On the Calogero–Moser space associated with dihedral groups

BLAISE PASCAL

Cédric Bonnafé

On the Calogero–Moser space associated with dihedral groups Volume 25, n^o2 (2018), p. 265-298.

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Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS

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On the Calogero–Moser space associated with dihedral groups

Abstract

We investigate some geometric properties of the Calogero–Moser spaces associated with a dihedral group. As a consequence, we check in this particular case some conjectures made by the author and Raphaël Rouquier about general Calogero–Moser spaces.

Using the geometry of the associated Calogero–Moser space, R. Rouquier and the author [6] 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 ofCalogero–Moser familiesdefined by Gordon [13]. If moreover Wis a Coxeter group, it is conjectured in [6, Chpt. 15] 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 wheneverWis adihedral group. Since they are all about the geometry of the Calogero–Moser space, we also study some conjectures in [6, Chpt. 16] about the fixed point subvariety under the action of a group of roots of unity, as well as some other aspects (presentation of the algebra of regular functions; cuspidal points as defined by Bellamy [3] and their associated Lie algebra). We do not prove all the conjectures but we get at least the following results (here, W is a dihedral group of order 2d, acting on a complex vector spaceVof dimension 2;

we denote byZits associated Calogero–Moser space as in [6] and byZthe algebra of regular functions onZ: see Section 3.3 for the definition):

• Calogero–Moser cellular characters and Kazhdan–Lusztig cellular characters coincide (this is [6, Conj. CAR]).

• Calogero–Moser families and Kazhdan–Lusztig families coincide (this is [14, Conj. 1.3]; see also [6, Conj. FAM]). This result is not new: it was already proved by Bellamy [2], but we propose a slightly different proof, based on the computation of cellular characters.

• We give a presentation ofC[V×V^∗]^W (this extends the results of [1] which deal with the case whered ∈ {3,4,6}). Using [7], we explain how one could derive from this a presentation ofZ: this is done completely only ford ∈ {3,4,6}.

The author is partly supported by the ANR (Project No ANR-16-CE40-0010-01 GeRepMod).

• Ifdis 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 groupS_W on the setW.

• Ifdis odd, then we prove [6, Conj. FIX] about the fixed point subvarietyZ^µd. We also investigate special cases using calculations with the softwareMAGMA[8], based on theMAGMApackageCHAMPdeveloped by Thiel [16], and a paper in preparation by Thiel and the author [7]. For instance, we get:

• Ifd ∈ {3,4,6}, then we prove [6, Conj. FIX] about the fixed point subvariety Z^µm (for anym).

• Ifd =4 and the parameters are equal and non-zero (respectively d =6 and the parameters are generic) and ifmis a Poisson maximal ideal ofZ, then we prove that the Lie algebram/m²is isomorphic tosl₃(C)(respectivelysp₄(C)).

We believe these intriguing examples have their own interest.

Notation. We setV =C²and we denote by(x,y)the canonical basis ofVand by(X,Y) the dual basis ofV^∗. We identifyGL_C(V)withGL₂(C). We also fix a non-zero natural numberd, as well as a primitived-th root of unityζ ∈C^×. Ifi∈Z/dZ, we denote byζⁱ the elementζⁱ⁰, wherei₀is any representative ofiinZ.

We denote byC[V]the algebra of polynomial functions onV(so thatC[V]=C[X,Y]is a polynomial ring in two variables) and byC(V)its fraction field (so thatC(V)=C(X,Y)).

We will denote by⊗the tensor product⊗_C. 1. The dihedral group

1.1. Generators Ifi∈Z, we set

si = 0 ζⁱ ζ⁻ⁱ 0

and

(s=s0, t=s₁.

Note thats_i =s_i+d is a reflection of order 2 for alli ∈ Z(so that we can writes_i for i∈Z/dZ). We set

W =hs,ti.

ThenW is a dihedral group of order 2d, and(W,{s,t})is a Coxeter system, where s² =t² =(st)^d =1.

If we need to emphasize the natural numberd, we will denote byW_dthe groupW. The Coxeter elementw_cis given by

w_c=ts= ζ 0

0 ζ⁻¹

,

so that the following equalities are easily checked (for alli,j ∈Z)

w_cs_iw⁻¹_c =s_i+2 and s_is_j =w^i−j_c . (1.1) It then follows that

sandtare conjugate inW if and only ifdis odd. (1.2) Note that

W ={w_cⁱ |i∈Z/dZ} Û∪ {s_i|i∈Z/dZ}. (1.3) The set Ref(W)of reflections ofW is equal to

Ref(W)={s_i|i∈Z/dZ}. (1.4)

Now, let

α_i^∨=ζⁱx−y and α_i =X−ζⁱY, so that

s_i(α_i^∨)=−α^∨_i and s_i(α_i)=−α_i. (1.5) Finally, we fix a primitive 2d-th root of unityξsuch thatξ²=ζ and we set

τ=

0 ξ ξ⁻¹ 0

.

Then it is readily seen that

τsτ⁻¹ =t, τtτ⁻¹ =s and τ² =1, (1.6) so thatτ∈NGL_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 ofW (the conjugacy byse). Ifdis even, thensandtare not conjugate in W and soτinduces a non-inner automorphism ofW.

1.2. Irreducible characters

We denote by1W the trivial character ofW and letε:W →C^×,w7→det(w). Ifdis even, then there exist two other linear charactersε_sandε_twhich are characterized by the following properties:

(ε_s(s)=ε_t(t)=−1, ε_s(t)=ε_t(s)=1.

Ifk∈Z, we set

ρ_k : W −→ GL2(C) s_i 7−→ s_ki wⁱ_c 7−→ w^ki_c.

It is easily checked from (1.1) thatρ_k is a morphism of groups (that is, a representation ofW). IfRis anyC-algebra, we still denote byρ_k :RW →Mat2(R)the morphism of algebras induced by ρ_k. The character afforded byρ_k is denoted byχ_k. The following proposition is well known:

Proposition 1.2. Letk∈Z. Then:

(1) χ_k = χ−k =χ_k+d.

(2) Ifdis odd andk.0 modd, thenχ_k is irreducible.

(3) Ifdis even andk.0ord/2 mod d, thenχ_k is irreducible.

(4) χ₀ =1W +εand, ifdis even, thenχ_d/2=ε_s+ε_t.

Corollary 1.3. Recall thatτis the element ofN_GLC(V)(W)defined in Section 1.1.

(1) Ifdis odd, then|Irr(W)|=(d+3)/2and

Irr(W)={1W, ε} Û∪ {χ_k|16k 6(d−1)/2}.

Moreover,τacts trivially onIrr(W).

(2) Ifdis even, then|Irr(W)|=(d+6)/2and

Irr(W)={1W, ε, ε_s, ε_t} Û∪ {χ_k|16k6(d−2)/2}.

Moreover,^τχ=χif χ∈Irr(W) \ {ε_s, ε_t}while^τε_s=ε_t.

1.3. Some fractions in two variables

We work in the fraction fieldC(V)=C(X,Y). If 16k 6d, then Õ

i∈Z/dZ

ζ^ki

X−ζⁱ = dX^k−1

X^d−1. (1.7)

Õ

i∈Z/dZ

ζ^ki

X−ζⁱY = dX^k−1Y^d−k

X^d−Y^d . (1.8)

Proof. Let us first prove (1.7). Since 16k6d, there exist complex numbers(ξ_i)_i∈Z/dZ such that

dX^k−1

X^d−1 = Õ

i∈Z/dZ

ξ_i X−ζⁱ. Then

ξ_i = lim

z→ζⁱ

dz^k−1(z−ζⁱ)

z^d−1 =dζ^(k−1)i Ö

j∈Z/dZ j,i

(ζⁱ−ζ^j)⁻¹

=dζ(k−1)i−(d−1)i d−1

Ö

j=1

(1−ζ^j)⁻¹ =ζ^ki, and (1.7) is proved.

Now, (1.8) follows easily from (1.7) by replacingXbyX/Y. Ifd =2eis even and 16k6e, then

Õ

i∈Z/eZ

ζ^2ki

X−ζ²ⁱY = eX^k−1Y^e−k

X^e−Y^e , (1.9)

Õ

i∈Z/eZ

ζ^k(2i+1)

X−ζ²ⁱ⁺¹Y =−eX^k−1Y^e−k

X^e+Y^e , (1.10)

Õ

i∈Z/eZ

ζ−(k−1)(2i+1)

X−ζ²ⁱ⁺¹Y = eX^e−kY^k−1

X^e+Y^e , (1.11)

Proof. The equality (1.9) follows from (1.8) by replacing ζ by ζ² and d bye. The equality (1.10) follows from (1.9) by replacingY byζY(note thatζ^e=−1). Finally, the equality (1.11) follows from (1.10) by replacingkbye−k+1 (note thatζ^{−(k−1)(2i+1)}=

−(ζ^e)²ⁱ⁺¹ζ−(k−1)(2i+1)=−ζ(e−k+1)(2i+1)).

2. Invariants

The aim of this section is to describe generators and relations for the invariant algebra C[V×V^∗]^W. Note that such results have been obtained ifd ∈ {3,4,6}in [1]. We set

q=xy, r =x^d+y^d, Q=XY and R=X^d+Y^d. Then

C[V]^W =C[Q,R] and C[V^∗]^W =C[q,r].

We setP•=C[V]^W ⊗C[V^∗]^W =C[q,r,Q,R] ⊂C[V×V^∗]^W. If 06i6d, we set a_i,0 =x^d−iYⁱ+y^d−iXⁱ.

Note that

a_0,0=r and a_d,0=R.

Finally, let

eu₀=x X+yY. Thena_i,0,eu0 ∈C[V×V^∗]^W.

We will now describe some relations between these invariants. For this, leteu⁽ⁱ⁾₀ = (x X)ⁱ+(yY)ⁱ. Then theeu⁽ⁱ⁾₀ ’s belong also toC[V×V^∗]^W. As they will appear in relations between generators ofC[V×V^∗]^W, we must explain how to express them as polynomials ineu₀. First of all,

euⁱ₀= Õi

j=0

i j

(x X)^j(yY)^i−j

= Õ

06j<₂ⁱ

i j

(qQ)^j (x X)^i−2j+(yY)^i−2j+













0 ifiis odd,

i

i 2

(qQ)^i/2 ifiis even.

Therefore,

euⁱ₀ = Õ

06j<₂ⁱ

i j

(qQ)^jeu⁽ⁱ⁻²₀ ^j)+













0 ifiis odd,

i

i 2

(qQ)^i/2 ifiis even.

So, by triangularity of this formula, an easy induction shows that there exists a family of integers(n_i,j)_06j6i/2such that

eu⁽ⁱ⁾₀ = Õ

06j6₂ⁱ

n_i,j(qQ)^jeu^i−2j₀ , (2.1) withn_i,0=1 for alli.

On the other hand, one can check that the following relations hold (for 16i6 j 6d−1):



















(Z⁰_i)eu0a_i,0=qa_i+1,0+Qa_i−1,0 (Z⁰_i,j)ai,0aj,0=q^d−jQⁱeu⁽₀^j−i)+













ra_i+j,0−q^d−i−jeu^(i+j)₀ if 26i+j 6d, r R−eu^(d)₀ ifi+j =d, Rai+j−d,0−Q^i+j−deu⁽₀^2d−i−j) ifd6i+j62d−2.

Using (2.1), these last relations can be viewed as relations betweenq,r,Q,R,eu0,a_1,0, a_2,0, . . . ,a_d−1,0.

Theorem 2.1. The algebra of invariantsC[V×V^∗]^W admits the following presentation:















Generators: q,r,Q,R,eu0,a_1,0,a_2,0, . . . ,a_d−1,0 Relations:

((Z⁰_i) for16i6d−1, (Z⁰_i,j) for16i6 j 6d−1.

This presentation is minimal, as well by the number of generators as by the number of relations (there ared+4generators and(d+2)(d−1)/2relations). Moreover,C[V×V^∗]^W is a freeP•-module with basis(1,eu0,eu²₀,eu^d₀,a_1,0,a_2,0, . . . ,a_d−1,0).

Proof. LetH^∗denote the subspace ofC[V]defined by H^∗=C⊕

d−1

Ê

i=1

CXⁱ ⊕CYⁱ

!

⊕C(X^d−Y^d).

Then H^∗ is a graded sub-C[W]-module of C[V]. If 0 6 i 6 d, let H_i^∗ denote the homogeneous component of degreeiofH^∗. Whenever 16i 6d−1, it affords χ_i for character whereasH₀^∗andH_d^∗ afford respectively1W andεfor characters. Similarly, we define

H=C⊕

d−1

Ê

i=1

Cxⁱ ⊕Cyⁱ

!

⊕C(x^d−y^d).

Then H is a graded sub-C[W]-module of C[V^∗]. If 0 6 i 6 d, let H_i denote the homogeneous component of degreeiofH. Whenever 16i 6d−1, it affords χ_i for character whereasH₀andH_d afford respectively1W andεfor characters. Moreover, the morphism ofC[V]^W-modulesC[V]^W ⊗H^∗→C[V]induced by the multiplication is a W-equivariant isomorphism and the morphism ofC[V^∗]^W-moduleC[V^∗]^W⊗H→C[V^∗] induced by the multiplication is aW-equivariant isomorphism. Consequently,

C[V×V^∗]^W =P•⊗ (H^∗⊗H)^W. (2.2)

An easy computation of the subspaces(H_i^∗⊗H_j)^W based on the previous remarks show that

(1,eu₀,eu⁽²⁾₀ , . . . ,eu^(d−1)₀ ,(X^d−Y^d)(x^d−y^d),a_1,0,a_2,0, . . . ,a_d−1,0)

is aC-basis of(H⊗H^∗)^W. By (2.2), it is also aP•-basis ofC[V×V^∗]^W. On the other hand,

(X^d−Y^d)(x^d−y^d)=2eu^(d)₀ −Rr,

so(1,eu0,eu⁽₀²⁾, . . . ,eu⁽₀^d),a1,0,a2,0, . . . ,ad−1,0)is aP•-basis ofC[V×V^∗]^W. By (2.1), C[V×V^∗]^W=P•⊕P•eu₀⊕P•eu²₀⊕ · · · ⊕P•eu^d₀ ⊕P•a_1,0⊕P•a_2,0⊕ · · · ⊕P•a_d−1,0, which shows the last assertion of the theorem.

It also proves thatC[V×V^∗]^W =C[q,r,Q,R,eu₀,a_1,0,a_2,0, . . . ,a_d−1,0]. LetE,A₁, A2, . . . ,Ad−1be indeterminates overC[q,r,Q,R]: we have a surjective morphism

C[q,r,Q,R,E,A₁,A₂, . . . ,A_d−1] −C[V×V^∗]^W

which sends q,r,Q, R, E, A1, A2, . . . ,Ad−1 onq,r,Q, R,eu0,a1,0,a2,0, . . . ,ad−1,0

respectively.

For 16i 6j 6d−1, letF_i (respectivelyF_i,j) denote the element of the polynomial algebraC[q,r,Q,R,E,A₁,A₂, . . . ,A_d−1]corresponding to the relation(Z⁰

i)(respectively (Z_i,⁰_j)). Let A denote the quotient of C[q,r,Q,R,E,A₁,A₂, . . . ,A_d−1] by the ideal a generated by theF_i’s and theF_i,j’s. We denote byq₀,r₀,Q₀,R₀,E₀,A_1,0,A_2,0, . . . ,A_d−1,0

the respective images ofq,r,Q,R,E,A₁,A₂, . . . ,A_d−1inA. We then have a surjective morphism of bigradedC-algebrasϕ: AC[V×V^∗]^W. We want to show thatϕis an isomorphism. For this, it is sufficient to show that the bi-graded Hilbert series coincide.

But,

dim^bigr

C (A)>dim^bigr

C (C[V×V^∗]^W), (2.3)

where an inequality between two power series means that we have the corresponding inequality between all the coefficients.

We setP₀ =C[q₀,r₀,Q₀,R₀]. Let

A⁰=P0+P0E0+P0E₀²+· · ·+P0E₀^d+P0A1,0+P0A2,0+· · ·+P0Ad−1,0. By construction,

dim^bigr

C (A⁰)6dim^bigr

C (C[V×V^∗]^W). (2.4)

We will prove that

A⁰is a subalgebra ofA. (2.5)

For this, taking into account the form of theF_i’s and theF_i,j’s, it is sufficient to show that E₀^d+1 ∈A⁰. But, by (2.1),

A_1,0A_d−1,0 =Q0q0©

­

« Õ

06j<(d−2)/2

n_d−2,jE₀^d−2−2jª

®

¬

+R0r0− Õ

06j<d/2

n_d,jE₀^d−2j,

andnd,0 =1. So

E₀^d =−A_1,0A_d−1,0+Q₀q₀©

­

« Õ

06j<(d−2)/2

n_d−2,jE₀^d−2−2jª

®

¬

+R₀r₀− Õ

16j<d/2

n_d,jE₀^d−2j

and so

E₀^d+1 =−E₀A_1,0A_d−1,0+Q₀q₀©

­

« Õ

06j<(d−2)/2

n_d−2,jE₀^d−1−2jª

®

¬

+R₀r₀− Õ

16j<d/2

n_d,jE₀^d+1−2j. It is then sufficient to show that E0A1,0Ad−1,0 ∈ A⁰. But E0A1,0Ad−1,0 = (q₀A2,0 + Q₀r₀)A_d−1,0, which concludes the proof of (2.5).

SinceA⁰containsq₀,r₀,Q₀,R₀,E₀,A_1,0,A_2,0, . . . ,A_d−1,0and sinceAis generated by these elements, we have A= A⁰. It follows from (2.3) and (2.4) that dim^bigr

C (A) = dim^bigr

C (C[V×V^∗]^W), which shows thatϕ: A→C[V×V^∗]^Wis an isomorphism. In other words, this shows that the presentation ofC[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, letp₀denote the bi-graded maximal ideal ofP•and setB=C[V×V^∗]^W/p₀C[V×V^∗]^W. We denote bye,a₁,a₂, . . . ,a_d−1the respective images ofeu₀,a_1,0,a_2,0, . . . ,a_d−1,0inB.

Hence,

B=C⊕Ce⊕Ce²⊕ · · · ⊕Ce^d⊕Ca₁⊕Ca₂⊕ · · · ⊕Ca_d−1 andBadmits the following presentation:



















Generators: e,a₁,a₂, . . . ,a_d−1 Relations (16i 6j 6d−1) :











 ea_i =0 aiaj =

(0 ifi+j ,d,

−e^d ifi+j =d.

It is sufficient to prove that the number of relations ofBis minimal. By reducing modulo the idealBe, we get that all the relations of the forma_ia_j =0 or−e^dare necessary. By reducing modulo the idealBa1+· · ·+Bai−1+Ba_i+1+· · ·Bad−1, we get that the relations

ea_i =0 are necessary.

Remark 2.2. It is easily checked that the elementτdefined in Section 1.1 satisfies

τq=q, ^τQ=Q, ^τeu₀ =eu₀ and ^τa_i,0=−a_i,0

for all 06i6d(this follows from the fact thatξ^d =−1).

Remark 2.3(Gradings). The algebraC[V ×V^∗]admits a natural(N×N)-grading, by putting elements ofV in bi-degree(1,0)and elements ofV^∗ in bi-degree(0,1). This bi-grading is stable under the action ofW, soC[V×V^∗]^W inherits this bi-grading. Note that the generators and the relations given by Theorem 2.1 are bi-homogeneous.

This(N×N)-grading induces aZ-grading such that any bi-homogeneous element of bi-degree(m,n)isZ-homogeneous of degreen−m(in other words, elements ofVhave Z-degree−1 while elements ofV^∗haveZ-degree 1).

3. Cherednik algebras

3.1. Definition

We denote byCtheC-vector space of maps Ref(W) →Cwhich are constant on conjugacy classes. Ifi ∈Z/dZ, we denote byC_ithe element ofC^∗which sendsc∈Ctoc_si. By (1.1), C_i =C_i+2. LetA=C₀andB=C₁. Ifdis odd, thenA=BandC[C]=C[A]whereas, if dis even, thenA,B(see (1.2)) andC[C]=C[A,B].

Thegeneric rational Cherednik algebra att =0 is theC[C]-algebraHdefined as the quotient ofC[C] ⊗ T(V⊕V^∗)oW

by the following relations (here, T(V ⊕V^∗)is the tensor algebra ofV⊕V^∗overC):











[u,u⁰]=[U,U⁰]=0, [u,U]=−2Í

i∈Z/dZC_i ^{hu,αii · hα}

∨ i,Ui

hα_i^∨,αii s_i, (3.1) forU,U⁰∈V^∗andu,u⁰∈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 ofC-algebrasC[V] →HsendingU∈V^∗⊂C[V] to the class ofU∈T(V⊕V^∗)oWinH.

• There is a unique morphism ofC-algebrasC[V^∗] →Hsendingu∈V ⊂C[V^∗] to the class ofu∈T(V ⊕V^∗)oW inH.

• There is a unique morphism ofC-algebrasCW →Hsendingw∈Wto the class ofw∈T(V⊕V^∗)oWinH.

• TheC-linear map C[C] ⊗C[V] ⊗CW ⊗C[V^∗] −→ Hinduced by the three morphisms defined above and the multiplication map is surjective. Note that it is C[C]-linear.

The last statement is strenghtened by the following fundamental result by Etingof and Ginzburg [11, Thm. 1.3] (see also [6, Thm. 4.1.2]).

Theorem 3.1(Etingof–Ginzburg).The multiplication mapC[C]⊗C[V]⊗CW⊗C[V^∗] −→

His an isomorphism ofC[C]-modules.

Remark 3.2. By [6, §3.5.C], the group NGL_C(V)(W) acts naturally onH. It follows from (1.6) that

τA=B and ^τB=A.

Here,τis the element of NGL_C(V)(W)defined in Section 1.1.

Remark 3.3(Gradings). The algebraHadmits a natural(N×N)-grading, by puttingVin bi-degree(1,0),V^∗in bi-degree(0,1),W in degree(0,0)andC^∗in degree(1,1)(see for instance [6, §3.2]).

This(N×N)-grading induces aZ-grading such that any bi-homogeneous element of bi-degree(m,n)isZ-homogeneous of degreen−m. In other words, deg(V) =−1, deg(V^∗)=1 and deg(W)=0=deg(C^∗)=0.

3.2. Specialization

Given c ∈ C, we denote by C_c the maximal ideal of C[C] defined byC_c = {f ∈ C[C] | f(c)=0}: it is the ideal generated by(C_i−c_si)_i∈Z/dZ. We set

H_c=(C[C]/C_c) ⊗_C[C]H=H/C_cH.

TheC-algebraH_cis the quotient of theC-algebra T(V ⊕V^∗)oW by the ideal generated by the following relations:











[u,u⁰]=[U,U⁰]=0, [u,U]=−2Í

i∈Z/dZc_si ^{hu,αii · hα}

∨ i,Ui

hα^∨_i,αii s_i, (3.2) forU,U⁰∈V^∗andu,u⁰∈V.

Remark 3.4(Grading). The idealC_c is not bi-homogeneous (except ifc =0) so the algebraH_cdoes not inherit fromHan(N×N)-grading. However,C_cisZ-homogeneous, soH_cstill admits a naturalZ-grading.

3.3. 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

P=C[C] ⊗C[V]^W ⊗C[V^∗]^W.

Similarly, ifc∈C, we denote byZ_cthe centre ofH_c: it turns out [6, Cor. 4.2.7] thatZ_c is the image of Zand that the image ofPinZ_cisP• =C[V]^W ⊗C[V^∗]^W. Recall also (see for instance [6, Cor. 4.2.7]) that

Z is a freeP-module of rank|W| (3.3) and that [11]

Z andZcare integral and integrally closed. (3.4) Since the C-algebraZ is finitely generated, we can associate to it an irreducible and normal (according to (3.4)) algebraic variety overC, called thegeneric Calogero–Moser space, and which will be denoted byZ. Ifc∈C, we denote byZ_cthe algebraic variety associated with theC-algebraZ_c.

3.4. About the presentation ofZ

We follow here the method of [7]. Ifh∈H, it follows from Theorem 3.1 that there exists a unique family of elements(h_w)_w∈W ofC[C] ⊗C[V] ⊗C[V^∗]such that

h= Õ

w∈W

h_ww.

We define theC[C]-linear map Trunc :H→C[C] ⊗C[V] ⊗C[V^∗]by Trunc(h)=h1.

The next lemma is proved in [7]:

Lemma 3.5. The restriction ofTrunctoZ yields an isomorphism of bi-gradedC[C]- modules

Trunc :Z −→^∼ C[C] ⊗C[V×V^∗]^W. We then seteu=Trunc⁻¹(eu₀)and, for 06i6d,

ai =Trunc⁻¹(a_i,0).

An explicit algorithm for computing the inverse map Trunc⁻¹is described in [7]. Note that Trunc⁻¹(p)= pfor p ∈ P, so thata₀ =a_0,0 =r anda_d =a_d,0 =R. By [7], the relations(Z⁰_i)_16i6d−1and(Z⁰_i,j)_16i6j6d−1can be deformed into relations(Zi)_16i6d−1and (Zi,j)_16i6j6d−1and it follows from Theorem 2.1 and [7] that:

Theorem 3.6. The centreZofHadmits the following presentation, as aC[C]-algebra:













Generators: q,r,Q,R,eu,a₁,a₂, . . . ,a_d−1 Relations:

((Zi) for16i 6d−1, (Z_i,j) for16i 6 j6d−1.

This presentation is minimal, as well by the number of generators as by the number of relations (there ared+4generators and(d+2)(d−1)/2relations). Moreover,

Z =P⊕Peu⊕Peu²⊕ · · · ⊕Peu^d ⊕Pa₁⊕Pa₂⊕ · · · ⊕Pa_d−1.

It must be said that we have no way to determine explicitly the relations (Zi,j)in general: we will describe them precisely only ford ∈ {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 theZ-grading on the algebraHconstructed in Remark 3.3 induce a bi-grading and aZ-grading onZ. 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 (Zi)and(Zi,j)given in Theorem 3.6 are bi-homogeneous.

Note. From now on, and until the end of this paper, we fix a parameterc∈Cand we set a=c_sandb=c_t. Note that, ifdis odd, thena=b.

3.5. Poisson bracket

Recall from [6, §4.4.A] that the algebraZis endowed with aC[C]-linear Poisson bracket { ·,· }:Z×Z −→Z,

which is a deformation of the Poisson bracket onZ₀=C[V×V^∗]^W obtained by restriction of theW-equivariant canonical Poisson bracket onC[V×V^∗]. This Poisson bracket induces a Poisson bracket onZ_c. It satisfies

{q,Q}=eu (3.5)

(see [9, §4] or [5, §3]).

4. Calogero–Moser cellular characters

The aim of this section is to determine, for all values ofc, the Calogero–Moserc-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 byC(V)W the group algebra ofW over the fieldC(V)and we set

D_x= Õ

i∈Z/dZ

ε(s_i)c_sihx, αii

α_i s_i =− Õ

i∈Z/dZ

c_si 1

X−ζⁱYs_i ∈C(V)W

and D_y= Õ

i∈Z/dZ

ε(s_i)c_sihy, α_ii

α_i s_i = Õ

i∈Z/dZ

c_si ζⁱ

X−ζⁱYs_i ∈C(V)W.

We denote by Gau(W,c)the sub-C(V)-algebra ofC(V)Wgenerated byD_xandD_y(it is commutative by [6, §13.4.B]). Note that this algebra is not necessarily split. IfL is a simple Gau(W,c)-module, and if χ∈Irr(W), we denote by mult^CM_{L, χ}the multiplicity ofL in a composition series of the module Res^C_Gau^(V)W_(W,c)C(V)E_χ, whereE_χis aCW-module affording the character χ. We then set

γ_L= Õ

χ∈Irr(W)

mult^CM_{L, χ} χ.

The set ofCalogero–Moserc-cellular charactersis

CellChar^CM_c (W)={γ_L|L ∈Irr(Gau(W,c))}.

We will denote byE₁^Gau(respectivelyE^Gau_ε , respectivelyL^Gau_k ) the restriction ofC(V)E_1W (respectivelyC(V)E_ε, respectivelyC(V)E_χk) to Gau(W,c). Ifdis even, then the restriction ofC(V)E_εs (respectivelyC(V)E_εt) to Gau(W,c)will be denoted byE^Gau_s (respectively E_t^Gau).

Remark 4.1. Note that, since Gau(W,c) ⊂ C(V)W, every simple Gau(W,c)-module occurs as a composition factor of some Res^C_Gau(W,c)^(V)W C(V)E_χ, for χrunning over Irr(W).

Notation 4.2. In order to simplify the computation in this section, we set D_x⁰ =Y^d−X^d

d D_x and D_y⁰ = X^d−Y^d d D_y,

so that Gau(W,c)is the sub-C(V)-algebra ofC(V)Wgenerated byD⁰_xandD_y⁰. 4.1. The case wherea=b=0

Whenever a = b = 0, there is only one Calogero–Moser 0-cellular character [6, Cor. 17.2.3], namely the regular characterÍ

χ∈Irr(W)χ(1)χ.

4.2. The case wherea,b=0

We will assume here, and only here, thatb=0,a: this forcesdto be even (and we write d =2e). LetW⁰be the subgroup ofW generated bys=s₀ands₂ =tst. ThenW⁰is a dihedral group of order 2eand

W=htinW⁰.

Let c⁰denote the restriction ofcto Ref(W⁰): thenc⁰is constant (and equal toa) and Gau(W,c)=Gau(W⁰,c⁰). It then follows from the definition that the Calogero–Moser c-cellular characters are all the characters of the form Ind^W_W0γ⁰, whereγ⁰is a Calogero–

Moserc⁰-cellular character ofW⁰. These charactersγ⁰will be determined in the next subsection and so it follows that the list ofc-cellular characters ofW is

1+ε_t, ε_s+ε,

(d−2)/2

Õ

k=1

χ_k. (4.1)

Remark 4.3. Ifa=0 andb,0, then one can use the elementτ∈N_GLC(V)(W)of order 2 such that^τs=tand^τt=sconstructed in Section 1.1 to be sent back to the previous case.

We then deduce from (4.1) that the list ofc-cellular characters ofWis 1+ε_s, ε_t+ε,

(d−2)/2

Õ

k=1

χ_k. (4.2)

Note that we have a semi-direct product decompositionW =hsinⁿW⁰. 4.3. The equal parameters case

We assume here, and only here, thata=b,0. Now, if 16k 6d−1, then ρ_k(D_x⁰)=−aY^d−X^d

d

Õ

i∈Z/dZ

1 X−ζⁱY

0 ζ^ki

ζ^−ki 0

=a

0 X^k−1Y^d−k X^d−k−1Y^k 0

by using (1.8). Similarly, ρ_k(D⁰_y)=a

0 X^kY^d−k−1

X^d−kY^k−1 0

= X

Y ρ_k(D_x⁰).

If we denote byMthe diagonal matrix ^X_{0 Y}⁰

, then it follows from the previous formulas that

∀16k6d−2, ∀D∈Gau(W,c), Mρ_k(D)M⁻¹=ρ_k+1(D). (4.3) This implies that

∀16k6d−2, L^Gau_k ' L^Gau_k+1. (4.4)

Since Tr(ρ_k(D⁰_x)) =Tr(ρ_k(D_y⁰)) =0, the nature of the restriction of ρ_k to Gau(W,c) depends on whether −det(ρ_k(D_x⁰)) = a²X^dY^d−2 is a square inC(V). Two cases may occur:

First case: assume thatdis odd

Then−det(ρ_k(D_x⁰))is not a square inC(V)(for 16k6d−1), so it follows thatL^Gau_k is simple (but not absolutely simple) and it follows from (4.4) that

Irr(Gau(W,c))={E₁^Gau,E^Gau_ε ,L₁^Gau}. (4.5) Moreover, the list ofc-cellular characters is given in this case by

1W, ε and

(d−1)/2

Õ

k=1

χ_k. (4.6)

Second case: assume thatdis even

In this case, it is easily checked thatE^Gau₁ ,E^Gau_ε ,E^Gau_s andE^Gau_t are four non-isomorphic simple Gau(W,c)-modules. Also, if 16k 6d−1, then

L^Gau_k ' E_s^Gau⊕ E_t^Gau, by (4.4). Therefore,

Irr(Gau(W,c))={E₁^Gau,E^Gau_ε ,E^Gau_s ,E^Gau_t }. (4.7) and the list ofc-cellular characters is given in this case by

1W, ε, ε_s+

(d−2)/2

Õ

k=1

χ_k and ε_t+

(d−2)/2

Õ

k=1

χ_k. (4.8)

4.4. The opposite parameters case

We assume here, and only here, thatb=−a,0. This forcesdto be even. Then, using the automorphism ofHinduced by the linear characterε_s(see [6, §3.5.B]), one can pass from the equal parameter case to the opposite parameter case by tensorizing byε_s. Therefore, the list ofc-cellular characters is given in this case by

ε_s, ε_t, 1W+

(d−2)/2

Õ

k=1

χ_k and ε+

(d−2)/2

Õ

k=1

χ_k. (4.9)

4.5. The generic case

We assume here, and only here, thatab(a²−b²) ,0 (so that we are not in the cases covered by the previous subsections). Note that this forcesdto be even. We will prove that the list ofc-cellular characters is given in this case by

1W, ε, ε_s, ε_t and

(d−2)/2

Õ

k=1

χ_k. (4.10)

Proof. We have, for 16k6e−1, ρk(D⁰_x)= X^d−Y^d

d a Õ

i∈Z/eZ

1 X−ζ²ⁱY

0 ζ^2ki ζ^−2ki 0

+b Õ

i∈Z/eZ

1 X−ζ²ⁱ⁺¹Y

0 ζ^k(2i+1)

ζ^−k(2i+1) 0 !

.

So it follows from (1.9), (1.10) and (1.11) that ρ_k(D_x⁰)= 1

2

0 X^k−1Y^e−k (a−b)X^e+(a+b)Y^e X^e−k−1Y^k (a+b)X^e+(a−b)Y^e

0

.

The matrixρ_k(D_y⁰)can be computed similarly and we can deduce that,

∀16k6e−2, ∀D∈Gau(W,c), Mρ_k(D)M⁻¹ =ρ_k+1(D).

Therefore,

∀16k6e−2, L^Gau_k ' L^Gau_k+1. (4.11) Moreover,

−det(ρ_k(D⁰_x))=1

4X^e−2Y^e (a−b)X^e+(a+b)Y^e

(a+b)X^e+(a−b)Y^e. Sinceab(a²−b²),0,−det(ρ_k(D_x⁰))is not a square inC(V), and soL^Gau

k is simple (but not absolutely simple) for 16k6e−1.

Moreover, an easy computation shows thatE₁^Gau,E_ε^Gau,E_s^GauandE^Gau_t are pairwise non-isomorphic simple Gau(W,c)-modules. So it follows from (4.11) that

Irr(Gau(W,c))={E^Gau₁ ,E^Gau_ε ,E^Gau_s ,E_t^Gau,L₁^Gau} (4.12)

and that (4.10) holds.

Parameters d=2e(even) d =2e−1 (odd)

a=b=0 Í

χ∈Irr(W)χ(1)χ Í

χ∈Irr(W)χ(1)χ a,b=0 1W +ε_t, ε_s+ε, Íe−1

k=1 χ_k XX

XXXX

XXXX a=0,b 1W +ε_s, ε_t+ε, Íe−1

k=1 χ_k XX

XXXX

XXXX a=b,0 1W, ε, ε_s+Íe−1

k=1 χ_k, ε_t+Íe−1

k=1 χ_k 1W, ε, Íe−1 k=1 χ_k a=−b,0 ε_s, ε_t, 1W +Íe−1

k=1χ_k, ε+Íe−1

k=1 χ_k XX

XXXX

XXXX ab(a²−b²),0 1W, ε_s, ε_t, ε, Íe−1

k=1χ_k XX

XXXX

XXXX Table 4.1. Calogero–Moser cellular characters ofW

4.6. Conclusion

The following Table 4.1 gathers all the possible list of cellular characters ofW, according to the values of the parametersaandb.

Remark 4.4. Whenevera,b∈R, the Kazhdan–Lusztig cellular characters for the dihedral groups are easily computable (see for instance [15]) and a comparison with Table 4.1 shows that they coincide with Calogero–Moser cellular characters: this is [6, Conj. CAR]

for dihedral groups.

5. Calogero–Moser families

The aim of this section is to compute theCalogero–Moserc-familiesofW (as defined in [6, §9.2]) for all values ofc. The result is given in Table 5.1. Note that this result is not new: the Calogero–Moser families have been computed by Bellamy in his thesis [2]. We provide here a different proof, which uses the computation of Calogero–Moser cellular characters.

5.1. Families

To any irreducible character χ, Gordon [13] associates a simpleH_c-moduleL_c(χ)(we follow the convention of [6, Prop. 9.1.3]). We denote by Ω^c_χ : Z →Cthe morphism defined by the following property: ifz∈Z, thenΩ^c_χ(z)is the scalar by whichzacts on

L_c(χ)(by Schur’s Lemma). We say that two irreducible characters χandχ⁰belong to the same Calogero–Moserc-familyifΩ^c_χ=Ω^c_χ0(see [13] or [6, Lem. 9.2.3]: note that Calogero–Moser families are calledCalogero–Moser blocksin [13]). We give here a different proof of a theorem of Bellamy [2]:

Theorem 5.1(Bellamy). Letc∈Cand let χandχ⁰be two irreducible characters ofW.

Thenχandχ⁰lie in the same Calogero–Moser family if and only ifΩ^c_χ(eu)=Ω^c_χ0(eu).

Consequently, the Calogero–Moser families are given by Table 5.1.

Proof. By [6, Prop. 7.3.2], the values ofΩ^c_χ(eu)are given as follows:

(a) Ifd=2e−1 is odd, then











 Ω₁^c

W(eu)=da, Ω^c_ε(eu)=−da, Ω^c_χ

k(eu)=0 if 16k6e−1.

(b) Ifd=2eis even, then























 Ω^c

1W(eu)=e(a+b), Ω^c_ε(eu)=−e(a+b), Ω^c_ε

s(eu)=e(a−b), Ω^c_ε

t(eu)=e(b−a),

Ω^c_χk(eu)=0 if 16k6e−1.

But two irreducible characters occuring in the same Calogero–Moserc-cellular character necessarily belong to the same Calogero–Moserc-family [6, Prop. 11.4.2]. So the Theorem

follows from (a), (b) and Table 4.1.

Remark 5.2. Whenevera,b∈R, the Kazhdan–Lusztig families for the dihedral groups are easily computable (see for instance [15]) and a comparison with Table 5.1 shows that they coincide with Calogero–Moser families: this is [6, Conj. FAM] for dihedral groups.

Note that this was already proved by Bellamy [2].

5.2. Cuspidal families

Recall that the algebrasZandZ_care endowed with a Poisson bracket{,}. This Poisson structure has been used by Bellamy [3] to define the notion ofcuspidalCalogero–Moser families. IfF is a Calogero–Moserc-family, we setm^c_F =Ker(Ω^c_χ) ⊂ Z_c, where χis some (or any) element ofF (note thatΩ^c_χfactorizes through the projectionZ Zc).

The Calogero–Moserc-familyF is calledcuspidalif{m^c_F,m^c_F} ⊂m^c_F. 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.

Parameters d=2e(even) d =2e−1 (odd)

a=b=0 Irr(W) Irr(W)

a,b=0 ^{{1W, εt}, {εs, ε},}

{χ1, . . ., χe−1}

aa aa

aa aa

aa

a=0,b ^{{1W, εs}, {εt, ε},}

{χ1, . . ., χe−1}

aa aa

aa aa

aa

a=b,0 _{ε ^{{1W}, {ε},}

s, εt, χ1, . . ., χe−1}

{1W}, {ε}, {χ1, . . ., χe−1}

a=−b,0 ^{{εs}, {εt},}

{1W, ε, χ1, . . ., χe−1}

aa aa

aa aa

aa

ab(a²−b²),0 ^{{1W}, {εs}, {εt}, {ε},}

{χ1, . . ., χe−1}

aa aa

aa aa

aa

Table 5.1. Calogero–Moser families ofW

Proposition 5.3. The list of cuspidal Calogero–Moser families is given by Table 5.2. The following properties hold:

(1) A Calogero–Moser familyF is cuspidal if and only if|F |>2and χ₁∈ F (and thenχ_k ∈ F for all16k<d/2).

(2) There is at most one cuspidal family. Ifd >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, thenΩ^c_χ(eu)=0. Since it follows from Table 5.1 that the Calogero–Moserc-families are determined by the values ofΩ^c_χ(eu), this implies that there is at most one cuspidal Calogero–Moserc-family, and that it must containχ₁(and

χ_k, for 16k<d/2).

Also, since a Calogero–Moserc-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 χ₁ ∈ F and that|F |>2. By Bellamy theory [3, Introduction], there exists a non-trivial

Parameters d =2e(even) d =2e−1 (odd)

a=b=0 Irr(W) Irr(W),e>2

a,b=0 {χ₁, . . . , χ_e−1},e>3 ``````````` a=0,b {χ<sub>1</sub>, . . . , χ<sub>e−1</sub>},e>3 ``````````` a=b,0 {ε_s, ε_t, χ₁, . . . , χ_e−1},e>2 {χ₁, . . . , χ_e−1},e>3 a=−b,0 {1W, ε, χ₁, . . . , χ_e−1},e>2 ``````````` ab(a<sup>2</sup>−b<sup>2</sup>),0 {χ1, . . . , χ<sub>e</sub>−1},e>3 ```````````

Table 5.2. Calogero–Moser cuspidal families ofW

parabolic subgroupW⁰ofWand a cuspidal Calogero–Moserc⁰-familyF⁰ofW⁰(here, c⁰denotes the restriction ofcto Ref(W⁰)) which are associated withF. Again, by [3, Main Theorem, Introduction],|F |=|F⁰|. We must show thatW =W⁰. So assume that W⁰,W. Then|W⁰|=2 and so|F⁰|62 andc⁰must be equal to 0. This forces|F |=2 andab=0 (andc,0). This can only occur in typeG₂: but the explicit computation of the Poisson bracket in typeG₂shows thatF is necessarily cuspidal.

IfF is a cuspidal Calogero–Moserc-family, then the Poisson bracket{ ·,· }stabilizes the maximal ideal m^c_F and so it induces a Lie bracket[ ·,· ] on the cotangent space Lie_c(F )=m^c_F/(m^c_F)². 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).

Theorem 5.4. LetF be a cuspidal Calogero–Moserc-family ofW. Then:

(1) Ifd=4(i.e. ifWis of typeB₂) anda=b,0, thenLie_c(F ) 'sl₃(C)is a simple Lie algebra of type A₂.

(2) Ifd=6(i.e. ifW is of typeG₂) andab(a²−b²),0, thenLie_c(F ) 'sp₄(C)is a simple Lie algebra of typeB2.

6. Calogero–Moser cells

Letc ∈C. The main theme of [6] is a construction of partitions ofW into Calogero–

Moser left, right and two-sidedc-cells, using a Galois closureMof the field extension