On the Calogero-Moser space associated with dihedral groups

Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a dihedral group.

Cet article est mis à disposition selon les termes de la licence C C -3.0 F .

Publication éditée par le laboratoire de mathématiques Blaise Pascal de l'université Clermont Auvergne, UMR 6620 du CNRS Clermont-Ferrand -France
• If d is odd, then Calogero-Moser (left, right or two-sided) cells coincide with the Kazhdan-Lusztig (left, right or two-sided) cells: this is a particular case of [6, Conjs. L and LR]. For proving this fact, we prove that the Galois group defined in [6,Chpt. 5] is equal to the symmetric group S W on the set W.
• If d is odd, then we prove [6,Conj. FIX] about the fixed point subvariety Z µ d .
We also investigate special cases using calculations with the software MAGMA [8], based on the MAGMA package CHAMP developed by Thiel [16], and a paper in preparation by Thiel and the author [7]. For instance, we get: • If d ∈ {3, 4, 6}, then we prove [6,Conj. FIX] about the fixed point subvariety Z µ m (for any m).
• If d = 4 and the parameters are equal and non-zero (respectively d = 6 and the parameters are generic) and if m is a Poisson maximal ideal of Z, then we prove that the Lie algebra m/m 2 is isomorphic to sl 3 (C) (respectively sp 4 (C)). We believe these intriguing examples have their own interest.
Notation. We set V = C 2 and we denote by (x, y) the canonical basis of V and by (X, Y ) the dual basis of V * . We identify GL C (V) with GL 2 (C). We also fix a non-zero natural number d, as well as a primitive d-th root of unity ζ ∈ C × . If i ∈ Z/dZ, we denote by ζ i the element ζ i 0 , where i 0 is any representative of i in Z.
We denote by C[V] the algebra of polynomial functions on V (so that C[V] = C[X, Y ] is a polynomial ring in two variables) and by C(V) its fraction field (so that C(V) = C(X, Y )). We will denote by ⊗ the tensor product ⊗ C .

Generators
If i ∈ Z, we set s i = 0 ζ i ζ −i 0 and s = s 0 , t = s 1 . Note that s i = s i+d is a reflection of order 2 for all i ∈ Z (so that we can write s i for i ∈ Z/dZ). We set W = s, t .
Then W is a dihedral group of order 2d, and (W, {s, t}) is a Coxeter system, where If we need to emphasize the natural number d, we will denote by W d the group W.
The Coxeter element w c is given by Finally, we fix a primitive 2d-th root of unity ξ such that ξ 2 = ζ and we set Then it is readily seen that τsτ −1 = t, τtτ −1 = s and τ 2 = 1, (1.6) so that τ ∈ N GL C (V ) (W).
Remark 1.1. If d = 2e − 1 is odd, then ξ = −ζ e and so τ = −s e induces an inner automorphism of W (the conjugacy by s e ). If d is even, then s and t are not conjugate in W and so τ induces a non-inner automorphism of W.

Irreducible characters
We denote by 1 W the trivial character of W and let ε : W → C × , w → det(w). If d is even, then there exist two other linear characters ε s and ε t which are characterized by the following properties: If k ∈ Z, we set It is easily checked from (1.1) that ρ k is a morphism of groups (that is, a representation of W). If R is any C-algebra, we still denote by ρ k : RW → Mat 2 (R) the morphism of algebras induced by ρ k . The character afforded by ρ k is denoted by χ k . The following proposition is well known: Proposition 1.2. Let k ∈ Z. Then: (1) χ k = χ −k = χ k+d .
(2) If d is odd and k 0 mod d, then χ k is irreducible.
(3) If d is even and k 0 or d/2 mod d, then χ k is irreducible.

Some fractions in two variables
We work in the fraction field C(V) = C(X, Y ). If 1 k d, then Proof. Let us first prove (1.7). Since 1 k d, there exist complex numbers (

Invariants
The aim of this section is to describe generators and relations for the invariant algebra Note that such results have been obtained if d ∈ {3, 4, 6} in [1]. We set Then Note that a 0,0 = r and a d,0 = R.
We will now describe some relations between these invariants. For this, let eu (i) 0 = (x X) i + (yY ) i . Then the eu (i) 0 's belong also to C[V × V * ] W . As they will appear in relations between generators of C[V × V * ] W , we must explain how to express them as polynomials in eu 0 . First of all, Therefore, So, by triangularity of this formula, an easy induction shows that there exists a family of integers (n i, j ) 0 j i/2 such that with n i,0 = 1 for all i.
On the other hand, one can check that the following relations hold (for 1 i j d−1): Using (2.1), these last relations can be viewed as relations between q, r, Q, R, eu 0 , a 1,0 , a 2,0 , . . . , a d−1,0 .
Generators: q, r, Q, R, eu 0 , a 1,0 , a 2,0 , . . . , a d−1,0 Relations: This presentation is minimal, as well by the number of generators as by the number of relations (there are d +4 generators and (d +2) Proof. Let H * denote the subspace of C[V] defined by Then H * is a graded sub-C induced by the multiplication is a W-equivariant isomorphism. Consequently, An easy computation of the subspaces (H * i ⊗ H j ) W based on the previous remarks show that On the other hand, which shows the last assertion of the theorem. It also proves that which sends q, r, Q, R, E, A 1 , A 2 , . . . , A d−1 on q, r, Q, R, eu 0 , a 1,0 , a 2,0 , . . . , a d−1,0 respectively. For 1 i j d − 1, let F i (respectively F i, j ) denote the element of the polynomial algebra C[q, r, Q, R, E, A 1 , A 2 , . . . , A d−1 ] corresponding to the relation (Z 0 i ) (respectively (Z 0 i, j )). Let A denote the quotient of C[q, r, Q, R, E, A 1 , A 2 , . . . , A d−1 ] by the ideal a generated by the F i 's and the F i, j 's. We denote by q 0 , r 0 , Q 0 , R 0 , E 0 , A 1,0 , A 2,0 , . . . , A d−1,0 the respective images of q, r, Q, R, E, A 1 , A 2 , . . . , A d−1 in A. We then have a surjective morphism of bigraded C-algebras ϕ : A C[V × V * ] W . We want to show that ϕ is an isomorphism. For this, it is sufficient to show that the bi-graded Hilbert series coincide. But, where an inequality between two power series means that we have the corresponding inequality between all the coefficients.
We will prove that A is a subalgebra of A. (2.5) For this, taking into account the form of the F i 's and the F i, j 's, it is sufficient to show that E d+1 0 ∈ A . But, by (2.1), , and n d,0 = 1. So and so which concludes the proof of (2.5).
Since A contains q 0 , r 0 , Q 0 , R 0 , E 0 , A 1,0 , A 2,0 , . . . , A d−1,0 and since A is generated by these elements, we have A = A . It follows from (2.3) and (2.4) that dim bigr In other words, this shows that the presentation of C[V × V * ] W given in Theorem 2.1 is correct.
It remains to prove the minimality of this presentation. The minimality of the number of generators follows from [7]. Let us now prove the minimality of the number of relations. For this, let p 0 denote the bi-graded maximal ideal of P • and set B = C[V ×V * ] W /p 0 C[V ×V * ] W . We denote by e, a 1 , a 2 , . . . , a d−1 the respective images of eu 0 , a 1,0 , a 2,0 , . . . , a d−1,0 in B. Hence, and B admits the following presentation: It is sufficient to prove that the number of relations of B is minimal. By reducing modulo the ideal Be, we get that all the relations of the form a i a j = 0 or −e d are necessary. By reducing modulo the ideal Ba 1 + · · · + Ba i−1 + Ba i+1 + · · · Ba d−1 , we get that the relations ea i = 0 are necessary.

Remark 2.2.
It is easily checked that the element τ defined in Section 1.1 satisfies for all 0 i d (this follows from the fact that ξ d = −1).

Definition
We denote by C the C-vector space of maps Ref(W) → C which are constant on conjugacy classes. If i ∈ Z/dZ, we denote by C i the element of C * which sends c ∈ C to c s i . By (1.1), The generic rational Cherednik algebra at t = 0 is the C[C]-algebra H defined as the quotient of C[C] ⊗ T(V ⊕ V * ) W by the following relations (here, T(V ⊕ V * ) is the tensor algebra of V ⊕ V * over C): for U, U ∈ V * and u, u ∈ V. Note that we have followed the convention of [6]. Given the relations (3.1), the following assertions are clear: • There is a unique morphism of C-algebras CW → H sending w ∈ W to the class induced by the three morphisms defined above and the multiplication map is surjective. Note that it is The last statement is strenghtened by the following fundamental result by Etingof Here, τ is the element of N GL C (V ) (W) defined in Section 1.1.

Specialization
Given c ∈ C, we denote by C c the maximal ideal of The C-algebra H c is the quotient of the C-algebra T(V ⊕ V * ) W by the ideal generated by the following relations: for U, U ∈ V * and u, u ∈ V.
Remark 3.4 (Grading). The ideal C c is not bi-homogeneous (except if c = 0) so the algebra H c does not inherit from H an (N × N)-grading. However, C c is Z-homogeneous, so H c still admits a natural Z-grading.

Calogero-Moser space
We denote by Z the centre of H. By [11], it contains C[V] W and C[V * ] W so, by Theorem 3.1, it contains the subalgebra and that [11] Z and Z c are integral and integrally closed.
Since the C-algebra Z is finitely generated, we can associate to it an irreducible and normal (according to (3.4)) algebraic variety over C, called the generic Calogero-Moser space, and which will be denoted by Z. If c ∈ C, we denote by Z c the algebraic variety associated with the C-algebra Z c .

About the presentation of Z
We follow here the method of [7]. If h ∈ H, it follows from Theorem 3.1 that there exists a unique family of elements We define the C[C]-linear map Trunc : The next lemma is proved in [7]: We then set eu = Trunc −1 (eu 0 ) and, for 0 i d, An explicit algorithm for computing the inverse map Trunc −1 is described in [7]. Note that Trunc −1 (p) = p for p ∈ P, so that a 0 = a 0,0 = r and a d = a d,0 = R. By [7], the relations Relations: This presentation is minimal, as well by the number of generators as by the number of relations (there are d + 4 generators and (d + 2)(d − 1)/2 relations). Moreover, It must be said that we have no way to determine explicitly the relations (Z i, j ) in general: we will describe them precisely only for d ∈ {3, 4, 6} in Section 8. Note that the information provided by Theorem 3.6 is sufficient enough to be able to prove Theorem 7.1 in Section 7.
Remark 3.7 (Gradings). The bi-grading and the Z-grading on the algebra H constructed in Remark 3.3 induce a bi-grading and a Z-grading on Z. Note that the map Trunc is bi-graded, so that the generators given in Theorem 3.6 are bi-homogeneous.
On the other hand, the deformation process for the relations described in [7] respects the bi-grading. So we may, and we will, assume in the rest of this paper that the relations (Z i ) and (Z i, j ) given in Theorem 3.6 are bi-homogeneous.
Note. From now on, and until the end of this paper, we fix a parameter c ∈ C and we set a = c s and b = c t . Note that, if d is odd, then a = b.

Calogero-Moser cellular characters
The aim of this section is to determine, for all values of c, the Calogero-Moser c-cellular characters as defined in [6, §11.1]. It will be given in Table 4.1 at the end of this section.
We will use the alternative definition [6,Thm. 13.4.2], which is more convenient for computational purposes (see also [7]). So, following [6, Chpt. 13], we denote by C(V)W the group algebra of W over the field C(V) and we set We will denote by E Gau 1 (respectively E Gau ε , respectively L Gau k ) the restriction of C(V)E 1 W (respectively C(V)E ε , respectively C(V)E χ k ) to Gau(W, c). If d is even, then the restriction of C(V)E ε s (respectively C(V)E ε t ) to Gau(W, c) will be denoted by E Gau s (respectively E Gau t ).   such that τ s = t and τ t = s constructed in Section 1.1 to be sent back to the previous case. We then deduce from (4.1) that the list of c-cellular characters of W is 1 + ε s , ε t + ε, Note that we have a semi-direct product decomposition W = s n W .

The equal parameters case
We assume here, and only here, that a = b 0. Now, if 1 k d − 1, then If we denote by M the diagonal matrix X 0 0 Y , then it follows from the previous formulas that This implies that Since Tr(ρ k (D x )) = Tr(ρ k (D y )) = 0, the nature of the restriction of ρ k to Gau(W, c) depends on whether − det(ρ k (D x )) = a 2 X d Y d−2 is a square in C(V). Two cases may occur: First case: assume that d is odd is not a square in C(V) (for 1 k d − 1), so it follows that L Gau k is simple (but not absolutely simple) and it follows from (4.4) that Moreover, the list of c-cellular characters is given in this case by 1 W , ε and Second case: assume that d is even In this case, it is easily checked that E Gau by (4.4). Therefore,

The generic case
We assume here, and only here, that ab(a 2 − b 2 ) 0 (so that we are not in the cases covered by the previous subsections). Note that this forces d to be even. We will prove that the list of c-cellular characters is given in this case by 1 W , ε, ε s , ε t and Proof. We have, for 1 k e − 1, So it follows from (1.9), (1.10) and (1.11) that The matrix ρ k (D y ) can be computed similarly and we can deduce that, Therefore, is not a square in C(V), and so L Gau k is simple (but not absolutely simple) for 1 k e − 1.
Moreover, an easy computation shows that E Gau 1 , E Gau ε , E Gau s and E Gau t are pairwise non-isomorphic simple Gau(W, c)-modules. So it follows from (4.11) that and that (4.10) holds.

Conclusion
The following Table 4 Remark 4.4. Whenever a, b ∈ R, the Kazhdan-Lusztig cellular characters for the dihedral groups are easily computable (see for instance [15]) and a comparison with

Calogero-Moser families
The aim of this section is to compute the Calogero-Moser c-families of W (as defined in [6, §9.2]) for all values of c. The result is given in  [2]. We provide here a different proof, which uses the computation of Calogero-Moser cellular characters.

Families
To any irreducible character χ, Gordon [13] associates a simple H c -module L c ( χ) (we follow the convention of [6, Prop. 9.1.3]). We denote by Ω c χ : Z → C the morphism defined by the following property: if z ∈ Z, then Ω c χ (z) is the scalar by which z acts on L c ( χ) (by Schur's Lemma). We say that two irreducible characters χ and χ belong to the same Calogero-Moser c-family if Ω c χ = Ω c χ (see [13] or [6, Lem. 9.2.3]: note that Calogero-Moser families are called Calogero-Moser blocks in [13]). We give here a different proof of a theorem of Bellamy [2]: Theorem 5.1 (Bellamy). Let c ∈ C and let χ and χ be two irreducible characters of W. Then χ and χ lie in the same Calogero-Moser family if and only if Ω c χ (eu) = Ω c χ (eu). Consequently, the Calogero-Moser families are given by But two irreducible characters occuring in the same Calogero-Moser c-cellular character necessarily belong to the same Calogero-Moser c-family [6,Prop. 11.4.2]. So the Theorem follows from (a), (b) and Table 4.1.
Remark 5.2. Whenever a, b ∈ R, the Kazhdan-Lusztig families for the dihedral groups are easily computable (see for instance [15]) and a comparison with

Cuspidal families
Recall that the algebras Z and Z c are endowed with a Poisson bracket {, }. This Poisson structure has been used by Bellamy [3] to define the notion of cuspidal Calogero-Moser families. If F is a Calogero-Moser c-family, we set m c F = Ker(Ω c χ ) ⊂ Z c , where χ is some (or any) element of F (note that Ω c χ factorizes through the projection Z Z c ).
They have been determined for most of the Coxeter groups by Bellamy and Thiel [4]. In our case, we recall here their result, as well as a proof for the sake of completeness. (1) A Calogero-Moser family F is cuspidal if and only if |F | 2 and χ 1 ∈ F (and then χ k ∈ F for all 1 k < d/2).
(2) There is at most one cuspidal family. If d 5, there is always exactly one cuspidal family.
Proof. The main observation is that {q, Q} = eu (see (3.5)). This implies that, if χ belongs to a cuspidal family, then Ω c χ (eu) = 0. Since it follows from Table 5.1 that the Calogero-Moser c-families are determined by the values of Ω c χ (eu), this implies that there is at most one cuspidal Calogero-Moser c-family, and that it must contain χ 1 (and χ k , for 1 k < d/2).
Also, since a Calogero-Moser c-family of cardinality 1 cannot be cuspidal [13, §5.2], this shows the "only if" part of (1). It remains to prove the "if" part of (1). So assume that χ 1 ∈ F and that |F | 2. By Bellamy theory [3,Introduction], there exists a non-trivial It is a question to determine in general the structure of this Lie algebra. We would like to emphasize here the following two particular intriguing examples (a proof will be given in Section 8, using explicit computations).
Lie algebra of type A 2 .
(2) If d = 6 (i.e. if W is of type G 2 ) and ab(a 2 − b 2 ) 0, then Lie c (F ) sp 4 (C) is a simple Lie algebra of type B 2 .

Calogero-Moser cells
Let c ∈ C. The main theme of [6] is a construction of partitions of W into Calogero-Moser left, right and two-sided c-cells, using a Galois closure M of the field extension Proof. We first prove an easy lemma about finite permutation groups. Recall that a subgroup Γ of the symmetric group S n is called primitive if it is transitive and if the stabilizer of an element of {1, 2, . . . , n} is a maximal subgroup of Γ.
Proof of Lemma 6.2. Since d is odd, the action of σ = ι(w c , w c ) on W is a cycle of length d (it fixes w c and acts by a cycle on W \ w c by (1.1)). Moreover, ι(w c , 1) and ι(1, w c ) belong to the centralizer of ι(w c , w c ) in Γ, so C Γ (σ) σ . Since Γ is primitive, it follows from [10,Ex. 7.4.12] But note that the action of ι(s, 1) is a product of d transpositions, so it is an odd permutation (because d is odd). Therefore, Γ A W and the Lemma is proved.
Assume that d is odd. Let us recall some results from [6, Chpt. 10] about Calogero-Moser two-sided cells. First, Calogero-Moser two-sided c-cells are defined as the orbits of some subgroup of S W . Also [6,Thm. 10.2.7], there is a bijection between Calogero-Moser c-families and Calogero-Moser two-sided c-cells and, if C is the two-sided c-cell corresponding to the c-family F , then Applied to the case where a = b 0, Table 5.1 shows that there is a subgroupĪ of G which has three orbits for its action on W, of respective lengths 1, 1 and 2d − 2. Since ι(W × W) is transitive on W, G is also transitive and we may assume that one of the two orbits of length 1 is the singleton {1}. Let ∆W denote the diagonal in W × W. Its action on W is by conjugacy: it has only one fixed point (because the center of W is trivial). This proves that the subgroup I, ι(∆W) acts transitively on W \ {1}. So G is 2-transitive and, in particular, primitive. The Theorem now follows from Lemma 6.2 above. Proof. Assume that c ∈ C takes real values. The computation of Calogero-Moser cfamilies and c-cellular characters shows that, if we choose randomly two prime ideals as in [6,Chpt. 15], then the associated Calogero-Moser two-sided or left c-cells have the same sizes as the Kazhdan-Lusztig two-sided or left c-cells respectively (see [6,Chpts. 10 and 11]). Since the Galois group G coincides with S W , we can manage to change the prime ideals so that Calogero-Moser and Kazhdan-Lusztig c-cells coincide.

Remark 6.4. Let
Note that, in our case, S B W (respectively S D W ) is a Weyl group of type B d (respectively D d ) and that S D W is a normal subgroup of S B W of index 2. Assume here, and only here, that d is even. It then follows from [6, Prop. 5.5.2] that G ⊂ S B W . We would bet a few euros (but not more) that G = S D W . This has been checked for d = 4 in [6, Thm. 19.6.1] and it will be checked in (8.4) whenever d = 6. Let us just prove a few general facts.
First, letW = W/Z(W) (it is a dihedral group of order d) and letῑ :W ×W → SW denote the morphism induced by the action by left and right translations. LetḠ denote the image of G in SW (indeed G ⊂ S B W and there is a natural morphism S B W → SW ). Then, if a = b 0, the Calogero-Moser two-sided c-cells have cardinalities 1, 1 and 2d − 2 (by [6, Thm. 10.2.7]) so it follows from the definition of Calogero-Moser cells that there exists a subgroup I 1 of G whose orbits have cardinalities 1, 1 and 2d − 2. Therefore, the imageĪ 1 of I 1 inḠ has orbits of cardinalities 1 and d − 1. Consequently,Ḡ is 2-transitive. Similarly, taking c such that ab(a 2 − b 2 ) 0, we get that there is a subgroupĪ 2 ofḠ whose orbits have cardinalities 1, 1 and d − 2. Therefore, G is 3-transitive. (

Fixed points
The Z-grading on the C-algebra H (defined in Remark 3.3) induces an action of the group C × on H as follows [6, §3.5.A]. If ξ ∈ C × then: • If y ∈ V, then ξ y = ξ −1 y.
So the center Z inherits an action of C × , which may be viewed as a C × -action on the Calogero-Moser space Z, which stabilizes all the fibers Z c (for c ∈ C). Now, if m ∈ N * , we denote by µ m the group of m-th root of unity in C × . In [6, Conj. FIX], R. Rouquier and the author conjecture that all the irreducible components of the fixed point variety Z µ m are isomorphic to the Calogero-Moser space of some other complex reflection groups (here, Z µ m is endowed with its reduced structure). This conjecture will be checked for d ∈ {3, 4, 6} and any m in Section 8. Proof. The case where d = 1 is not interesting, so we assume that d 3. Let I denote the ideal of Z generated by { ζ z − z | z ∈ Z }. Then the algebra of regular functions on Z µ d is Z/ √ I. We will describe Z/I, and this will prove that I = √ I in this case. Therefore, I = q, Q, a 1 , a 2 , . . . , a d−1 , and so Z/I is generated by the images of A, r, R and eu. In the quotient Z/I, all the equations of Z-degree which is not divisible by d are automatically fulfiled, so it only The only bi-homogeneous monomials in A, r, R and eu of bi-degree (d, d) are r R and the eu k A d−k (for 0 k d). Therefore, the above equation implies that there exist complex numbers λ 0 , . . . , λ d−1 such that On the other hand, it follows from [6,Cor. 9.4.4] that Since all the C × -fixed points belong to Z µ d , this implies that t − dA and t + dA both divide and so there remains only one relations in the quotient Z/I, namely (eu − dA)(eu + dA)eu d−2 ≡ r R mod I, as desired.

Examples
We are interested here in the cases where d ∈ {3, 4, 6}. These are the Weyl groups of rank 2 (of type A 2 , B 2 or G 2 ). For each of these cases, we give a complete presentation of the centre Z of H (using the algorithms developed in [7]). For the sake of completeness, we give the minimal polynomial of eu over P, as the Galois group G is defined as the Galois group of this polynomial over the fraction field of P (see [6, §5.1.D]): it will only be used in type G 2 for proving that G = S D W in this case. We use these explicit computations to check some of the facts that have been stated earlier in this paper. Most of the computations are done using MAGMA [8].
In this section, we denote by z χ the point of Z c corresponding to the maximal ideal Ker Ω c χ of Z c .

Type A 2
We work here under the following hypothesis: Note. We assume in this subsection, and only in this subsection, that d = 3. In other words, W is a Weyl group of type A 2 .
Using MAGMA and [7], we can compute effectively the generators of the C[C]-algebra Z and we obtain the following presentation for Z (note that C[C] = C[A] because A = B): -algebra Z admits the following presentation: Generators: q, r, Q, R, eu, a 1 , a 2

Relations:
The minimal polynomial of eu is given by We conclude by proving [6,Conj. FIX] in this case (about the variety Z µ m ). Note that the only interesting case is where m divides the order of an element of W. So m ∈ {1, 2, 3}. The case m = 1 is stupid while the case m = 3 is treated in Theorem 7.1: admits the following presentation: Relations: In particular, if a 0, then the variety Z (2) An isolated point, which is equal to z χ 1 .

Type B 2
We work here under the following hypothesis: Note. We assume in this subsection, and only in this subsection, that d = 4. In other words, W is a Weyl group of type B 2 .
Using MAGMA and [7], we can compute effectively the generators of the C[C]-algebra Z and we obtain the following presentation for Z (note that A B): -algebra Z admits the following presentation: Generators: q, r, Q, R, eu, a 1 , a 2 , a 3 Relations:

Then
The minimal polynomial of eu is f 4 (t 2 ). (8.2)

Cuspidal point
We aim to prove here Theorem 5.4 (1). By Table 5 Then there is only one cuspidal family F in Z c (the one containing χ 1 ). It is easily checked that m c F = q, r, Q, R, eu, a 1 , a 2 , a 3 Z c and it is readily checked from the presentation given in Proposition 8.3 that the cotangent space Lie c (F ) = m c F /(m c F ) 2 has dimension 8: so a basis is given by the images q, r, Q, R, eu, a 1 , a 2 , a 3 of q, r, Q, R, eu, a 1 , a 2 , a 3 respectively.
The computation of the Poisson bracket can be done using the MAGMA package CHAMP: writing the result modulo (m c F ) 2 gives the Lie bracket on Lie c (F ). We can then deduce that: Proposition 8.4. Assume here that a = b. Then the linear map ℵ c : Lie c (F ) −→ sl 3 (C) defined in Table 8.1 is a morphism of Lie algebras. It is an isomorphism if a 0.

Type G 2
We work here under the following hypothesis: Note. We assume in this subsection, and only in this subsection, that d = 6. In other words, W is a Weyl group of type G 2 .
Using MAGMA and [7], we can compute effectively the generators of the C[C]-algebra Z and we obtain the following presentation for Z (note that A B): Proposition 8.6. The C[A, B]-algebra Z admits the following presentation: Generators: r, R, eu, a 1 , a 2 , a 3 , a 4 , a 5 Relations: see Table 8.2 Recall from Proposition 5.3 that there is a unique cuspidal Calogero-Moser c-family F : it is the one which contains χ 1 (this fact does not depend on the parameter c; however, the cardinality of F depends on the parameter). It corresponds to the maximal ideal m = q, r, Q, R, eu, a 1 , a 2 , a 3 , a 4 , a 5 of Z c . It follows from the presentation of Z given by Proposition 8.6 that the cotangent space Lie c (F ) = m c F /(m c F ) 2 has dimension10 (a basis is given by the images q, r, Q, R, eu, a 1 , a 2 , a 3 , a 4 and a 5 of q, r, Q, R, eu, a 1 , a 2 , a 3 , a 4 , a 5 in m respectively). The Poisson bracket (and so the Lie bracket in Lie c (F )) can then be computed explicitly using the MAGMA package CHAMP. We can then deduce the following result: (1) If a 2 b 2 , then ℵ c is an isomorphism of Lie algebras.
(2) If a 2 = b 2 , then its image is isomorphic to sl 2 (C) (with basis ℵ c (q), ℵ c (Q) et ℵ c (eu)) and its kernel is commutative, of dimension7 (as a module for sl 2 (C), it is irreducible).