanishes, it is sufficient to check that all its homogeneous components of degree ≤ d_{mn} vanish. Ob_{2} : To show that a polynomial Q(x) ∈ QI_{m}(G) belongs to J_{[a]_{G}}(m), it is

sufficent to show that γ_{Q}(x, ∂_{x}) kills the element Θ_{[a]_{G}}(m) defined in (4) of Proposition 6.1.

Proof Suppose that Φ ∈ Γ_{[a]_{G}} satisfies

Φ^{(k)} = 0 (for k ≤ d_{mn}),  (6.43)

and suppose if possible that Φ ≠ 0. Now we have shown that that for any Φ ∈ Γ_{[a]_{G}} we have μ(Φ) ∈ H_{G}(m). Thus (6.43) gives μ(Φ) = 0. But this is absurd since by definition Φ ≠ 0 implies μ(Φ) ≠ 0. This proves Ob_{1}. Note next that the equation γ_{Q}Θ_{[a]_{G}} = 0 together with (6.12) (b) yields the identity

∑_{b∈[a]_{G}} ϵ(b)Q(b)Ψ_{m}(x, b) = 0 .

However, the independence of {Ψ_{m}(x, b)})_{b∈[a]_{G}} then forces Q_{(}b) = 0 for all b ∈ [a]_{G}, But that is Q ∈ J_{[a]_{G}}(m). This proves Ob_{2}.

We shall now establish Theorem I.2 by proving a bit more. Theorem 6.2

Let B ∈ QI_{m}(G) yield a homogeneous basis for the quotient QI_{m}(G)/J_{G}(m), and let B_{k} be the subset of elements of degree k in B. Then the collection

{γ_{b}(x, ∂_{x})Π_{G}(x)^{2m+1}}_{b∈B_{k}}

is a basis of H_{=k}(H_{G}(m)). Proof

We shall have to use two facts, namely that

d_{mn} = degree(Π_{G}(x)^{2m+1})  (6.44)

and that the Hilbert series F_{H_{G}(m)}(t) is palindromic. That is, we have

dim(H_{k}(H_{[a]_{G}}) = dim(H_{d_{mn}−k}(H_{[a]_{G}}) .  (6.45)

We should note that (6.44) immediately follows from the definition of the polynomial ψ_{b}(x) given in (1.8), since we proved there that the maximum degree of the ordinary G-harmonics is equal to the degree of the discriminant Π_{G}(x).

As for (6.45) we have to refer to the paper of Felder–Veselov [5] for a proof. This given, we shall prove our result by induction on k. More precisely we

shall assume that

1_{k} : P (∂_{x})Φ_{[a]}) |_{<d_{mn}−k} = 0 implies that P is congruent mod J_{[a]_{G}}(m)

to a polynomial Q of degree ≤ k;

and

2_{k} : {γ_{b}(x, ∂_{x})Π_{G}(x)^{2m+1}}_{b∈B_{k}} is a basis for H_{=d_{mn}−k}(H_{G}(m));

hold true for all k < k_{o} and complete the induction by showing that 1_{k_{o}} and 2_{k_{o}} must hold as well. To this end, note first that, since dim(H_{0}(H_{[a]_{G}}) = 1, it follows from (6.39) that we also have

dim(H_{d_{mn}}(H_{[a]_{G}})) = 1 .  (6.46)

Since the polynomial Π_{G}(x)^{2m+1} is in QI_{m}(G) and is clearly killed by all Ginvariant differential operators, it follows that it lies in H_{d_{mn}}(H_{[a]_{G}}). But then (6.44) yields that every element of H_{d_{mn}}(H_{[a]_{G}}) is necessarily a multiple of Π_{G}(x)^{2m+1}. This proves 2_{0}. To start, we need also check the validity of 1_{0}. Note that 1_{0} says that any P such that

γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{<d_{mn}} = 0  (6.47)

must be congruent to a constant mod J_{[a]_{G}}(m). Note that if γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{=d_{mn}} also vanishes then Ob_{1} gives that γ_{P} (x, ∂_{x}) kills Θ_{[a]_{G}}, and then Ob_{2} yields that P is congruent to zero mod J_{[a]_{G}}(m). On the other hand, if γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{=d_{mn}} does not vanish, then, since the homogeneous elements of degree d_{mn} in H_{[a]} are all multiples of Π_{G}(x)^{2m+1}, from (4) b) of Proposition 6.1 we derive that

γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{=d_{mn}} = c μ(Θ_{[a]_{G}})

for a suitable constant c. But then all the homogeneous components of degree d_{mn} or less in

(γ_{P} (x, ∂_{x}) − c) Θ_{[a]_{G}}

must vanish, and Ob_{1} and Ob_{2} again yield that P − c ∈ J_{[a]_{G}}(m). This gives 1_{0}.

We are thus in a position to proceed with our induction, and we shall assume that 1_{k} and 2_{k} hold for all k < k_{o}. We start by proving 2_{k_{o}}. To this end, note that by (6.45) for k = k_{o} we need only show that {γ_{b}(x, ∂_{x})Π_{G}(x)^{2m+1}}_{b∈B_{ko}} is an independent set. So, let there be constants c_{b} such that

∑_{b∈B_{ko}} c_{b} γ_{b}(x, ∂_{x})Π_{G}(x)^{2m+1} = 0 ,  (6.48)

and set

P (x) = ∑_{b∈B_{ko}} c_{b} b(x) .  (6.49)

Now, (4.48) implies that

γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{≤d_{mn}−k_{o}} = 0 .

However, this brings us into 1_{k_{o}−1} and by induction we can find a polynomial Q of degree < k_{o} congruent to P modulo J_{[a]_{G}}(m). But now, since degree P = k_{o} > degree Q and P is homogeneous, we deduce that P ∈ gr J_{[a]_{G}}(m). Now we have seen in (5.25) that gr J_{[a]_{G}}(m) = J_{G}(m), so it follows that P ∈ J_{G}(m). But this together with (6.49) contradicts the independence of B mod J_{G}(m). This forces the vanishing of all the coefficients c_{b} in (6.49). Thus 2_{k_{o}} must hold true as desired.

Next we show 1_{k_{o}}. So let

γ_{P} (x, ∂_{x})Θ_{[a]_{G}} |_{<d_{mn}−k_{o}} = 0 .  (6.50)

If γ_{P} (x, ∂_{x})Θ_{[a]_{G}} = 0, then by Ob_{2} we must have P ∈ J_{[a]_{G}}(m) and we are done. If γ_{P} (x, ∂_{x})Θ_{[a]_{G}} does not vanish, then (6.50) gives that

degree μ(γ_{P} (x, ∂_{x})Θ_{[a]_{G}}) = d_{mn} − k_{1} ( with k_{1} ≤ k_{o} ) .

By 2_{k_{1}}, which is now available up to and including k_{o}, we can find a homogeneous polynomial Q_{1} of degree k_{1} such that

μ(γ_{P} (x, ∂_{x})Θ_{[a]_{G}}) = γ_{Q_{1}}(x, ∂_{x}) Π_{G}(x)^{2m+1} .  (6.51)

This in turn implies that for a suitable constant c

γ_{P −cQ_{1}}(x, ∂_{x})Θ_{[a]_{G}} |_{≤d_{mn}−k_{1}} = 0 .

However, since k_{1} ≤ k_{o}, this brings us down into the domain of 1_{k_{o}−1}, so we can use the induction hypothesis and conclude that P − cQ_{1} is congruent mod J_{[a]_{G}}(m) to a polynomial Q_{2} of degree at most k_{o} − 1. In other words, we have shown that P is congruent mod J_{[a]_{G}}(m) to the polynomial Q = cQ_{1} + Q_{2} which is of degree at most k_{o}, which is precisely what we needed to show. This completes the induction and our proof.

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Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, Dept 0112, La Jolla, CA 92093-0112 USA.