Symmetry, Integrability and Geometry: Methods and Applications Dunkl Operators and Canonical Invariants of Reflection Groups ⋆

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.


Introduction and main results
Let V R be a real vector space with a scalar product, and W ⊂ O(V R ) be a finite group generated by reflections. In this paper we construct a family of W -invariants (which we refer to as canonical invariants) in S(V ), where V = C ⊗ V R , by means of Dunkl operators (see [8]). These canonical invariants form a basis in S(V ) W (depending on a continuous parameter c) and, as such, include both the c-elementary and c-quasiharmonic invariants introduced in our earlier paper [2]. Using this technique, we prove that for W = S n the c-elementary invariants are deformations of the elementary symmetric polynomials in the vicinity of c = 1/n. Dunkl operators ∇ y , y ∈ V * , are differential-difference operators first introduced by Charles Dunkl in [8] and given (for any W ) by: where S is the set of all reflections in W , c : S → C is a W -invariant function on S, and α s ∈ V is the root of the reflection s. In particular, for W = S n , where s ij ∈ S n is the transposition switching x i and x j . The remarkable result by Charles Dunkl that all ∇ y commute allows to define the operators ∇ p , p ∈ S(V * ), by ∇ p+q = ∇ p + ∇ q , ∇ pq = ∇ p ∇ q for all p, q ∈ S(V * ).
In what follows we will mostly think of c as a formal parameter in the affine space A S/W , where S/W is the set of W -orbits in S. Using the notation S c (V ) = C(c) ⊗ S(V ), consider each ⋆ This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html Dunkl operator ∇ p as a C-linear map S(V ) → S c (V ) (or, extending scalars, as a C(c)-linear endomorphism of S c (V )).
Thus, the association p → ∇ p defines an action of S c (V * ) on S c (V ) by differential-difference operators. In turn, this action and the isomorphism V ∼ = V * (hence S c (V * ) ∼ = S c (V )) given by the scalar product define the bilinear form (·, ·) c : S c (V ) × S c (V ) → C(c) by for all f, g ∈ S c (V ) where x → [x] 0 is the constant term projection S c (V ) → C(c). Clearly, (f, g) c = 0 if f , g are homogeneous and deg f = deg g. The form (1.1) is symmetric and its specialization at generic c : S → C and c = 0 is nondegenerate. Understanding the values of c when the specialization of the form is degenerate and the structure of the radical is crucial for the study of representations of the rational Cherednik algebra H c (W ) (see e.g. [9,11,4]).
A classical Chevalley theorem [6] says that the algebra S(V ) W of W -invariants in S(V ) is isomorphic to the algebra of polynomials C[u 1 , . . . , u ℓ ] of certain homogeneous elements u 1 , . . . , u ℓ , where ℓ def = dim V . Throughout the paper we will call such u 1 , . . . , u ℓ homogeneous generators or, collectively, a homogeneous generating set of S(V ) W . The homogeneous generators u 1 , . . . , u ℓ are not unique, but their degrees d 1 , . . . , d ℓ (which we traditionally list in the increasing order) are uniquely defined for each group W ; they are called the exponents of the group. In particular, d 1 = 2 iff V W = {0}; the largest exponent h def = d ℓ is called the Coxeter number of W . The monomials u a def = u a 1 1 · · · u a ℓ ℓ where a 1 , . . . , a ℓ ∈ Z ≥0 form an additive basis in S(V ) W . Let ≺ be the inverse lexicographic order on Z ℓ ≥0 : for a, a ′ ∈ Z ℓ ≥0 we write a ′ ≺ a if the last non-zero coordinate of the vector a − a ′ is positive. The following is our first result asserting the existence and uniqueness of canonical invariants: a ∈ S c (V ) W = C(c)⊗ S(V ) W unique up to multiplication by a complex constant and such that for any homogeneous generating set u 1 , . . . , u ℓ of S(V ) W one has: We will prove Theorem 1.1 in Section 3.1. We will refer to each element b a as a canonical W -invariant in S c (V ) W and to the set B = {b a | a ∈ Z ℓ ≥0 }, as the canonical basis of S c (V ) W . By the construction, the canonical basis B is orthogonal with respect to the form (1.1). Remark 1.1. We can extend the theorem to the case when d k = d k+1 for some k. If V is irreducible, then this happens only when W is of type D ℓ with even ℓ and k = ℓ/2. In this case V = ℓ i=1 C · x i , the positive roots are of the form x i ± x j , and let σ : V → V be the involution given by σ( , i.e., σ is acting on roots as the symmetry of the Dynkin diagram. Then Theorem 1.1 holds verbatim for any choice of homogeneous generators u 1 , . . . , u ℓ of S(V ) W such that σ(u ℓ/2 ) = −u ℓ/2 (i.e., u ℓ/2 ∈ C · x 1 · · · x ℓ ) and σ(u j ) = u j for all j = ℓ/2. An equality d k = d k+1 can also happen when V is reducible, i.e., V = V 1 ⊕ V 2 , W = W 1 × W 2 and each W i is a reflection group of V i . This case can be handled by induction because Remark 1.2. Theorem 1.1 generalizes to all complex reflection groups if one replaces the symmetric bilinear form on V with the Hermitian one that canonically extends the W -invariant Hermitian form on S c (V ) (provided that c(s −1 ) = c(s) for all complex reflections s). The case of equal degrees d k can be treated along the lines of Remark 1.1. More precisely, the phenomenon d k = d k+1 occurs only for the following irreducible complex reflection groups (see e.g., [7,5]): 1. The series G(m, p, ℓ) with ℓ ≥ 2, p|ℓ, p|m, and d k = d k+1 = mℓ/p, k = ℓ/p.
In the case 1, similarly to Remark 1.1, one has , where ζ is an m-th primitive root of unity. Then Theorem 1.1 holds verbatim for any choice of homogeneous generators u 1 , . . . , u ℓ of S(V ) W such that σ(u k ) = ζ m/p u k (i.e., u k ∈ C · (x 1 · · · x ℓ ) m/p ) and σ(u i ) = u i for all i = k.
In the case 2 one can use various embeddings of rank 2 complex reflection groups (see e.g., [7,Section 3]) to acquire canonical invariants. For instance, G 5 is a normal subgroup of index 2 in G 7 and G 5 has degrees (6,12), which implies that if {b a make sense for all complex reflection groups. See Section 3.2 for the proof. The elements b (0,a 2 ,...,a ℓ ) of the canonical basis are more elusive, however we compute them completely when W is a dihedral group.  ) we have: We will prove the theorem in Section 3.5 by explicitly reducing the Dunkl Laplacians to the Jacobi operators. In fact, it is easy to see that the formula (1.2) is equivalent to: (y) is the k-th Jacobi polynomial (see e.g. [1, Section 6.3] or formula (3.7) below). This and other of our arguments bear some similarity with methods of the seminal papers [8] and [11] where Jacobi polynomials were first studied in the context of Dunkl operators.
Returning to the general case, note that deg b a = d k a k . For each d ∈ Z ≥0 such that The following result was essentially proved in our previous paper [2].
(c) For each k = 1, 2, . . . there is a unique, up to a multiple, element e (c) We will give a new proof of Theorem 1.3 in Section 3.2. The proof will rely on the construction of canonical invariants in Theorem 1.1.
Following [2], we refer to each e The elementary invariants for c = 0 were, most apparently, defined by Dynkin (see e.g. [16]) and later explicitly computed by K. Iwasaki in [15]. We extend the results of [15] to all c in Theorem 1.4 below.
We will also construct elementary invariants for W = S n , V = C n with the natural S n -action. It is convenient to identify S c (V ) with the algebra C(c)[x 1 , . . . , x n ] of polynomials in n variables depending rationally on c. The degrees d k are here d k = k, k = 1, . . . , n, so Theorem 1.1 and Theorem 1.3 are applicable.
To give the explicit formula for the invariants e (c) We prove Theorem 1.4 in Section 3.3. Our proof (as well as the formula (1.3)) is very similar to the one by K. Iwasaki who (using ∂ p instead of Dunkl operators ∇ p ) computed e (0) k in [15]. Following his argument, one can construct the elementary canonical invariants e (c) d k for other classical groups as well.
Note that the formula (1.3) resembles the polynomial expansion of the elementary symmetric polynomial e k = e k (x 1 , . . . , x n ): The following main result demonstrates that this observation is not a mere coincidence.
This result allows to introduce the elementary invariant polynomials for other reflection groups via e d k = lim where h is the Coxeter number. We will prove Theorem 1.5 in Section 3.4 by analyzing the behaviour of the form (1.1) near c = 1/n. Note, however, that we could not derive the theorem directly from the explicit formula (1.3).
It is easy to see that e  (see e.g. [19] for definition). Direct computations show, though, that these polynomials are not the same. [17, equation (7)] shows, in particular, that the expression of J (α) λ via elementary symmetric polynomials e i does not depend on n; for instance, J (α) (11...1) = e k for all n and k (the partition contains k units). Formulas for b a , on the contrary, contain n explicitly (see e.g. (1.4)). So, the relation between b a and Jack polynomials is yet to be clarified.

The Dunkl Laplacian and the scalar product
Throughout the section we assume that 2 = d 1 < · · · < d ℓ = h and denote where x 1 , . . . , x ℓ is any orthonormal basis in the real space V R . Obviously, e 2 is a unique (up to a scalar multiple) quadratic W -invariant in S 2 (V ). The operator L = ∇ e 2 = i ∇ 2 x i , called the Dunkl Laplacian, is independent of the choice of the basis x i ; it equals the ordinary Laplacian if c = 0.
The operator L plays a key role in the theory of Dunkl operators for W . As the following result shows, an action of any Dunkl operator can be expressed via L: where p in the right-hand side means the operator of multiplication by p.
The operator E of multiplication by e 2 , Dunkl Laplacian L, and the operator H In particular, where the direct summands are orthogonal with respect to (·, ·) c . In particular, the restriction of (·, ·) c to each e k 2 · U d is nondegenerate.
Proof . First, note that S c (V ) is an sl 2 -module, locally finite with respect to L, and ⊕ d≥0 U (c) d is the highest weight space, so that decomposition (2.1) takes place. Furthermore, note that the operator H from Proposition 2.1 is scalar on the space of polynomials of any given degree and therefore self-adjoint; the operators L and E are adjoint to one another with respect to (·, ·) c . Therefore, for This proves the orthogonality of the decomposition. In particular, this implies that the restriction of the nondegenerate form (·, ·) c to each e k 2 · U d is nondegenerate. The lemma is proved.
Using this, we compute the form (·, ·) c as follows. Denote by ϕ c a (unique) linear function S c (V ) → C(c) such that: Proposition 2.2. We have: Proof . Assume first that the function c takes only negative real values and define ϕ c by equa- On the other hand, decomposition (2.1) guarantees that (f, g) c = (f ,g k ) c . Therefore, to verify (2.2) for any k it suffices to take g = e k 2g for g ∈ U (c) d . Assume that k > 0. Then Proposition 2.1 implies that Therefore, by induction on k, which finishes the proof for c negative real. Now (2.2) implies that for c negative real the value (f, g) c depends only on the product f g (provided d and k are fixed). Since (f, g) c is a rational function of the values of c, this holds true for all c as well -so, one can use (2.2) to define ϕ c in the general case.
The following is the main result of the section. Denote by S(V ) + the kernel of the constant term projection u → [u] 0 , see (1.1). Define a symmetric bilinear form Φ c : Proof . Prove (a) by induction on the degree of u. Indeed, it follows from [4, Proposition 2.1] that for any x, y ∈ V one has: Proof . Clearly, the sl 2 -action from Proposition 2.1 preserves both R[c] ⊗ S(V R ) and S A (V R ), so that the orthogonal decomposition (2.1) is valid for S A (V R ) ⊂ S c (V ). Therefore, it suffices to verify (2.4) only forũ ∈ e k 2Ũ d , wherẽ For every suchũ it follows from (2.2) that Since the product in the right-hand side is not divisible by (1 − h c ), we see that ϕ c (ũ 2 ) ∈ A for allũ ∈ S A (V R ). Implication (2.4) is now equivalent to the following one: . Note first that for each a ∈ Z ℓ ≥0 the spaces S c (V ) W ≺a , S c (V ) W a do not depend on the choice of generators u 1 , . . . , u ℓ of S(V ) W . Indeed, let u ′ 1 , . . . , u ′ ℓ be another set of generators of S(V ) W . Since d 1 < d 2 < · · · < d ℓ , one has u ′ i = α i u i + P i (u 1 , . . . , u i−1 ), where α i ∈ C \ {0} and P i is a polynomial of i − 1 variables for i = 1, 2, . . . , ℓ.
We are going to define the canonical invariant b a ∈ S c (V ) W a as the unique (up to a multiple) vector orthogonal to the subspace S c (V ) W ≺a . However, the uniqueness of such an element requires more arguments.  Proof . We need the following general fact. Let A be a unital commutative local ring with no zero-divisors, m its maximal ideal, k = A/m the residue field. In what follows we assume that k ⊂ A so that the restriction of the canonical projection A → k to k is the identity homomorphism k → k. Let U be a vector space over k and let Φ : U × U → A be a k-bilinear symmetric form on U ; denote by Φ 0 : U × U → k the residual form given by Φ 0 def = π • Φ, where π : A → k is the canonical quotient map.
Lemma 3.1. In the notation as above assume that:

Then the natural A-bilinear extension of
Proof . We proceed by induction on k. For k = 0, m = {0}, F = A = k, and we have nothing to prove. Assume that k ≥ 1. Define the quotient ring Clearly, the ring A ′ and its ideals m ′ i satisfy the assumptions of the lemma for k − 1; therefore, the inductive hypothesis holds in the following form: if Φ(ũ,ũ) ∈ m 1 for someũ ∈ A ⊗ U, thenũ ∈ m 1 ⊗ U. (
The lemma is proved.
We apply the lemma in the case when F = C(c) = C(c 1 , . . . , c k ) is the field of rational functions in the variables c 1 , . . . , c k (where k = |S/W | is the number of conjugacy classes of reflections in W ), A ⊂ F is the local ring of all rational functions regular at c = 0, and m i is the ideal of A generated by c 1 , . . . , c i for i = 0, 1, . . . , k. Clearly, the ideals m i satisfy condition 2 of Lemma 3.1. Take U to be any subspace of S(V R ) and let Φ : U × U → A ⊂ R(c) be the restriction of the form (1.1) to U . Since the specialization Φ 0 of Φ at c = 0 is a positive definite form on U , condition 1 of Lemma 3.1 holds as well.
Therefore, the restriction of the form (1.1) to R(c) ⊗ U is non-degenerate. By extending the coefficients from R(c) to C(c) this immediately proves assertion (a) of Proposition 3.1.
To prove assertion This completes the proof of Proposition 3.1.
Now we are ready to finish the proof of Theorem 1.1. For each a ∈ Z ℓ ≥0 denote by S c (V R ) ≺a = S c (V R ) ∩ S c (V ) ≺a (see (3.1)) the real forms of S(V ) ≺a . Fix u 1 , . . . , u ℓ to be a homogeneous generating set of S(V R ) W so that (3.1) Therefore, Proposition 3.1 is applicable to this situation with U = S c (V R ) W ≺a , u = u a , and there exists a unique (up to a complex multiple) element . In other words, b a satisfies both conditions of Theorem 1.1, and the theorem is proved.  1 guarantees that for any a = (a 1 , . . . , a ℓ ), Equivalently, taking into account that (e r+1 ≥0 . This proves that B r = {b a | a 1 ≤ r} is a (linearly independent) subset of the kernel of L r+1 Sc(V ) W . On the other hand, since e 2 and L form a representation of sl 2 by Proposition 2.1, we obtain isomorphisms of graded spaces: where K is the kernel of L| Sc(V ) W . In particular, the Hilbert series of K is for each a = (a 1 , . . . , a ℓ ) ∈ Z ℓ ≥0 . Sinceb a satisfies condition 1 of Theorem 1.1, to prove that b a =b a it suffices to verify that the elementsb a satisfy condition 2 of the same theorem. This is equivalent to the elementsb a being pairwise orthogonal, i.e., (b a ,b a ′ ) c = 0 whenever a = a ′ . Part (a) guarantees that both b 0,a 2 ,...,a ℓ and b 0,a ′ 2 ,...,a ′ ℓ are in the kernel of L, so by Lemma 2.2 we obtain: 0,a 2 ,...,a ℓ ) , b (0,a ′ 2 ,...,a ′ ℓ ) ) c · C(c) = δ a,a ′ · C(c) because the elements b (0,a 2 ,...,a ℓ ) and b (0,a ′ 2 ,...,a ′ ℓ ) of the canonical basis B are orthogonal unless (a 2 , . . . , a ℓ ) = (a ′ 2 , . . . , a ′ ℓ ). This proves (b). Proposition 1.1 is proved.
To prove part (b) let µ (c) k ∈ S c (V ) W be any element satisfying its conditions. Then for any a ≺ δ k one has (u a , µ Since d 1 < · · · < d k < · · · < d ℓ , one has deg b q > d k for every q. Therefore, the homogeneity condition implies that µ Part (c), given here for completeness, is proved in our previous paper [2]. Theorem 1.3 is proved.

Proof of Theorem 1.4
The argument follows almost literally the original proof for c = 0 given in [15]; we put it here mostly for reader's convenience.

Proof of Theorem 1.5
We need the following result. we see that π(b a ) ∈ U def = a ′ ≺a,a ′ =0 k · u a ′ , where k = A/(1 − h c ). But since Φ c (b a , u a ′ ) = 0 for all a ′ ≺ a, we see that Φ c (π(b a ), U ) = 0 which contradicts Theorem 2.1(c). The contradiction obtained proves that ℓ = 0, i.e., π(b a ) is well-defined. The orthogonality of these elements is obvious.
Let W = S n , V = C n . Here d k = k, k = 1, . . . , n. Recall that e k = e k (x 1 , . . . , x n ) ∈ C[x 1 , . . . , x n ] is the k-th elementary symmetric polynomial andē k = e k (x 1 − e 1 (x) n , . . . , x n − e 1 (x) n ). The elementsē k , k = 2, . . . , n generate a subalgebra of Lemma 3.3. Using the notation p r def = y r 1 + · · · + y r n , we obtain Proof . Denote . . , x n − e 1 (x) n (the i-th argument omitted). Easy calculations show that and therefore the Dunkl operator An induction on r (where (3.4) is the base) gives then Summation over i using (3.3) finishes the proof.
On the other hand, by Theorem 1.3(b) one has ∇ P (e (c) k ) = 0 for any homogeneous symmetric polynomial P of degree deg P < k. Therefore, Φ 1/n (ē a ′ ,ē k ) = 0 = Φ 1/n ē a ′ , e (1/n) k for any a ′ ≺ δ k . Here we abbreviated Φ 1/n def = Φ c with c = 1/n in the notation of Theorem 2.1 and e k , which is well-defined by Proposition 3.2. Consequently, Therefore, e (1/n) k = αē k because, on the one hand, e (1/n) k − αē k ∈ U for some α = 0, and on the other hand, the restriction of Φ 1/n to U is non-degenerate by Theorem 2.1(c). Theorem 1.5 is proved.

Canonical invariants of dihedral groups and proof of Theorem 1.2
Throughout the section we deal with the dihedral group W = I 2 (m) = s 2 0 = s 2 1 = (s 0 s 1 ) m = 1 of order 2m.
We denote by {z, z} the basis of V such that s j (z) = −ζ jz , s j (z) = −ζ −j z.
where C def = c(s 1 ) + c(s 2 ) and δ def = c(s 2 ) − c(s 1 ) (so that δ = 0 when m is odd); ∂ e 2 and ∂ em mean here differentiation with respect to e 2 and e m , respectively, in the ring C[e 2 , e m ].
Proof . Clearly, s • ∂ y = ∂ s * (y) • s for any linear automorphism s of V and any y ∈ V * , where ∂ y : S(V ) → S(V ) is the directional derivative. So, L can be rewritten in the form L = i D i s i where D i are differential operators (of order at most 2) and s i are reflections. Thus, on the space of invariant functions L is a second order differential operator. To determine its coefficients it suffices to compute L on monomials of degree 1 and 2 in e 2 , e m . This is done in [10]; see also [2]. Proof . Since ∂ e 2 e 2 = e 2 ∂ e 2 + 1, one obtains 4e 2 L = (2e 2 ∂ e 2 ) 2 + 4m(e 2 ∂ e 2 )(e m ∂ em ) + 4m 2 e m 2 ∂ 2 em − 2mCe 2 ∂ e 2 + 2m 2 δe m/2 2 ∂ em .
To finish the proof it remains to find the value of the normalization coefficients n k = n k (c) in (1.2). To do this, substitute e 2 = 0 into (1.2), so that e ′ m = e m /4. Under this specialization, b (0,k) becomes a (complex) multiple of e k m and the right-hand side of (1.2) becomes (with the abbreviation α = C−δ−1 2 ):