Computational aspects of Calogero–Moser spaces

We present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the C×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^\times $$\end{document}-fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}-factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.


Introduction
Let V be a finite-dimensional complex vector space and let W ⊆ GL(V ) be a finite subgroup generated by reflections in V .The rational Cherednik algebras defined by Etingof and Ginzburg [EtGi] yield a flat family of deformations H c of the algebra C[V × V * ] ⋊ W . Here, C[V × V * ] is the coordinate ring of the space V × V * considered as an algebraic variety and C[V × V * ] ⋊ W denotes the semi-direct product with the group algebra of W , which incorporates the action of W on V ×V * .Moreover, c is a parameter from a complex vector space Cwhose dimension is equal to the number of conjugacy classes of reflections contained in W .We note that in [EtGi] a further deformation parameter t ∈ C is considered but here we focus on the case t = 0. Since their introduction, rational Cherednik The first author is partly supported by the ANR: Projects No ANR-16-CE40-0010-01 (GeRepMod) and ANR-18-CE40-0024-02 (CATORE).This work is a contribution to the SFB-TRR 195 'Symbolic Tools in Mathematics and their Application' of the German Research Foundation (DFG).We would like to thank Gwyn Bellamy and Gunter Malle for comments on a preliminary version of this paper.
algebras have attracted a tremendous amount of interest.This stems from their diverse connections to other fields and problems, especially in algebraic geometry and representation theory.We will mention two such connections that serve as motivation for this paper.
The spectrum Z c of the center Z c of H c is the Calogero-Moser space originating from physics [Cal, Mos].It is a Poisson deformation of the quotient variety (V × V * )/W , the latter being a symplectic singularity in the sense of Beauville [Bea].The representation theory of H c is closely tied to the geometry of Z c and plays a key role in the question of whether (V × V * )/W admits a symplectic (equivalently, crepant) resolution, see [EtGi,GiKa,Nam1,BST1], and in determining the chamber decomposition of the movable cone of a Q-factorial terminalization (i.e. a relative minimal model) of (V × V * )/W , see [Nam2,BST2].
The representation theory of H c is furthermore (conjecturally) linked to (parts of) Kazhdan-Lusztig theory [KaLu] and provides a candidate for extending this theory from finite Coxeter groups to complex reflection groups following the philosophy of the "spetses" program [BrMaMi1].It was noticed in [Gor] that Z c contains the subalgebra This inclusion defines a finite C × -equivariant morphism Υ c : Z c → P. The quotient of H c by the ideal generated by the origin in P is the restricted rational Cherednik algebra H c .Its blocks are in bijection with the fiber of Υ c at the origin-which in turn is precisely the set of C × -fixed points of Z c -and the set of simple modules of H c is naturally in bijection with the set Irr(W ) of irreducible complex characters of W .In particular, the blocks of H c yield a partition of Irr(W ).These are the Calogero-Moser families.They are (conjecturally) related to Rouquier families defined by the associated Hecke algebra, see [GoMa,Mar1,Chl4], which generalize Lusztig's families of unipotent characters associated with finite Coxeter groups [Lus2,Lus3].Rouquier and the first author [BoRo1,BoRo2] noticed that the Galois group of the Galois closure R c → P of the covering Z c → P acts (non-canonically) on the set W .By considering orbits in W under various subgroups of this Galois group, they obtained a decomposition of W .These are the Calogero-Moser cells which are conjectured to coincide with Kazhdan-Lusztig cells and thus potentially provides a way to generalize them to complex reflection groups.This conjecture was recently proven to be true for the symmetric group [BrGoWh].Associated to each Calogero-Moser left cell [BoRo2] constructed a cellular character: several descriptions are available and, in this paper, we will use the one involving Gaudin operators.They are expected to coincide with Lusztig's constructible characters [Lus3] for Coxeter groups.
1.A. Algorithms.-According to the Shephard-Todd classification [ShTo], the class of irreducible complex reflection groups splits into two classes: the infinite series of groups G(d, m, n), which are normal subgroups of the wreath product C d ≀ S n , and 34 exceptional groups G 4 , . . ., G 37 .Whereas the infinite series shows a common combinatorial pattern, each of the exceptional groups is essentially unique and often needs separate treatment.
It is especially because of the exceptional groups that we wanted to find algorithms allowing us to explicitly compute invariants of the associated Calogero-Moser spaces and rational Cherednik algebras so that we can test conjectures and gather data to drive the development of the theory.An important aspect of our approach is that we wanted to compute invariants for all parameters c.This is why we usually work with the generic rational Cherednik algebra H (and its center Z) defined over a polynomial ring, i.e. our parameters c are indeterminates.
In this paper we present algorithms for computing the following: 1.The inverse of the natural truncation map Trunc : Z → C[ C× V × V * ] W , which is an isomorphism of C[ C]-modules (Algorithm 3.7).Our algorithm iteratively deforms an element of C[ C × V × V * ] W to an element of Z. 2. A minimal system of algebra generators of Z (Algorithm 3.10).3. A presentation of Z and thus of the Calogero-Moser space (Algorithm 3.14).4. The Calogero-Moser families (Algorithm 5.7). 5. ).The complement of this hyperplane arrangement is precisely the locus where the Calogero-Moser families remain generic.These hyperplanes play a similar role as the essential hyperplanes for Rouquier families [Chl4].6. Cuspidal families (Algorithm 5.17).These are the families lying on a zero-dimensional symplectic leaf of Z c , see [Bel1, BelThi].7. Cellular characters (Algorithm 6.2).Our approach uses the description of cellular characters via Gaudin operators from [BoRo2].
1.B.Results and applications.-Our algorithms are not just theoretical.We have implemented all of them in the Cherednik Algebra Magma Package (CHAMP) created by the second author [Thi1], which is based on the computer algebra system MAGMA [Magma] and is available at https://github.com/ulthiel/Champ .Because of the complexity and wealth of data-there are many invariants and all of them depend on parameters-we can in this paper only give a brief summary of what is now actually computable.Some of the data is organized in a database which is accessible from within CHAMP.The most striking results and applications presented in this paper are the following: 1.An explicit minimal system of algebra generators of Z is now known for all exceptional complex reflection groups except G 16 − G 22 and G 27 − G 37 ; moreover, all algebra generators of Z-degree 0 are known also for G 27 and G 28 (see Section 3.E).Knowing the degree-0 generators is the key to the following results on families and hyperplanes.2. The Calogero-Moser hyperplanes are now known for all exceptional complex reflection groups except G 16 − G 19 , G 21 , and G 32 (see Table 5.C).Applying results of [Nam2,BST2] we thus know in all these cases the chamber decomposition of the movable cone of a Q-factorial terminalization (i.e. a relative minimal model) of the associated symplectic singularity (V × V * )/W and we know the number of nonisomorphic relative minimal models.Particularly exciting is that we were able to determine this for the Weyl group G 28 = F 4 (the computation took several days).3. The Calogero-Moser families are now known for all exceptional complex reflection groups (and all parameters) except G 16 − G 19 , G 21 , G 29 − G 37 .In all cases we confirmed a conjecture by Martino [Mar1] stating that the Calogero-Moser families are unions of the Rouquier families (Theorem 5.15).Particularly exciting is again the case G 28 = F 4 where the Rouquier families are the same as the Lusztig families and the equality to Calogero-Moser families was conjectured previously by We recall some basics about Calogero-Moser spaces and rational Cherednik algebras from [EtGi], keeping the notation of [BoRo2].Throughout this paper, we abbreviate ⊗ C by ⊗.All varieties are complex algebraic, and for an affine variety X we denote its coordinate ring by C[X].For a group W we denote by Irr(W ) the set of complex irreducible characters of W .

2.A. Reflection groups.
-We fix in this paper a complex vector space V of finite dimension n and a finite subgroup W of GL(V ).For an element w ∈ W we denote by V w the fixed space of w.By we denote the set of reflections in W .We assume throughout this paper that W is generated by its reflections, i.e.W is a reflection group.We denote by ε : W → C × , w → det(w) , the determinant character of W . Considering V and V * as algebraic varieties, we identify the coordinate ring Here, (V s ) ⊥ denotes the space of linear forms whose kernel contains V s , and (V * s ) ⊥ is defined analogously.
We denote by A the set of reflecting hyperplanes of W , namely We set e H = |W H |. Note that W H is cyclic of order e H and that If Ω is a W -orbit of reflecting hyperplanes, we write e Ω for the common value of all the e H , where H ∈ Ω.We denote by ℵ(W ) the set of pairs (Ω, j) where Ω ∈ A/W and 0 j e Ω − 1.Let K denote the space of maps

2.B. Parameters
is an isomorphism and we denote by κ : C → K its inverse.Throughout the paper, we will identify C and Kthrough these isomorphisms.

2.C.
Rational Cherednik algebras at t = 0. -We follow the convention of [BoRo2] and define the generic rational Cherednik algebra H (at t = 0) to be the quotient of the , the second factor being the semi-direct product of the tensor algebra T(V × V * ) with the group W , by the relations Here , : V × V * → C is the standard pairing.From (2.1), one deduces an equivalent presentation, namely The first commutation relations imply that we have morphisms of algebras [EtGi,Theo. 1.3] we have an isomorphism of C-vector spaces induced by multiplication.This is the so-called PBW-decomposition of H.

2.D. Calogero-Moser space.
-We denote by Z the center of the algebra H.It is wellknown [EtGi] that Z is an integral domain which is integrally closed.Moreover, Z contains C[V ] W and C[V * ] W as subalgebras [Gor], hence it contains , and Z is a free P-module of rank |W |.We denote by Z the affine algebraic variety whose ring of regular functions C[Z] is Z.This is the generic Calogero-Moser space associated with (V, W ). It is irreducible and normal.
We set For c ∈ C we denote by 2.E.Extra-structures on the center.-The center Z is endowed with extra-structures (a bi-grading, a Poisson bracket, and a special central element) which are described below.

2.E.1. Gradings and C
graded in such a way that the generators have the following bi-degrees This descends to an (N × N)-grading on H because the defining relations (2.2) are bihomogeneous.As a consequence, Z is also (N × N)-graded.This bi-grading induces a Z-grading on H and Z, by saying that a bi-homogeneous element of bi-degree (i, j) has Z-degree i − j.If c ∈ C, then the ideal C c is not bihomogeneous (except if c = 0) but is Z-homogeneous.Therefore, H c and Z c do not inherit a bi-grading, but they inherit a Z-grading.Note finally that, by definition, The Z-grading on Z and on P induces a C × -action on Z and on P, and the morphism Υ : 2.E.2.Poisson structure.-Let t ∈ C. One can define a deformation H t of H as follows: Recall that eu does not depend on the choice of the basis of V .Also (2.8) eu ∈ Z, Frac(Z) = Frac(P) [eu] , and (2.9) {eu, z} = dz if z ∈ Z is Z-homogeneous of degree d (see for instance [BoRo2,Prop. 3.3.3]).

2.F. Symplectic singularities and symplectic leaves. -
The following concept has been introduced by Beauville [Bea]: a normal variety X is said to have symplectic singularities if its smooth locus X sm carries a symplectic form ω and for some (thus any) resolution of singularities π : Y → X the pull-back π * (ω) defined on π −1 (X sm ) ⊆ Y extends to a (possibly degenerate) 2-form on all of Y .
For any c ∈ C the Calogero-Moser space Z c is a variety with symplectic singularities by [Gor].Moreover, Z c admits a stratification into smooth symplectic subvarieties called the symplectic leaves of Z c , see [BrGo].If m denotes the maximal ideal in Z c corresponding to a point p ∈ Z c , then the symplectic leaf L c (p) containing p is the subvariety defined by the largest ideal I contained in m which is Poisson, i.e. {Z c , I} ⊂ I.
The zero-dimensional symplectic leaves are also called cuspidal points.A cuspidal point p ∈ Z c corresponds to a maximal ideal m in Z c such that {m, m} ⊂ m.It follows that the cotangent space m/m 2 in p inherits a structure of a Lie algebra from the Poisson bracket.We will denote this Lie algebra by Lie c (p).

2.G. CHAMP. -
The rational Cherednik algebra is implemented in CHAMP.This means concretely that it is realized as an algebra with a basis given by the PBW-decomposition (2.4) and that products of elements will be rewritten in the basis (the algorithm is described in [Thi1]).Instead of working over the complex numbers it is (mostly) sufficient to work over the defining field of the complex reflection group W , which is a number field.CHAMP comes along with an extensive documentation and we can just give a minimal impression with the simple examples included in this paper.We note that because all actions in MAGMA are from the right, it was conceptually more consistent to implement in CHAMP the opposite algebra of the rational Cherednik algebra as defined here-this is only relevant when translating elements between paper and CHAMP.
Example 2.10.-We verify that the Euler element in the generic rational Cherednik algebra for the Weyl group of type B 2 is indeed central (this simply works by checking whether the element commutes with the generators x j , y j and with a set of generators of the group).

Generators and presentation of the center
We describe algorithms for computing a minimal system of algebra generators and a presentation of the center Z of H. Since Z c = Z/C c Z, we automatically obtain in this way also a (minimal) system of generators and a presentation of Z c for any c ∈ C. We use these algorithms in the sequel to obtain new results about the Calogero-Moser space for G 4 (Section 4) and to compute families and hyperplanes (Section 5) for many exceptional complex reflection groups.

3.A.
The truncation map.-The backbone of our algorithms is the computation of the inverse of the truncation map: by the PBW-decomposition there is for any h ∈ H a unique family of elements then there exists a unique element z = w∈W z w w of Z (where W is a P-linear map between two free P-modules of rank |W |.Since Z/C 0 Z = Z 0 , it is surjective modulo C 0 .By the graded Nakayama lemma (see for instance [Bro,Lemma 2.24]) applied to graded modules over the graded ring C[ C], this shows that it is surjective.As any surjective map between finitely generated free modules of the same rank is an isomorphism, the proof is complete.
Note for future reference the following consequence: Proof.-The map Trunc is C[ C]-linear, so it is sufficient to prove the result for m = 0.But this follows from the fact that Z/C 0 Z = Z 0 .
We shall now describe how to compute the inverse map A proof of this relation can be found in [Gor,§3.6]We define Let y reg ∈ V reg and use the notation of Lemma 3.1.Then [y reg , z] = 0 and, by (3.3), we get Therefore, for all w ∈ W .If we denote by z <r> w the reduction modulo for all w ∈ W .This formula yields the following algorithm for computing (1) For all w ∈ W , we define inductively a sequence (z Note that, at each step, z

[r]
w is bi-homogeneous of bi-degree (d, e).
(2) The formula (3.5) implies that the right-hand side of the above formula is divisible by y reg − w(y reg ), which is non-zero.It also implies that z [r] The result is the Euler element.Recall that CHAMP implements the opposite of the rational Cherednik algebra and therefore the variables in f are reversed.-Now that we can compute Trunc −1 it is easy to find a system of generators of Z. Proposition 3.9.-Let (z i and so the result follows from the graded Nakayama Lemma.

Algorithm 3.10. Minimal system of generators
A minimal system of generators of Z is obtained as follows: (1) Compute a minimal system of bi-homogeneous generators (z this is "just" computational invariant theory and there are algorithms for this in MAGMA already, specifically the algorithm [Kin] by King which is implemented via the command FundamentalInvariants).
(2) Use Algorithm 3.7 to compute Then (z i ) i∈I is a minimal system of bi-homogeneous generators of Z.
Example 3.11.-We continue Example 2.10 and compute a minimal system of generators of Z.In CHAMP, a minimal system of bi-homogeneous generators of C[V × V * ] W is computed via the command SymplecticDoublingFundamentalInvariants and the computation of the generators of Z via Algorithm 3.10 proceeds and stores the elements in the same order.In this example there are 8 generators, their bi-degrees are as listed below.
Remark 3.12.-We discovered a bug in the computation of fundamental invariants in MAGMA that resulted in the return value not necessarily being a minimal system of generators (which contradicts the definition of a system of fundamental invariants and also contradicts the algorithm in [Kin]).After reporting this bug, we were informed by MAGMA developer Allan Steel that this bug existed since V2.22 (May 2016).It was fixed after our report in V2.26-9 (October 2021).

3.C. Presentation. -
We now come to the computation of a presentation of Z, i.e. to the computation of the relations between a (minimal) system of generators of Z.
We fix a minimal system of bi-homogeneous generators (z We fix a family of indeterminates (z i ) i∈I and we denote by We endow C[(z i ) i∈I ] with the (N × N)-grading such that z i has the same bi-degree as z i (or z (0) i ), so that π 0 and π are bi-graded morphisms.By definition, π 0 is surjective and by Proposition 3.9, π is surjective.Our aim is to compute a (minimal) set of generators of Ker(π).We first explain how to compute preimages under the surjective morphism π.

Algorithm 3.13. Preimages under π
Let z ∈ Z be bi-homogeneous of bi-degree (d, e) and let δ = min(d, e).A bihomogeneous element of bi-degree (d, e) in π −1 (z) is computed by induction on δ as follows: ( bi-homogeneous such that π 0 (F 0 ) = z 0 (this is "just" computational commutative algebra and there are algorithms for this in MAGMA already, specifically one can compute the relations between a system of generators of given by primary invariants and irreducible secondary invariants via the command Relations and can translate this to relations between the fundamental invariants using the function HomogeneousModuleTest).
. By induction, we can find a family e) and its image under π is equal to z.
Example 3.16.-We continue Example 2.10 and compute a presentation of Z.In CHAMP, the command SymplecticDoublingInvariantRingPresentation computes the relations of the invariant ring and the computation of the relations for Z via Algorithm 3.14 proceeds and stores the relations in the same order.In this example, Z is a quotient of a polynomial ring in 8 variables (corresponding to the 8 elements in the minimal system of generators that we computed previously) by an ideal generated by 9 relations.(1) Compute the commutator [u t , v t ] in H t , see Section 2.E.2, and project to H by specializing t = 0. Denote this element by w.Note that w ∈ Z.
(2) Compute w ′ = π −1 (z 3 ) using Algorithm 3.13 Then w ′ is an expression of {u, v} in terms of the generators (z i ) i∈I of Z.
Example 3.18.-We continue Example 2.10 and compute the Poisson brackets {z i , z j } of all generators z i , z j of Z.This is done in CHAMP with the command PoissonMatrix.We note that because of the opposite algebra convention, the Poisson brackets in CHAMP are the negatives of what we have on paper.Since z 2 = eu by Example 3.11, the second row of the matrix gives the (negatives) of the Poisson brackets {z i , eu}.Recall that {z i , eu} = deg Z z i by (2.9).
3.E.Summary of what is computable.-Our Algorithm 3.7 for computing the inverse of the truncation map performs very well in practice.In particular, we were able to compute a complete minimal system of generators of Z for all exceptional complex reflection groups except G 16 − G 22 and G 27 − G 37 .All these elements are stored in the database of CHAMP and can be loaded quickly.To give an impression of the performance-but also of the complexity involved in these computations-we note that we were able to compute, e.g., the inverse image of a generator z (0) i of (N × N)-degree (6, 6) for the Weyl group of type F 4 (equal to the exceptional complex reflection group G 28 ) which has 6,010 monomials and the total number of monomials in the various coefficients of the inverse image z i in Z is 569,936.This single computation took about 6 hours.
The higher the degree of the element of which we want to compute the inverse image, the more iterations Algorithm 3.7 will take-and each iteration will be more complicated than the previous.This is one of the reasons why we were not able to compute a complete system of generators of Z for all the exceptional complex reflection groups.Another problem arises even before we can come to the iteration part of the algorithm: the invariant theory of the action of W on V × V * is very complicated and for the higher rank cases we were not even able to compute a system of generators of C[V × V * ] W .The first in the series of exceptional complex reflection groups where this was not possible anymore is the group G 30 , which is also the Coxeter group of type H 4 and therefore very interesting.
In our algorithm for computing families and hyperplanes that we present in Section 5 we do not need a complete (minimal) system of generators of Z but only the sub-system consisting of generators of Z-degree 0. We were able to compute this also for G 27 and G 28 .Unfortunately, for the interesting case G 30 we are so far not able to do this because we do not even know how many generators of Z-degree 0 there are in the invariant ring.We think this case poses a very interesting problem-both theoretically and computationally.
Finally, we note that computing a presentation of Z via Algorithm 3.14 is computationally very challenging and works only in the smallest cases: dihedral groups up to order 16 and the exceptional group G 4 .Nonetheless, even these few cases led to important insight and results, see Section 3.F and Section 4. In Table 3.E we provide an overview of the performance of the algorithms in the case of small dihedral groups.

3.F. Case of dihedral groups.
-We mention some new results about the Calogero-Moser space for dihedral groups that grew out of the fact that we were able to compute an explicit presentation of Z in small cases (see Table 3.E).
Assume that dim C V = 2 and that W is the dihedral group of order 2d, with d 3. Write W = s, t where s, t are Coxeter generators.Fix c ∈ C and write a = c s and b = c t .
F G 3 0.02s 7 1 0.02s 0.03s 5 0.05s 4 0.02s 8 2 0.02s 0.04s 9 0.42s 5 0.03s 9 2 0.15s 0.09s 14 4.33s 6 0.01s 10 3 0.12s 0.18s 20 29.9s 7 0.07s 11 3 1.38s 0.39s 27 211s 8 0.06s 12 4 4.00s 15.5s 35 9222s  We assume that ab = 0. Recall that Z c has dimension 4 and that a = b whenever d is odd.Note the following facts: • If d 4 and a = b, then Z c has a unique singular point p, which is necessarily cuspidal [Bon1,Tab. 5.2].• If d 6 is even and a = b, then Z c has a unique singular point q, which is necessarily cuspidal [Bon1,Tab. 5.2].Recall from Section 2.F that we denote by Lie c (p) the Lie algebra in a cuspidal point p of Z c .If g is a simple Lie algebra, we denote by O min (g) the minimal nilpotent orbit of g.The following facts have been proved in [Bon1,Prop. 8.4 and 8.8] and [BBFJLS], and rely on the fact that we were able to compute an explicit presentation of the center Z: The main application of our algorithms in case of dihedral groups is given by the next theorem [BBFJLS].It has been obtained through an explicit description of a presentation of Z c whenever a = b for any d 3 (see [Bon2]) which was obtained after computing the cases 3 d 8 with our algorithms and finding some general patterns.So, even though this does not appear finally in the proof of the main results in [Bon2] and [BBFJLS], it is fair to say that these two papers owe their existence to the algorithms developed here.Before stating the result, let us introduce some notation: we set This gives a new family of examples of isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville [Bea,(4.3)].

The Calogero-Moser space for G 4
In this section, we use our algorithms to investigate Calogero-Moser spaces for the exceptional complex reflection group G 4 : we give an explicit presentation, identify the symplectic singularity in the origin, and confirm a conjecture about symplectic leaves.

4.A. The group and parameters
where ζ is a primitive third root of unity.Then W is a primitive complex reflection group of type G 4 .Note that (4.1) We denote by (y 1 , y 2 ) the canonical basis of C 2 and by (x 1 , x 2 ) its dual basis.Let H s denote the reflecting hyperplane of s (note that |A/W | = 1 and e Hs = 3).For simplification, we set K j = K Hs,j .Recall that K 0 + K 1 + K 2 = 0. We also fix c ∈ C and set k = κ(c) for the corresponding parameter in K (see Section 2.B).For simplification, we set k j = k Hs,j .Note that k 0 + k 1 + k 2 = 0.

4.B. Presentation of Z.
-The computation of a presentation of Z in CHAMP takes about 2 minutes and is performed by the following commands: > W:=ComplexReflectionGroup(4); > H:=RationalCherednikAlgebra(W,0 : Type:="BR-K"); > Zpres:=CenterPresentation(H); The presentation is also stored in the database and can be retrieved in an instance.This returns a family (z j ) 1 j 8 of 8 generators of Z k and a family (E ′ j ) 1 j 9 of 9 equations.To describe this presentation explicitly, we rename and reorder the generators as follows: Moreover, we replace the equations (E ′ j ) by the equations (E j ) defined as follows: We then obtain the following theorem.
Theorem 4.2.-We have: (b) The Z-degrees of the generators are given by The C[ C]-algebra Z admits the following presentation: (0, 0, 0).Since the equations are symmetric in the k i 's, we must focus on the case where k 1 + k 2 = 0 and, after multiplying by a scalar, the only possible case is the one considered in the above Theorem.
Proof.-The proof is obtained via explicit computations.First, we specialize the presentation of Z in the fixed parameter c to obtain a presentation of Z c and then create Z c as a scheme: Next, we compute the singular locus of Z c as a (reduced) scheme: > Zcsing := SingularSubscheme(Zc); > Zcsing := ReducedSubscheme(Zcsing); > Zcsing := Scheme(A8,MinimalBasis(Zcsing)); Now, we show that 0 ∈ Z c ⊂ C 8 is the unique singular point of Z c .

4.D.
Confirming a conjecture about symplectic leaves.-For the moment, let W be an arbitrary complex reflection group.For a variety X we denote by X nor its normalization.If moreover X is affine and irreducible, then X nor is also affine and C[X nor ] is the integral closure of C[X] in its fraction field C(X).
We fix an element of finite order τ of N GL C (V ) (W ) and a parameter c ∈ C such that τ c = c.Then τ also acts on the Calogero-Moser space Z c and we denote by Z τ c the closed subvariety of fixed points under the action of τ (endowed with its reduced structure).As explained in [Bon3], the variety Z τ c inherits a partition into symplectic leaves and we recall a conjecture about their structure [Bon3, Conj.B]: Conjecture 4.6 (Bonnafé).-Let L be a symplectic leaf of Z τ c and let L denote its closure in Z τ c .Then L nor is isomorphic, as a Poisson variety endowed with a C × -action, to some Calogero-Moser space associated with another pair (V L , W L ) and some parameter k L ∈ ℵ(W L ).
In fact, [Bon3, Conj.B] is somewhat more precise, as it explains how to recover the pair (V L , W L ).However, the parameter k L is quite a mystery.Note that, even if τ = Id V , this conjecture is still unproved.Let us give some known cases: c is also smooth and its symplectic leaves are its irreducible components.It has been proved in [BoMa,Theo. 1.3] that, whenever τ is the scalar multiplication by a root of unity, then these irreducible components are isomorphic, as varieties endowed with a C × -action, to Calogero-Moser spaces.This proves part of Conjecture 4.6 (it remains to check that the isomorphism constructed in [BoMa,Theo. 1.3] respects the Poisson structure).Note however that precise formulas are given for the value of k L in [BoMa,Theo. 4.21(b)].
• If W is a Weyl group of type B and τ = Id V , then Conjecture B is proved by Bellamy-Maksimau-Schedler in [BeMaSc].As explained in [Bon3,Coro. 10.7], we can easily deduce from their result that this also implies Conjecture B for Weyl groups of type D, with τ being the identity or a non-trivial involutive graph automorphism.
We will now add one more example to this list: Theorem 4.7.-Assume that W is of type G 4 .Then Conjecture 4.6 holds.
Proof.-We assume that W is the group G 4 as in Section 4.A.Note that W acts trivially on Z c , so replacing τ by wτ for some w ∈ W , does not change the fixed point subvariety.Therefore, since the normalizer of W is W • C × , we may, and we will, assume that τ is the scalar multiplication by a root of unity.Since − Id V ∈ W , we may also assume that the order of τ is even: let us denote it by 2d.Then and so Z τ c is zero-dimensional.In this case, there is nothing to prove.This means that we may, and we will, assume that d ∈ {1, 2, 3}.
If d ∈ {2, 3}, then the computation of Z τ c has been done in [BoMa,§5] and it has been checked in both cases that there is only one irreducible component of dimension 2 (the other being of dimension 0) and it has been checked that it is isomorphic, as a variety endowed with a C × -action, to the Calogero-Moser space associated with the pair (C, µ 2d ), as expected from [Bon3,Conj. B].However, it has not been checked in [BoMa,§5] that the isomorphism respects the Poisson bracket, but this can be easily done in CHAMP.We will explain how precisely in the more difficult situation where d = 1, see below.
Consequently, this means that we may assume that d = 1, so that Z τ c = Z c .We will now prove that the symplectic leaves of Z c satisfy Conjecture 4.6.
Let L be a symplectic leaf of Z c .First, note that dim(Z c ) = 4 so that L has dimension 0, 2 or 4. If dim(L) = 4, then L = Z c and there is nothing to prove.If dim(L) = 0, then L is a point and there is again nothing to prove.So we may assume that dim(L) = 2.Note also that we may assume that k = 0 since the case k = 0 has been treated for any group.
By the theory of Bellamy [Bel1,Prop. 4.9], L is parametrized by a pair (P, p) where P is a parabolic subgroup of W and p is a cuspidal point of Z c (V /V P , P ).Since dim(L) = 2, [Bel1,Prop. 4.7] forces dim(V P ) = 1 and so we may assume that P = s .But, by [BoRo2,§18.5.A], Z c (V /V P , P ) admits a cuspidal point if and only if at least two of the k j 's are equal.As the isomorphism class of Z c depends on k 0 , k 1 , k 2 only up to permutation, we may assume that k 1 = k 2 .Since k = 0 and k 0 + k 1 + k 2 = 0 and since we can rescale the parameters, this means that we may assume that As in the proof of Theorem 4.4 and continuing with these computations, we create Z c as a scheme in CHAMP as follows: We conclude that Z sing c is irreducible over Q but not necessarily geometrically irreducible.However, this also implies at least that it is of pure dimension 2. But its irreducible components (over C) are symplectic leaves of dimension 2: since Z c (V /V P , P ) has only one cuspidal point, [Bel1,Prop. 4.9] implies that Z c has only one symplectic leaf of dimension 2, and this forces Z sing c to be geometrically irreducible.In other words, Z sing c = L. Since N W (P )/P ≃ µ 2 acting on V P ≃ C by scalar multiplication, we need to find k L ∈ ℵ(N W (P )/P ) = ℵ(µ 2 ) and a C × -equivariant isomorphism of Poisson varieties Looking at the equations of Z c in CHAMP we see that among them we have the two following ones: We create the affine scheme L 0 with coordinate ring A in CHAMP as follows: > A6<A,B,C,X,Y,E>:=AffineSpace(Rationals(),6); > L0:=Scheme(A6,[Evaluate(f,[X^2,Y^2,X^3,Y^3,A,B,C,E]) : f in Basis(Ideal(Zcsing))]); We see that L 0 is not reduced and define L 1 to be its reduced subscheme: We see that A, B and C can be expressed as polynomials in X, Y and eu.Therefore, L 1 ≃ {(x, y, e) ∈ C 3 | (e − 6)(e + 12) = xy}.As it is a smooth, hence normal, variety this shows that L 1 = L nor .In other words, setting e ′ = e + 3 we obtain: Note also that X and Y have degree 2 and −2 respectively.Using the description of Calogero-Moser spaces in rank 1 given by [BoRo2,Theo. 18.2.4],we get that we have a C × -equivariant isomorphism of varieties (4.9) L nor ∼ −→ Z k L (C, µ 2 ).It remains to show that this isomorphism respects the Poisson bracket.For degree reasons, it respects the Poisson bracket with the Euler element, see (2.9).So, it remains to compute the Poisson bracket {X, Y } on both sides.On the right-hand side of (4.9), we have {X, Y } = −4eu # + 12, where eu # denotes the image of eu under the isomorphism (4.9).On the left-hand side of (4.9), we have ] is an integral domain, it is sufficient to show that ].The matrix whose entries are the Poisson brackets {u, v} for u and v running over the generators X 1 , Y 1 , X 2 , Y 2 , A, B, C, eu of Z c is obtained in CHAMP using Algorithm 3.17 as follows (recall that Poisson brackets in CHAMP are the negatives of our Poisson brackets because of the opposite algebra convention): ,8,8,phi)) : f in Eltseq(pmatZ) ]); //Permute rows and columns to account for change in ordering of generators > pmatZc := Permute(pmatZc, [2,5,6,8,3,4,7,1]); Finally, we confirm that equation (?) indeed holds: This completes the proof.

Families and hyperplanes
In this section we describe algorithms for computing the Calogero-Moser families (which correspond to the C × -fixed points of Z c ) and an associated hyperplane arrange- ment which plays an important role in geometry (Section 5.D) and representation theory (Section 5.E).We furthermore give an algorithm for computing the cuspidal points (i.e. the zero-dimensional symplectic leaves) of the Calogero-Moser space.

5.A.
Restricted algebras, families, and hyperplanes.-Throughout, let W be an arbitrary complex reflection group.Let m 0 be the maximal ideal in The algebras H c were studied by Gordon [Gor].
The PBW-decomposition of H induces a triangular decomposition of H c and this leads to a theory of standard modules similar to the theory for finite-dimensional complex semisimple Lie algebras.Specifically, for each λ ∈ Irr(W ) there is an associated H cmodule ∆ c (λ) which has simple head L c (λ) and {L c (λ) | λ ∈ Irr W } is a complete set of representatives of isomorphism classes of simple H c -modules.The details are given in Gordon [Gor].Since H c is a finite-dimensional C-algebra, it has a block decomposition and each simple H c -module belongs to exactly one block.Via the bijection λ → L c (λ) this block decomposition yields a partition of Irr W that we denote by CM c .The parts of this partition are called the Calogero-Moser c-families.
Let Υ c : Z c → P c be the fiber of the morphism Υ : Z → P from (2.5) in c, i.e.Υ c is the morphism defined by the inclusion P c ⊆ Z c .This morphism is C × -equivariant and by [Gor] the blocks of H c , and thus the Calogero-Moser c-families, are naturally in bijection with the C × -fixed points of Z c .Denoting by Z C × c the set of C × -fixed points of Z c we define (5.1) The set C CM is known to be a union of (finitely many) hyperplanes (see below): these are the Calogero-Moser hyperplanes.
Remark 5.2.-The original purpose of CHAMP, as presented in [Thi1], was to compute decomposition matrices of standard modules for restricted rational Cherednik algebras-from which one can easily deduce the Calogero-Moser families.The database of CHAMP contains a wealth of such information for several exceptional complex reflection groups.We note that the algorithm for computing Calogero-Moser families we will present below (Algorithm 5.7) is much more efficient and allows to cover many more cases than the computation via decomposing standard modules (which, on the other hand, yields much more information).

5.B. Families and hyperplanes via central characters. -
The Calogero-Moser families and hyperplanes admit an explicit description via central characters that-combined with the fact that we can compute generators of Z via Algorithm 3.10-immediately yields an algorithm for computing these invariants.We denote by linear map which, through the isomorphism (2.4) of vector spaces given by the PBW-decomposition, sends an element ), and which is the identity on C[ C] and CW .This map is W -equivariant, so its restriction to Z induces a map then the C-families are unions of C ′ -families, i.e. specializing yields coarser families.In particular, the number of C ′ -families is less than or equal to the number of C-families.Let (5.4) be the locus of all C where the C-families are not equal to (and thus coarser than) the generic families.It follows from [Thi2,Theorem 1.3] that this locus is a closed subset of Spec(C[ C]) which is either empty or pure of codimension 1.It is obviously contained in for some χ, χ ′ }.This set is the union of the zero sets of the polynomials Ω χ (eu It thus follows that C Eu is a union of (finitely many) hyperplanes: we call them the Euler hyperplanes.In particular, C Eu is non-empty and thus pure of codimension 1.This shows that there is c ∈ C such that the c-families are equal to the generic families and this implies that the set C CM from (5.1) is equal to (the set of closed points of) C CM .It has been proven in [BoRo2,Coro. 7.8.3] that C CM is a union of Euler hyperplanes.In particular, C CM is a union of hyperplanes.
Given two partitions P, P ′ of Irr W we denote by P ∧ P ′ the partition of Irr W whose parts are the unions of all parts of P and P ′ having non-empty intersection.Let H 1 , . . . ,H r be the Calogero-Moser hyperplanes and for each j let C j be the corresponding prime ideal in C[ C].It follows from [Thi2,Lemma 3.6] that for c ∈ C CM we have (5.5) i.e. the c-families are obtained by taking the meet of the C j -families for all i such that c is contained in H i .
Our discussion yields an immediate algorithm for computing the Calogero-Moser hyperplanes and the c-families for all parameters c ∈ C at once.Before stating this we record an elementary fact that simplifies the computations.
Proof.-Indeed, the morphism Algorithm 5.7.Families and hyperplanes (1) Let (z (2) Compute the generic Calogero-Moser families: these are the fibers of the map (3) Compute the Calogero-Moser hyperplanes: for any two characters χ, χ ′ with Ω χ = Ω χ ′ compute the primary decomposition of the radical of the ideal in C[ C] generated by the elements Ω χ (z i ) − Ω χ ′ (z i ) for i ∈ I 0 .In MAGMA this can be done via the command RadicalDecomposition, which is more efficient than first computing the radical and then computing its primary decomposition.The collection of the hyperplanes obtained in this way are the Calogero-Moser hyperplanes.(4) Compute the Calogero-Moser families on the hyperplanes: let H 1 , . . ., H r be the Calogero-Moser hyperplanes and for each j let C j be the corresponding prime ideal in C[ C].The C j -families are the fibers of the map (5) For arbitrary c ∈ Cthe Calogero-Moser c-families are computed as follows.Since C CM was computed in (3), we can test whether c ∈ C CM .If c / ∈ C CM , the c- families are equal to the generic families computed in (2).If c ∈ C CM , the c- families are given by where CM C j was computed in (4).

5.C. Summary of what is computable.
-Remember from Section 3.E that we could compute (z i ) i∈I 0 for all exceptional complex reflection groups except G 16 − G 22 and G 29 − G 37 .In all these cases, we could successfully run Algorithm 5.7 and we thus know the Calogero-Moser hyperplanes and the c-families for all c ∈ C. The results are contained in the database of CHAMP.It is particularly exciting that we were able to compute this for G 28 because it is a Weyl group (of type F 4 ).Table 5.C gives a brief summary of results about the Calogero-Moser hyperplane arrangement for exceptional complex reflection groups (we have published this table previously in [BST2]).In the following two subsections we mention important results that follow from these computations.

Group #Hyp
Poincaré polynomial . Some data about the Calogero-Moser hyperplane arrangement for exceptional complex reflection groups: the number of hyperplanes, the Poincaré polynomial of the arrangement, and the number of Q-factorial terminalizations of the associated symplectic singularity (see Section 5.D).We note that the groups G 12 , G 22 -G 24 , G 27 , G 29 -G 31 , and G 33 -G 37 have a single conjugacy class of reflections so that C is 1-dimensional and C CM is just the origin.Hence, only for the exceptional complex reflection groups G 16 -G 19 , G 21 , and G 32 the Calogero-Moser hyperplanes are still unknown.

5.D.
Applications in birational geometry.-Let X = (V × V * )/W .This variety is normal.Recall that the vector space V × V * carries a natural symplectic form and the action of W on this space is symplectic.Since the symplectic group is contained in the corresponding special linear group, it follows from the Reid-Tai criterion that X has canonical singularities (see, e.g., [Kol,Theorem 3.21]).An application of [BCHM,Corollary 1.4.3]thus shows that X admits a Q-factorial terminalization, i.e. a crepant projective birational morphism π : Y → X from a normal Q-factorial variety Y with terminal singularities (the projectivity of the morphism follows from the proof of [BCHM,Corollary 1.4.3]).Since X has trivial canonical class by a result by Watanabe (see, e.g.[Bens,Theorem 4.6.2]), the Q-factorial terminalizations are the same as relative minimal models of X (relative to a log-resolution of X).Now, choose such a Q-factorial terminalization π : Y → X.For more details on the following we refer to [Nam2].The cohomology group formed by the π-movable line bundles.This cone decomposes into the ample cones of the various other Q-factorial terminalizations of X, and the codimension-1 faces of each of these ample cones generate a hyperplane arrangement in K R decomposing Mov(π) into various chambers.
It is shown in [Bel2] that K R can naturally be identified with the real vector space in- side the complex parameter space K spanned by the K Ω,j , i.e., K R = (Ω,j)∈ℵ(W ) RK Ω,j .Recall from Section 5.B that the Calogero-Moser hyperplanes are Euler hyperplanes.It is then clear that in the parameters K Ω,j the Calogero-Moser hyperplanes have rational coefficients and so they define a hyperplane arrangement in K R as well.Using the Poisson-geometric description of the chamber decomposition of the movable cone given in [Nam2], it was shown in [BST2] that these chambers coincide with orbits of the chambers of the Calogero-Moser hyperplane arrangement under the action of the Namikawa Weyl group on K R .This group was shown in [Bel2] to be simply the product of the full permutation groups on the indices j of the variables K R = (Ω,j) for fixed Ω.
To summarize, when we denote by E CM (W ) the number of chambers of the Calogero-Moser hyperplane arrangement inside K R and by QFT(W ) the number of Q-factorial terminalizations of (V × V * )/W , then (5.9) Recall that the number of chambers of a real hyperplane arrangement is equal to the evaluation at −1 of its Poincaré polynomial [Zas].Algorithms for computing the Poincaré polynomial are implemented in, e.g., the computer algebra system Sage [Sage].In all cases of the exceptional complex reflection groups where we could compute the Calogero-Moser hyperplane arrangement we were also able to compute the number QFT(W ) as well, see Table 5.C.We note that the computation of the Poincaré polynomial in case of G 11 in Sage took three weeks.
Remark 5.10.-It would be very interesting to explicitly construct a Q-factorial terminalization of (V × V * )/W .Among the exceptional groups, this has so far only been done for G 4 , see [LeSO], which is very special because it is the only exceptional group where the associated symplectic singularity admits a symplectic resolution, i.e. the Q-factorial terminalizations are smooth.Recently, an explicit presentation of a Q-factorial terminalization in the case of odd dihedral groups was constructed in [Sch,Corollary 7.3.13].
5.E.Applications in representation theory.-Several notions of families have been associated with real or complex reflection groups: they all form a partition of Irr(W ) and it is a natural question to understand, or compare, all these partitions.Fix c ∈ C and let k = κ(c) ∈ K. Let k ♯ = (k ♯ Ω,j ) (Ω,j)∈ℵ(W ) denote the element of K defined by k ♯ Ω,j = k Ω,−j (the indices j being viewed modulo e Ω ).
• If W is real (i.e. a Coxeter group) and c has real values (since reflections have order 2 in this case, this is equivalent to requiring that k has real values; note also that k = k ♯ in this case), there are two different notions: (L) The Lusztig c-families are defined using the notion of c-constructible characters of W built on the Lusztig a-function (see for instance [BoRo2,§15.3.B]).They have been explicitly computed by Lusztig in all cases [Lus3].Note that they are related to the representation theory of finite reductive groups [Lus2].(KL) The Kazhdan-Lusztig c-families are associated with the partition of W into Kazhdan-Lusztig c-cells, built from the Kazhdan-Lusztig basis of the Hecke algebra (see for instance [BoRo2,§8.6]).Lusztig conjectures that both notions coincide [Lus3] and this conjecture holds in many cases (for instance, if c is constant or if W is of type F 4 ).
• If W is a general complex reflection group, there are also two notions: (R) Broué-Kim [BrKi] have associated a partition of Irr(W ) into Rouquier k-families as blocks of the Hecke algebras of W with parameter k over a suitable ring (see for instance [BoRo2,Def. 6.5.1] for an extension of this definition to the case where k takes complex values: they are called Hecke k-families there).
Chlouveraki [Chl1,Chl2,Chl3,Chl4,Chl5] computed Rouquier k-families in all cases: as a consequence, she proved that, for W real, Rouquier k-families coincide with Lusztig c-families.(CM) The Calogero-Moser k-families defined by Gordon [Gor] and studied in this paper.Note that this last notion of families does not involve the Hecke algebra, in contrast to the first three ones.The Martino conjecture is known to be true for the whole infinite series of complex reflection groups G(m, p, n), see [Mar1,Mar2].The Gordon-Martino conjecture is known to be true for all Coxeter groups except possibly H 4 and E 6 -E 8 by [EtGi,GoMa,Gor,Mar2,Thi3]. Therefore, Rouquier families and Calogero-Moser families are two candidates for extending the notion of (Kazhdan-)Lusztig families to general complex reflection groups.However, it must be pointed out that Martino's Conjecture does not claim that Calogero-Moser families coincide with Rouquier families: indeed, there are cases where it is known that both notions do not coincide [Thi3].
We will show that both conjectures are true for all the exceptional complex reflection groups for which we could compute the Calogero-Moser families (see Section 5.C).Note that both conjectures involve an infinite range of parameters.To be able to algorithmically test the conjectures, we need to reduce this to a finite problem.This is possible due to the semi-continuity property of both types of families: in both cases there is a finite hyperplane arrangement (the Calogero-Moser hyperplanes for the Calogero-Moser families and the so-called essential hyperplanes for the Rouquier families [Chl4]) such that for all parameters outside these hyperplanes the respective families are always the same and for an arbitrary parameter the families are the smallest unions of the generic families on the hyperplanes containing the parameter.For the Calogero-Moser families this is precisely (5.5) and for the Rouquier families this property was established in [BoRo2].We thus arrive at the following algorithm: (2) For each Calogero-Moser hyperplane defined by a linear polynomial H in the variables k Ω,j consider the polynomial H ♯ obtained by replacing k Ω,j by k ♯ Ω,j = k Ω,−j .If H ♯ is an essential hyperplane, test whether the generic Rouquier families on H ♯ are unions of the generic Calogero-Moser families on H; otherwise test if the generic Rouquier families are unions of the generic Calogero-Moser families on H.If all tests are true, then the Martino conjecture is true (for all parameters).
By testing for equality instead of union we can also test the Gordon-Martino conjecture in the same way.We note that the Rouquier families and the essential hyperplanes for the exceptional complex reflection groups have been computed in [Chl4] and this data has been added to the database of CHAMP.

5.F. Cuspidal families.
-Recall from Section 2.F that the zero-dimensional symplectic leaves of Z c are also called cuspidal points.It was shown in [BelThi] that the cuspidal points are contained in Z C × c .Accordingly, we call a Calogero-Moser family cuspidal if the corresponding C × -fixed point in Z c is cuspidal.The cuspidal families are important because there is a kind of Harish-Chandra theory, developed by Bellamy [Bel1], which reduces the study of Calogero-Moser families to the cuspidal ones (for parabolic subgroups of W ).
For a Calogero-Moser c-family F we denote by Ω c F the map Ω c χ for one (any) χ ∈ F. F ({z i , z j })) = 0 for all i, j ∈ I.Note that, thanks to Lemma 5.6, we only need to compute the Poisson brackets {z i , z j } for the pairs (i, j) such that deg Z (z i ) + deg Z (z j ) = 0. We thus arrive at the following algorithm: Algorithm 5.17.Cuspidal families Let c ∈ C.
(1) Compute a system of generators (z i ) i∈I of Z via Algorithm 3.10.
(3) Compute the Poisson brackets {z i , z j } i,j∈I for all (i, j) such that deg Z (z i ) + deg Z (z j ) = 0 via Algorithm 3.17.Then the cuspidal Calogero-Moser c-families are precisely those families F for which Example 5.18.-We compute the cuspidal Calogero-Moser families for the dihedral group of order 8 in CHAMP and verify that in the equal parameter case there is a unique cuspidal family (see also Section 3.F):

Cellular characters
We fix c ∈ C. Calogero-Moser c-cellular characters have been defined in [BoRo2,Def. 1.8.4] but we will use here the equivalent definition given by [BoRo2,Theo. 8.3.2] in terms of Gaudin algebras.
generated by the D y 's (y ∈ V ): it will be called the generic Gaudin algebra associated with W . Recall that it is commutative Then γ Gau L is a Calogero-Moser c-cellular character of W , and they can all be obtained in this way [BoRo2,Theo. 13.2.10].Note moreover that W acts on Gau c (W ), so it acts on Irr(C(V )Gau c (W )) and (6.1) ) and all w ∈ W .

6.B.
Algorithm.-Following Appendix I (see Remark I.12), the following algorithm allows us to compute the Calogero-Moser c-cellular characters.For this, we fix a number field k and a k-form V k of V which is stabilized by W (this is done in order to perform exact computations).

Algorithm 6.2. Cellular characters
Let c ∈ C having values in k.The set of Calogero-Moser c-cellular characters is computed as follows: ( is the specialization at y of D (for all y ∈ V ).
(2) Let Π D denote the quotient of the characteristic polynomial ΠD of D by the greatest common divisor of ΠD and Π′ Hypothesis and notation.We fix in this appendix a finite dimensional k-vector space E, a split subalgebra A of End k (E) and an integral and integrally closed k-algebra P .Its field of fraction is denoted by K.
If R is a commutative k-algebra, we set RE = R⊗E and RA = R⊗A.We also fix a positive integer n and n commuting elements D 1 ,. . ., D n of End P (P E) (for 1 i n) which also commute with the action of P A. We aim to define a notion of cellular characters of A and study its behavior under specialization or field extension.The results presented here are certainly neither new nor original: they are just adapted to our situation, and stated here with proof for the sake of completeness.

I.
A. Cellular characters.-The K-algebra KA is split, so there is a bijection [CuRe,Propositions 3.56 and 7.7] where, for a finite dimensional algebra B, K 0 (B) is the Grothendieck group of the cate- gory of B-modules.But the K-vector space KE inherits an action of K where γ p S ∈ K 0 (KA) ≃ K 0 (A) (because A is split).
Definition I.2.-The element γ P S of K 0 (A) is called a D-cellular character.
It might be the case that the map is not injective.But we will only be interested in the image of γ P , and its behavior under specialization or field extension.This last situation is the simplest: Proposition I.3.-Let k ′ be an extension field of k and let P ′ = k ′ ⊗ P .We assume that P ′ is integral and integrally closed and we set K ′ = k ′ ⊗ K (it is then the fraction field of P ′ ).Then the images of the maps γ P and γ P ′ are equal.
I.B.The case n = 1.-We will first study here the case where n = 1.As it will be explained in the next subsection, the general case can in fact be reduced to this case.
Hypothesis and notation.We assume in this subsection, and only in this subsection, that n = 1 and we write D = D 1 .
The characteristic polynomial char D of D has coefficients in P .Using Euclid's algorithm in K[t], one can compute the polynomial char rad D (recall that it is the greatest common divisor of char D and its derivative char ′ D ) and so the polynomial char sem D = char D /char rad D is computable.We then write ) are all isomorphic to L i .Hence, if we denote by [ Ker(Π i (D) n i ) ] KA the image of Ker(Π i (D) n i ) in K 0 (KA), then this image is a multiple of dim(L i ) = deg(Π i ) and (I.7) Therefore: Proposition I.8.-With the above notation, the set of D-cellular characters of A is equal to 1 deg(Π i ) [ Ker(Π i (D) n i ) ] KA 1 i r .
We denote by ∆ the discriminant of the polynomial char D : then ∆ ∈ P and, if p is a prime ideal of P , we denote by ∆(p) the image of ∆ in P/p.Note that (I.9) ∆ = 0 because the factorization of char D into irreducible polynomials is multiplicity-free (and since k has characteristic 0).where π i,j ∈ k P (p)[t] is irreducible, e i,j 1 and π i,j = π i,j ′ if 1 j < j ′ d i .Then Since ∆(p) = 0, this means that e i,j = 1 for all i, j and that π i,j = π i ′ ,j ′ if and only if (i, j) = (i ′ , j ′ ).Now, let E = {(i, j) | 1 i r and 1 j d i } and, if (i, j) ∈ E, let .
Since k has characteristic 0, an equality between elements of the Grothendieck group is equivalent to an equality between the corresponding (virtual) characters.If we denote by χ i (resp.χ p i,j ) the character of L i (resp.L p i,j ), then χ i,j mod p.
Since χ i,j = χ i ′ ,j ′ if and only if (i, j) = (i ′ , j ′ ), the result follows from (I.1) by taking associated characters and reduction modulo p.
I.C.Back to the general case.-We come back to the initial set-up of this appendix.

Hypothesis and notation.
Until the end of this appendix, we no longer assume that n = 1.
The main result of this subsection (see Proposition I.11) shows that the computation of the D-cellular characters can be reduced to the case where n = 1, by extending the ring of definition P .Let X 1 ,. . ., X n be indeterminates over P and let P [X] = P [X 1 , . . ., X n ].We set Proof.-Let ∆ X denote the discriminant of char sem D (it is an element of P [X]).Since ∆ X = 0, there exists x = (x 1 , . . ., x n ) ∈ k n such that ∆ X (x 1 , . . ., x n ) = 0. Let D(x) = x 1 D 1 + • • • + x n D n ∈ End P (P E).It follows from Proposition I.10 that the set of D-cellular characters coincide with the set of D(x)-cellular characters (one must replace, in Proposition I.10, the ring P by P [X] and take p = X 1 − x 1 , . . ., X n − x n ).So it remains to prove that the set of D(x)-cellular characters coincides with the set of D-cellular characters.
Now, let M be an element of K[D], and let m j denote its j-th diagonal coefficient.If j, j ′ are such that λ(j) = λ(j ′ ), then necessarily m j = m j ′ .It then follows from ( * ) that there exists a polynomial f ∈ K[t] such that f ( n i=1 x i λ (i) j ) = m j for all j ∈ {1, 2, . . ., d}.Therefore, M − f (D(x)) belongs to K[D] and has all its diagonal coefficients equal to 0. This forces M − f (D(x)) ∈ Rad(K[D]), as desired.
Remark I.12.-Proposition I.11 together with Proposition I.8 provides a way to reduce the problem of computing D-cellular characters to the case n = 1.In this former case, using Proposition I.10, one can reduce the problem to the case where D = D 1 has coefficients in a finite extension of k (if P is finitely generated over k).
Remark I.13.-In this remark, and only in this remark, the D-cellular characters will be called the (k, D)-cellular characters.Let k ′ be a field extension of k such that P ′ = k ′ ⊗ k P is still an integrally closed domain.Let K ′ be the fraction field of P ′ .Since the k ′ -algebra k ′ A is still split, we can define the (k ′ , D)-cellular characters γ P ′ S ′ attached to a simple K ′ [D]-module S ′ .In fact, through the isomorphism K 0 (A) . -Let C denote the C-vector space of maps c : Ref(W ) −→ C , s → c s , which are invariant under W -conjugation.We denote by Ref(W )/ ∼ the set of conjugacy classes of reflections of W .For s ∈ Ref(W ) we denote by C s ∈ C * the linear form defined by C s (c) = c s .Of course, C s = C t if s and t are conjugate.We identify C[ C] = C[(C s ) s∈Ref(W )/∼ ].If c ∈ C, we denote by C c the ideal of C[ C] of functions vanishing at c.This ideal is generated by (C s − c s ) s∈Ref(W )/∼ .
equivariant and P-linear.Recall that C 0 denotes the ideal of C[ C] generated by (C s ) s∈Ref(W )/W .Lemma 3.1.-The restriction of the map Trunc to Z induces an isomorphism of bi-graded C[ C]modules Trunc :

i
) i∈I .B: cardinality of I. C: maximum of δ among the z (0) i .D: computation time of (z i ) i∈I via Algorithm 3.10.E: computation time of the relations between the z (0) i .F: number of relations.G: computation time of the relations between the z i via Algorithm 3.14.
), and we endow it with the Lie algebra structure such that Sym d (C 2 ) is a commutative ideal of g d and the adjoint action of sl 2 (C) on Sym d (C 2 ) coincides with the natural action.Theorem 3.20.-If d 5 and a = b, then Lie c (p) ≃ g d and the isolated symplectic singularity (Z c , p) has trivial local fundamental group.
4.C.The symplectic singularity in the origin.-Further to the dihedral group case mentioned in Section 3.F, we are able to identify a symplectic singularity in the Calogero-Moser space for G 4 .Theorem 4.4.-Assume the parameter is (k 0 , k 1 , k 2 ) = (0, 1, −1).Then the origin is the unique singular point of Z c and the symplectic singularity (Z c , 0) is equivalent to (O min (sl 3 (C)), 0).Remark 4.5.-By the Harish-Chandra theory of symplectic leaves [Bel1], the Calogero-Moser space Z c may admit an isolated singularity only if

Conjecture 5 .
11 (Martino).-Let c ∈ C and let k = κ(c) ∈ K = C. Then any Calogero-Moser c-family is a union of Rouquier k ♯ -families.Conjecture 5.12 (Gordon-Martino).-Assume that W is a Coxeter group and assume c ∈ C is real-valued.Then Calogero-Moser c-families coincide with Lusztig c-families.
Test if the generic Rouquier families are unions of the generic Calogero-Moser families.
We recall here the definition of cellular characters which involves the Gaudin algebra [BoRo2, §13.2].First, recall that C[ C× V reg ][W ] denotes the group algebra of W over the algebra C[ C× V reg ], and not the semi- Finally, we set D = (D 1 , . . ., D n ) and we denote by P [D] the commutative algebra generated by D 1 ,. . ., D n .If p is a prime ideal of P , we denote by D i (p) ∈ End P/p ((P/p)E) the reduction of D i modulo p and we set D(p) = (D 1 (p), . . ., D n (p)).

r
where Π i belongs to K[t] and is irreducible, n i 1 andΠ i = Π j if i = j.By [BoRo2, Proposition B.4.1], (I.4) char sem D , Π 1 , . . ., Π r ∈ P [t].If 1 i r, we set L i = K[t]/ Π i , and we view it as a P [D]-module, where D acts by multiplication by the indeterminate t.Then (I.5)Irr(K[D]) = {L 1 , . . ., L r } and L i ≃ L j if i = j.i (D) n i ) is a decomposition of KE into K[D] ⊗ K KA-modules.As a K[D]-module, the chief factors of Ker(Π i (D) n i Proposition I.10.-Let p be a prime ideal of P such that ∆(p) = 0 and P/p is integrally closed.Then the set of D-cellular characters coincides with the set of D(p)-cellular characters.Proof.-Assume that ∆(p) = 0. Let us write Π i (p) = = k P (p)[t]/ π i,j , viewed as a k P (p)[D(p)]-module, where D(p) acts by multiplication by t.Then, by (I.5), the map E −→ Irr(k P (p)[D(p)]) (i, j) P [X] (P [X]E).Since P [X] is integrally closed, we are in the same set-up as in the previous section (the case n = 1), with P replaced by P [X] and D replaced by D. So we can define similarly D-cellular characters of A. Proposition I.11.-The set of D-cellular characters of A coincides with the set of D-cellular characters of A.
K[D(x)] ⊂ K[D], it remains to prove that K[D] = K[D(x)]+Rad(K[D]).By [CuRe, Corollary 7.8 and Theorem 7.9], we may assume that K is algebraically closed.In this case, there exists a basis of KE such that D 1 ,. . ., D n are represented by commuting upper triangular matrices.Let d = dim k (E) and let (λ (i) j ) 1 j d denote the (ordered) sequence of diagonal coefficients of D i and let λ : {1, 2, . . ., d} −→

TABLE 3 .
18. Overview of the performance of the algorithms in the case of dihedral groups of order 2d for 3 d 8.The meaning of the columns is as follows.A: computation time of (z

TABLE 4 .
3. Relations for Z in type G 4 .
The corresponding point m c F of Z C × c is then equal to the kernel of Ω c F .If (z i ) i∈I denotes asystem of algebra generators of Z then m c F is generated by the elements (z i − Ω c F (z i )) i∈I .We thus obtain: Fis cuspidal if and only if Ω c