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groups
Cédric Bonnafé
To cite this version:
Cédric Bonnafé. On the Calogero-Moser space associated with dihedral groups. Annales Math-ématiques Blaise Pascal, Université Blaise-Pascal - Clermont-Ferrand, 2018, 25 (2), pp.265-298. �10.5802/ambp.377�. �hal-01579445v2�
WITH DIHEDRAL GROUPS
by
CÉDRIC BONNAFÉ
Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author [BoRo] have attached to any finite complex reflection groupW several notions
(Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon [Gor]. If moreoverW is a
Cox-eter group, it is conjectured in [BoRo, Chapter 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 wheneverW is a dihedral group. Since they are all about the geometry of the Calogero-Moser space, we also study some conjectures in [BoRo, Chapter 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 [Bel2] 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 order2d, acting on a complex vector spaceV of dimension
2; we denote byZ its associated Calogero-Moser space as in [BoRo] and byZ the algebra
of regular functions onZ: see §3.C for the definition):
• Calogero-Moser cellular characters and Kazhdan-Lusztig cellular characters
coin-cide (this is [BoRo, Conjecture CAR]).
• Calogero-Moser families and Kazhdan-Lusztig families coincide (this is [GoMa,
Conjecture 1.3]; see also [BoRo, Conjecture FAM]). This result is not new: it was already proved by Bellamy [Bel1], 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 [AlFo] which deal
with the case whered ∈ {3, 4, 6}). Using [BoTh], we explain how one could derive
from this a presentation ofZ: this is done completely only ford∈ {3, 4, 6}.
• If d is odd, then Calogero-Moser (left, right or two-sided) cells coincide with the
Kazhdan-Lusztig (left, right or two-sided) cells: this is a particular case of [BoRo, Conjectures L and LR]. For proving this fact, we prove that the Galois group defined in [BoRo, Chapter 5] is equal to the symmetric groupSW on the setW.
• Ifd is odd, then we prove [BoRo, Conjecture FIX] about the fixed point subvariety
Zµd.
We also investigate special cases using calculations with the software MAGMA [Mag], based on theMAGMA packageCHAMP developed by Thiel [Thi], and a paper in prepa-ration by Thiel and the author [BoTh]. For instance, we get:
• Ifd ∈ {3, 4, 6}, then we prove [BoRo, Conjecture FIX] about the fixed point subvariety Zµm (for anym).
• If d = 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/m2 is isomorphic tosl
3(C)(respectivelysp4(C)). We believe these intriguing examples have their own interest.
Notation.— We setV = C2and we denote by(x , y )the canonical basis ofV and by(X , Y )
the dual basis of V∗. We identify GLC(V ) with GL2(C). We also fix a non-zero natural numberd, as well as a primitived-th root of unityζ∈ C×. Ifi∈ Z/d Z, we denote byζi
the elementζi0, wherei
0is any representative ofi inZ.
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.A. Generators. — Ifi∈ Z, we set
and si= 0 ζ i ζ−i 0 and ¨ s = s0, t = s1.
Note that si = si +d is a reflection of order 2 for all i ∈ Z (so that we can write si for
i∈ Z/d Z). We set
W =〈s , t 〉.
ThenW is a dihedral group of order2d, and(W ,{s , t })is a Coxeter system, where
s2= t2= (s t )d= 1.
If we need to emphasize the natural numberd, we will denote byWd the groupW.
We set
c = t s =ζ 0
0 ζ−1
,
so that the following equalities are easily checked (for alli, j∈ Z)
(1.1) c sic−1= si +2 and sisj = ci− j.
It then follows that
(1.2) s andt are conjugate inW if and only ifd is odd. Note that
(1.3) W ={ci | i ∈ Z/d Z} ˙∪ {si | i ∈ Z/d Z}.
The setRef(W )of reflections ofW is equal to
Now, let α∨ i=ζ ix − y and αi= X − ζiY , so that (1.5) si(α∨i) =−αi∨ and si(αi) =−αi.
Finally, we fix a primitive2d-th rootξsuch thatξ2=ζand we set τ = 0 ξ
ξ−1 0
.
Then it is readily seen that
(1.6) τs τ−1= t , τt τ−1= s and τ2= 1,
so thatτ∈ NGLC(V )(W ).
Remark 1.7. — Ifd = 2e− 1is odd, thenξ =−ζe and soτ =−s
e induces an inner
auto-morphism ofW (the conjugacy by se). Ifd is even, thens andt are not conjugate inW
and soτinduces a non-inner automorphism ofW.
1.B. Irreducible characters. — We denote by 1W the trivial character ofW and letǫ :
W → C×,w 7→ det(w ). Ifd is even, then there exist two other linear charactersǫ
s andǫt
which are characterized by the following properties:
¨ ǫs(s ) =ǫt(t ) =−1, ǫs(t ) =ǫt(s ) = 1. Ifk∈ Z, we set ρk: W −→ GL2(C) si 7−→ sk i ci 7−→ ck i.
It is easily checked from (1.1) thatρk is a morphism of groups (that is, a representation
of W). If R is any C-algebra, we still denote by ρk : RW → Mat2(R ) the morphism of
algebras induced byρk. The character afforded by ρk is denoted byχk. The following
proposition is well-known:
Proposition 1.8. — Letk∈ Z. Then:
(a) χk=χ−k=χk +d.
(b) Ifd is odd andk6≡ 0 mod d, thenχk is irreducible.
(c) Ifd is even andk6≡ 0ord/2 mod d, thenχk is irreducible.
(d) χ0= 1W +ǫand, ifd is even, thenχd/2=ǫs+ǫt.
Corollary 1.9. — Recall thatτis the element ofNGLC(V )(W )defined in §1.A.
(a) Ifd is odd, then|Irr(W )| = (d + 3)/2and
Irr(W ) ={1W,ǫ} ˙∪ {χk | 1 ¶ k ¶(d − 1)/2}.
(b) Ifd is even, then|Irr(W )| = (d + 6)/2and
Irr(W ) ={1W,ǫ, ǫs,ǫt} ˙∪ {χk | 1 ¶ k ¶(d − 2)/2}.
Moreover,τχ = χ ifχ∈ Irr(W ) \ {ǫ
s,ǫt}while τǫs=ǫt.
1.C. Some fractions in two variables. — We work in the fraction fieldC(V ) = C(X , Y ).
If1 ¶ k ¶ d, then (1.10) X i∈Z/d Z ζk i X− ζi = d Xk−1 Xd− 1. (1.11) X i∈Z/d Z ζk i X− ζiY = d Xk−1Yd−k Xd− Yd .
Proof. — Let us first prove (1.10). Since1 ¶ k ¶ d, there exist complex numbers(ξi)i∈Z/d Z such that d Xk−1 Xd− 1= X i∈Z/d Z ξi X− ζi. Then ξi= lim z→ζi d zk−1(z− ζi) zd− 1 = dζ (k−1)i Y j∈Z/d Z j6=i (ζi− ζj)−1= dζ(k−1)i −(d −1)i d−1 Y j =1 (1− ζj)−1=ζk i, and (1.10) is proved.
Now, (1.11) follows easily from (1.10) by replacingX byX/Y.
Ifd = 2e is even and1 ¶ k ¶ e, then
(1.12) X i∈Z/e Z ζ2k i X − ζ2iY = e Xk−1Ye−k Xe− Ye , (1.13) X i∈Z/e Z ζk (2i +1) X− ζ2i +1Y =− e Xk−1Ye−k Xe+ Ye , (1.14) X i∈Z/e Z ζ−(k −1)(2i +1) X − ζ2i +1Y = e Xe−kYk−1 Xe+ Ye ,
Proof. — The equality (1.12) follows from (1.10) by replacingζby ζ2 andd by e. The equality (1.13) follows from (1.12) by replacingY byζY (note thatζe =−1). Finally, the
equality (1.14) follows from (1.13) by replacing k by e− k + 1 (note that ζ−(k −1)(2i +1) = −(ζe)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 [AlFo]. We set
q = x y , r = xd+ yd, Q = X Y et R = Xd+ Yd. 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. If0 ¶ i ¶ d, we set ai ,0= xd−iYi+ yd−iXi. Note that a0,0= r and ad ,0= R . Finally, let eu0= x X + y Y . Thenai ,0,eu0∈ C[V × V∗]W.
We will now describe some relations between these invariants. For this, let eu(i )
0 =
(x X )i+ (y Y )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. Firt of all, eui 0 = i X j =0 i j (x X )j(y Y )i− j = X 0 ¶ j< i2 i j (qQ )j (x X )i−2 j+ (y Y )i−2 j+ 0 ifi is odd, i i 2 (qQ )i/2 ifi is even. Therefore, eui 0= X 0 ¶ j<i2 i j (qQ )jeu(i−2 j ) 0 + 0 ifi is odd, i i 2 (qQ )i/2 ifi is even.
So, by triangularity of this formula, an easy induction shows that there exists a family of integers(ni , j)0 ¶ j ¶ i/2such that
(2.1) eu(i ) 0 = X 0 ¶ j ¶2i ni , j(qQ )jeui−2 j 0 , withni ,0= 1for alli.
On the other hand, one can check that the following relations hold (for1 ¶ i ¶ j ¶ d−1): (Z0i) eu0ai ,0= q ai +1,0+ Q ai −1,0 (Z0i , j) ai ,0aj ,0= qd− jQieu( j−i ) 0 + r ai + j ,0− qd−i − jeu (i + j ) 0 if2 ¶ i + j ¶ d , r R− eu(d )0 ifi + j = d , R ai + j−d ,0−Qi + j−deu (2d−i − j ) 0 ifd ¶ i + j ¶ 2d− 2. Using (2.1), these last relations can be viewed as relations betweenq, r,Q,R,eu0,a1,0, a2,0,. . . ,ad−1,0.
Theorem 2.2. — The algebra of invariantsC[V × V∗]W admits the following presentation: Generators: q , r,Q , R , eu0, a1,0, a2,0, . . ., ad−1,0 Relations: ¨ (Z0 i) for1 ¶ i ¶ d− 1, (Z0i , j) for1 ¶ i ¶ j ¶ d− 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 = P•⊕ P•eu0⊕ P•eu2
0⊕ ··· ⊕ P•eud0 ⊕ P•a1,0⊕ P•a2,0⊕ ··· ⊕ P•ad−1,0.
Proof. — LetE∗denote the subspace ofC[V ]defined by
E∗= C⊕
d−1
M
i =1
CXi⊕ CYi⊕ C(Xd− Yd).
ThenE∗ is a graded sub-C[W ]-module ofC[V ]. If0 ¶ i ¶ d, letE∗
i denote the
homoge-neous component of degreei of E∗. Whenever1 ¶ i ¶ d − 1, it affords χi for character
whereasE0∗andEd∗afford respectively1W andǫfor characters. Similarly, we define
E = C⊕
d−1
M
i =1
Cxi⊕ Cyi ⊕ C(xd− yd).
ThenE is a graded sub-C[W ]-module ofC[V∗]. If 0 ¶ i ¶ d, letEi denote the
homoge-neous component of degreei of E. Whenever 1 ¶ i ¶ d − 1, it affords χi for character
whereasE0andEd afford respectively1W andǫfor characters. Moreover, the morphism
ofC[V ]W-modulesC[V ]W ⊗ E∗→ C[V ]induced by the multiplication is aW-equivariant
isomorphism and the morphism ofC[V∗]W-moduleC[V∗]W ⊗ E → C[V∗]induced by the
multiplication is aW-equivariant isomorphism. Consequently,
(♣) C[V × V∗]W = P•⊗ (E∗⊗ E )W.
An easy computation of the subspaces(Ei∗⊗ Ej)W based on the previous remarks show
that
(1, eu0, eu(2)0 , . . ., eu(d0 −1), (Xd− Yd)(xd− yd), a1,0, a2,0, . . ., ad−1,0)
is aC-basis of(E⊗ E∗)W. By(♣), it is also aP
•-basis ofC[V × V∗]W. On the other hand,
(Xd− Yd)(xd− yd) = 2eu(d )0 − R r, so(1, eu0, eu(2)0 , . . ., eu(d )0 , 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•eu2 0⊕ ··· ⊕ P•eu d 0 ⊕ P•a1,0⊕ P•a2,0⊕ ··· ⊕ P•ad−1,0, which shows the last assertion of the Theorem.
It also proves thatC[V × V∗]W = C[q , r,Q , R , eu0, a1,0, a2,0, . . ., ad
−1,0]. LetA1,A2, . . . ,Ad−1 be indeterminates overC[q , r,Q , R ]: we have a surjective morphism
C[q , r,Q , R , E , A1, A2, . . ., Ad−1]−։ C[V × V∗]W
which sendsq, r,Q,R,E,A1, A2, . . . ,Ad−1 onq,r,Q,R,eu0,a1,0,a2,0,. . . ,ad−1,0 respec-tively.
For1 ¶ i ¶ j ¶ d−1, letFi (respectivelyFi , j) denote the element of the polynomial
alge-braC[q , r,Q , R , E , A1, A2, . . ., Ad
−1]corresponding to the relation(Zi0)(respectively (Zi , j0 )).
Fi’s and the Fi , j’s. We denote by q0, r0, Q0, R0, E0, A1,0, A2,0, . . . , Ad−1,0 the respective images ofq, r,Q, R, E, A1, A2, . . . , Ad−1 in A. We then have a surjective morphism of
bigradedC-algebrasϕ : A ։ C[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,
(♦) dimbigrC (A) ¾ dim
bigr
C (C[V × V∗]
W),
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
A′= P0+ P0E0+ P0E02+··· + P0E0d+ P0A1,0+ P0A2,0+··· + P0Ad−1,0. By construction,
(♥) dimbigrC (A′) ¶ dim
bigr
C (C[V × V∗]
W).
We will prove that
(♠) A′is a subalgebra ofA.
For this, taking into account the form of theFi’s and theFi , j’s, it is sufficient to show that E0d +1∈ A′. But, by (2.1), A1,0Ad−1,0= Q0q0 X 0 ¶ j<(d−2)/2 nd−2, jE d−2−2 j 0 + R0r0− X 0 ¶ j<d /2 nd , jE d−2 j 0 , andnd ,0= 1. So E0d=−A1,0Ad−1,0+ Q0q0 X 0 ¶ j<(d−2)/2 nd−2, jE d−2−2 j 0 + R0r0− X 1 ¶ j<d /2 nd , jE d−2 j 0 and so E0d +1=−E0A1,0Ad−1,0+ Q0q0 X 0 ¶ j<(d−2)/2 nd−2, jE0d−1−2 j + R0r0− X 1 ¶ j<d /2 nd , jE d +1−2 j 0 .
It is then sufficient to show thatE0A1,0Ad−1,0∈ A′. ButE0A1,0Ad−1,0= (q0A2,0+ Q0r0)Ad−1,0, which concludes the proof of(♠).
SinceA′containsq0,r0,Q0,R0,E0,A1,0,A2,0, . . . ,Ad−1,0and sinceAis generated by these
elements, we haveA = A′. It follows from(♦)and(♥)thatdimbigrC (A) = dim bigr
C (C[V×V∗]W), which shows thatϕ : A→ C[V × V∗]W is an isomorphism. In other words, this shows that
the presentation ofC[V × V∗]W given in Theorem 2.2 is correct.
It remains to prove the minimality of this presentation. The minimality of the number of generators follows from [BoTh]. Let us now prove the minimality of the number of relations. For this, let p0 denote the bi-graded maximal ideal of P• and set B = C[V × V∗]W/p0C[V × V∗]W. We denote bye,a1,a2,. . . ,ad−1 the respective images ofeu0,a1,0,
a2,0,. . . ,ad−1,0inB. Hence,
andB admits the following presentation: Generators: e , a1, a2, . . ., ad−1 Relations (1 ¶ i ¶ j ¶ d− 1) : e ai= 0 aiaj = ¨ 0 sii + j6= d, −ed sii + j = d.
It is sufficient to prove that the number of relations ofBis minimal. By reducing modulo
the idealB e, we get that all the relations of the formaiaj = 0or−ed are necessary. By
reducing modulo the idealB a1+··· + B ai−1+ B ai +1+··· B ad−1, we get that the relations e ai= 0are necessary.
Remark 2.3. — It is easily checked that the elementτdefined in §1.A satisfies
τq = q , τQ = Q , τeu
0= eu0 and τai ,0=−ai ,0
for all0 ¶ i ¶ d (this follows from the fact thatξd=−1).
Remark 2.4 (Gradings). — The algebra C[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 2.2 are bi-homogeneous.
This (N× N)-grading induces a Z-grading such that any bi-homogeneous element of
bi-degree(m, n)isZ-homogeneous of degreen− m (in other words, elements ofV have Z-degree−1while elements ofV∗haveZ-degree1).
3. Cherednik algebras
3.A. Definition. — We denote byC theC-vector space of mapsRef(W )→ Cwhich are
constant on conjugacy classes. Ifi ∈ Z/d Z, we denote by Ci the element of C∗ which
sendsc∈ C tocsi. By (1.1),Ci= Ci +2. LetA = C0andB = C1. Ifd is odd, thenA = B and C[C ] = C[A]whereas, ifd is even, thenA6= B (see (1.2)) andC[C ] = C[A, B ].
The generic rational Cherednik algebra att = 0is theC[C ]-algebraHdefined as the
quo-tient ofC[C ]⊗ T(V ⊕ V∗) ⋊ W by the following relations (here,T(V ⊕ V∗)is the tensor
algebra ofV ⊕ V∗overC): (3.1) [u, u′] = [U ,U′] = 0, [u,U ] =−2 X i∈Z/d Z Ci 〈u, αi〉 · 〈α∨i,U〉 〈α∨ i,αi〉 si,
forU,U′∈ V∗ andu,u′∈ V. Note that we have followed the convention of [BoRo].
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∗) ⋊ W inH.
• There is a unique morphism ofC-algebrasC[V∗]→ Hsendingu∈ V ⊂ C[V∗]to the
• There is a unique morphism ofC-algebrasCW → Hsendingw ∈ W to the class of
w∈ T(V ⊕ V∗) ⋊ W inH.
• TheC-linear mapC[C ]⊗ C[V ] ⊗ CW ⊗ C[V∗]−→ Hinduced by the three morphisms
defined above and the multiplication map is surjective. Note that it isC[C ]-linear.
The last statement is strenghtened by the following fundamental result by Etingof and Ginzburg [EtGi, Theorem 1.3] (see also [BoRo, Theorem 4.1.2]).
Theorem 3.2 (Etingof-Ginzburg). — The multiplication mapC[C ]⊗C[V ]⊗CW ⊗C[V∗]−→
His an isomorphism ofC[C ]-modules.
Remark 3.3. — By [BoRo, §3.5.C], the groupNGLC(V )(W )acts naturally onH. It follows
from (1.6) that
τA = B and τB = A.
Here,τis the element ofNGLC(V )(W )defined in §1.A.
Remark 3.4 (Gradings). — The algebraHadmits a natural(N×N)-grading, by puttingV
in bi-degree(1, 0),V∗ in bi-degree(0, 1),W in degree(0, 0)andC∗ in degree(1, 1)(see for
instance [BoRo, §3.2]).
This(N×N)-grading induces aZ-grading such that any homogeneous element of
bi-degree(m, n)isZ-homogeneous of degreen−m. In other words,deg(V ) =−1,deg(V∗) = 1
anddeg(W ) = 0 = deg(C∗) = 0.
3.B. Specialization. — Givenc∈ C, we denote byCc the maximal ideal ofC[C ]defined
byCc={f ∈ C[C ] | f (c ) = 0}: it is the ideal generated by(Ci− csi)i∈Z/d Z. We set Hc = (C[C ]/Cc)⊗C[C ]H= H/CcH.
TheC-algebraHc is the quotient of theC-algebraT(V ⊕ V∗) ⋊ W by the ideal generated
by the following relations:
(3.5) [u, u′] = [U ,U′] = 0, [u,U ] =−2 X i∈Z/d Z csi 〈u, αi〉 · 〈α∨i,U〉 〈α∨ i,αi〉 si, forU,U′∈ V∗ andu,u′∈ V.
Remark 3.6 (Grading). — The ideal Cc is not bi-homogeneous (except if c = 0) so the
algebraHc does not inherit fromHan(N× N)-grading. However,Cc isZ-homogeneous,
3.C. Calogero-Moser space. — We denote by Z the centre ofH. By [EtGi], it contains C[V ]W andC[V∗]W so, by Theorem 3.2, it contains the subalgebra
P = C[C ]⊗ C[V ]W⊗ C[V∗]W.
Similarly, ifc ∈ C, we denote byZc the centre ofHc: it turns out [BoRo, Corollary 4.2.7]
thatZc is the image ofZ and that the image ofP inZc isP•= C[V ]W×C[V∗]W. Recall also
(see for instance [BoRo, Corollary 4.2.7]) that
(3.7) Z is a freeP-module of rank|W |.
Since theC-algebraZ is finitely generated, we can associate to it an algebraic variety
overC, called the generic Calogero-Moser space, and which will be denoted byZ. Ifc ∈ C,
we denote byZc the algebraic variety associated with theC-algebraZc.
3.D. About the presentation ofZ. — We follow here the method of [BoTh]. Ifh ∈ H,
it follows from Theorem 3.2 that there exists a unique family of elements (hw)w∈W of
C[C ]⊗ C[V ] ⊗ C[V∗]such that
h = X
w∈W hww.
We define theC[C ]-linear mapTrunc : H→ C[C ] ⊗ C[V ] ⊗ C[V∗]by Trunc(h ) = h1.
The next lemma is proved in [BoTh]:
Lemma 3.8. — The restriction ofTrunctoZ yields an isomorphism of bi-gradedC[C ]-modules
Trunc : Z −→ C[C × V × V∼ ∗]W.
We then seteu= Trunc−1(eu0)and, for0 ¶ i ¶ d, ai= Trunc−1(ai ,0).
An explicit algorithm for computing the inverse map Trunc−1 is described in [BoTh].
Note thatTrunc−1(p ) = p forp ∈ P, so thata0= a0,0= r andad= ad ,0= R. By [BoTh], the
relations (Z0i)1 ¶ i ¶ d−1 and (Zi , j0 )1 ¶ i ¶ j ¶ d−1 can be deformed into relations (Zi)1 ¶ i ¶ d−1 and(Zi , j)1 ¶ i ¶ j ¶ d−1and it follows from Theorem 2.2 and [BoTh] that:
Theorem 3.9. — The centreZ ofHadmits the following presentation, as aC[C ]-algebra:
Generators: q , r,Q , R , eu, a1, a2, . . ., ad−1 Relations: ¨ (Zi) for1 ¶ i ¶ d− 1, (Zi , j) for1 ¶ i ¶ j ¶ d− 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 ⊕ P eu ⊕ P eu2⊕ ··· ⊕ P eud⊕ P a1⊕ P a2⊕ ··· ⊕ P ad−1.
It must be said that we have no way to determine explicitly the relations(Zi , j)in
gen-eral: we will describe them precisely only ford∈ {3, 4, 6}in §8. Note that the information
Remark 3.10 (Gradings). — The bi-grading and the Z-grading on the algebra H
con-structed in Remark 3.4 induce a bi-grading and aZ-grading onZ. Note that the map Truncis bi-graded, so that the generators given in Theorem 3.9 are bi-homogeneous.
On the other hand, the deformation process for the relations described in [BoTh] re-spects 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 3.9 are bi-homogeneous.
From now on, and until the end of this paper, we fix a parameter c ∈ C and we seta = cs andb = ct. Note that, ifd is odd, thena = b.
3.E. Poisson bracket. — Recall from [BoRo, §4.4.A] that the algebraZ is 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 restric-tion of theW-equivariant canonical Poisson bracket onC[V × V∗]. This Poisson bracket
induces a Poisson bracket onZc. It satisfies
(3.11) {q ,Q } = eu
(see [De, §4] or [BEG, §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 [BoRo, §11.1]. It will be given in Table 4.14 at the end of this sec-tion. We will use the alternative definition [BoRo, Theorem 13.4.2], which is more con-venient for computational purposes (see also [BoTh]). So, following [BoRo, Chapter 13], we set Dx = X i∈Z/d Z ǫ(si)csi 〈x , αi〉 αi si=− X i∈Z/d Z csi 1 X− ζiY si∈ C(V )[W ] and Dy = X i∈Z/d Z ǫ(si)csi 〈y , αi〉 αi si= X i∈Z/d Z csi ζi X − ζiY si∈ C(V )[W ].
We denote byGau(W , c )the sub-C(V )-algebra ofC(V )[W ]generated by Dx andDy (it is
commutative by [BoRo, §13.4.B]). Note that this algebra is not necessarily split. IfL is a
simpleGau(W , c )-module, and ifχ∈ Irr(W ), we denote bymultCML ,χ the multiplicity ofLin
a composition series of theResCGau(W ,c )(V )W C(V )Eχ, whereEχ is aCW-module affording the
characterχ. We then set
γL=
X
χ∈Irr(W )
multCML ,χ χ.
The set of Calogero-Moserc-cellular charactersis
We will denote byEGau
1 (respectivelyEǫGau, respectivelyLkGau) the restriction ofC(V )E1W
(respectivelyC(V )Eǫ, respectivelyC(V )Eχ
k) toGau(W , c ). Ifd is even, then the restriction
ofC(V )Eǫ
s (respectivelyC(V )Eǫt) toGau(W , c )will be denoted byE
Gau
s (respectivelyEtGau).
Remark 4.1. — Note that, sinceGau(W , c )⊂ C(V )W, every simpleGau(W , c )-module
oc-curs as a composition factor of someResCGau(W ,c )(V )W C(V )Eχ, forχ running overIrr(W ).
Notation. — For simplifying the computation in this section, we set
Dx′=Y d− Xd d Dx and D ′ y = Xd− Yd d Dy,
so thatGau(W , c )is the sub-C(V )-algebra ofC(V )W generated byD′
x andDy′.
4.A. The case wherea = b = 0. — Whenevera = b = 0, there is only one Calogero-Moser
0-cellular character [BoRo, Corollary 17.2.3], namely the regular characterPχ∈Irr(W )χ(1)χ.
4.B. The case wherea 6= b = 0. — We will assume here, and only here, thatb = 06= a:
this forcesd to be even (and we writed = 2e). LetW′be the subgroup ofW generated
bys = s0ands2= t s t. ThenW′is a dihedral group of order2e and W =〈t 〉 ⋉ W′.
Let c′ denote the restriction of c to Ref(W′): then c′ is constant (and equal toa) and
Gau(W , c ) = Gau(W′, c′). It then follows from the definition that the Calogero-Moserc
-cellular characters are all the characters of the formIndWW′γ′, whereγ′is a Calogero-Moser c′-cellular character ofW′. These charactersγ′will be determined in the next subsection
and so it follows that the list ofc-cellular characters ofW is
(4.2) 1 +ǫt, ǫs+ǫ,
(d−2)/2
X
k =1
χk.
Remark 4.3. — Ifa = 0 andb 6= 0, then one can use the elementn∈ N of order2such
thatns = t andnt = s to be sent back to the previous case. We then deduce from (4.2) that
the list ofc-cellular characters ofW is
(4.4) 1 +ǫs, ǫt+ǫ,
(d−2)/2
X
k =1
χk.
4.C. The equal parameters case. — We assume here, and only here, thata = b 6= 0. Then, if1 ¶ k ¶ d− 1, then ρk(Dx′) =−a Yd− Xd d X i∈Z/d Z 1 X − ζiY 0 ζk i ζ−k i 0 = a 0 Xk−1Yd−k Xd−k −1Yk 0 by using (1.11). Similarly, ρk(Dy′) = a 0 XkYd−k −1 Xd−kYk−1 0 =X Y ρk(D ′ x).
If we denote byM the diagonal matrixX 0
0 Y
, then it follows from the previous formu-las that
(4.5) ∀ 1 ¶ k ¶ d − 2, ∀ D ∈ Gau(W , c ), M ρk(D )M−1=ρk +1(D ).
This implies that
(4.6) ∀ 1 ¶ k ¶ d − 2, LkGau≃ L
Gau
k +1.
SinceTr(ρk(Dx′)) = Tr(ρk(Dy′)) = 0, the nature of the restriction ofρk toGau(W , c )depends
on whether− det(ρk(Dx′)) = a2XdYd−2is a square inC(V ). Two cases may occur:
First case: assume thatdis odd. — Then− det(ρk(Dx′))is not a square inC(V )(for1 ¶ k ¶ d−
1), so it follows thatLkGauis simple (but not absolutely simple) and it follows from (4.6)
(4.7) Irr(Gau(W , c )) ={E1Gau,EǫGau,L1Gau}.
Moreover, the list ofc-cellular characters is given in this case by
(4.8) 1W, ǫ and
(dX−1)/2
k =1
χk.
Second case: assume thatd is even. — In this case, it is easily checked thatE1Gau,EǫGau,EsGau
andEGau
t are four non-isomorphic simpleGau(W , c )-modules. Also, if1 ¶ k ¶ d− 1, then
LkGau≃ E Gau s ⊕ E Gau t , by (4.6). Therefore, (4.9) Irr(Gau(W , c )) ={E1Gau,E Gau ǫ ,E Gau s ,E Gau t }.
and the list ofc-cellular characters is given in this case by
(4.10) 1W, ǫ, ǫs+ (d−2)/2 X k =1 χk and ǫt+ (d−2)/2 X k =1 χk.
4.D. The opposite parameters case. — We assume here, and only here, thatb =−a 6= 0. This forcesd to be even. Then, using the automorphism ofHinduced by the linear
char-acterǫs (see [BoRo, §3.5.B]), one can pass from the equal parameter case to the opposite
parameter case by tensorizing byǫs. Therefore, the list ofc-cellular characters is given in
this case by (4.11) ǫs, ǫt, 1W+ (d−2)/2 X k =1 χk and ǫ + (d−2)/2 X k =1 χk.
4.E. The generic case. — We assume here, and only here, thata b (a2− b2)6= 0(so that
we are not in the cases covered by the previous subsections). Note that this forcesd to be
even. We will prove that the list ofc-cellular characters is given in this case by
(4.12) 1W, ǫ, ǫs, ǫt and
(d−2)/2
X
k =1
χk.
Proof. — We have, for1 ¶ k ¶ e− 1,
ρk(Dx′) = Xd− Yd d a X i∈Z/e Z 1 X− ζ2iY 0 ζ2k i ζ−2k i 0 +b X i∈Z/e Z 1 X− ζ2i +1Y 0 ζk (2i +1) ζ−k (2i +1) 0 .
So it follows from (1.12), (1.13) and (1.14) that
ρk(Dx′) = 1 2 0 Xk−1Ye−k (a− b )Xe+ (a + b )Ye Xe−k −1Yk (a + b )Xe+ (a− b )Ye 0 .
The matrixρk(Dy′)can be computed similarly and we can deduce that,
∀ 1 ¶ k ¶ e − 2, ∀ D ∈ Gau(W , c ), M ρk(D )M−1=ρk +1(D ). Therefore, (∗) ∀ 1 ¶ k ¶ e − 2, LkGau≃ Lk +1Gau. Moreover, − det(ρk(Dx′)) = 1 4X e−2Ye (a − b )Xe+ (a + b )Ye(a + b )Xe+ (a− b )Ye. Sincea b (a2− b2)6= 0,− det(ρ
k(Dx′))is not a square inC(V ), and soLkGauis simple (but not
absolutely simple) for1 ¶ k ¶ e− 1.
Moreover, an easy computation shows that EGau
1 , EǫGau, EsGau and EtGau are pairwise
non-isomorphic simpleGau(W , c )-modules. So it follows from(∗)that
(4.13) Irr(Gau(W , c )) ={E1Gau,E Gau ǫ ,E Gau s ,E Gau t ,L Gau 1 } and that (4.12) holds.
Parameters d = 2e (even) d = 2e− 1(odd) a = b = 0 X χ∈Irr(W ) χ(1)χ X χ∈Irr(W ) χ(1)χ a6= b = 0 1W +ǫt, ǫs+ǫ, e−1 X k =1 χk ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = 06= b 1W +ǫs, ǫt+ǫ, e−1 X k =1 χk ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = b6= 0 1W, ǫ, ǫs+ e−1 X k =1 χk, ǫt+ e−1 X k =1 χk 1W, ǫ, e−1 X k =1 χk a =−b 6= 0 ǫs, ǫt, 1W+ e−1 X k =1 χk, ǫ + e−1 X k =1 χk ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a b (a2− b2)6= 0 1 W, ǫs, ǫt, ǫ, e−1 X k =1 χk ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛
TABLE4.14. Calogero-Moser cellular characters of W
4.F. Conclusion. — The following Table 4.14 gathers all the possible list of cellular
char-acters ofW, according to the values of the parametersaandb.
Remark 4.15. — Whenevera,b∈ R, the Kazhdan-Lusztig cellular characters for the
di-hedral groups are easily computable (see for instance [Lus]) and a comparison with Ta-ble 4.14 shows that they coincide with Calogero-Moser cellular characters: this is [BoRo, Conjecture CAR] for dihedral groups.
5. Calogero-Moser families
The aim of this section is to compute the Calogero-Moser c-families of W (as defined
in [BoRo, §9.2]) for all values ofc. The result is given in Table 5.2. Note that this result
is not new: the Calogero-Moser families have been computed by Bellamy in his the-sis [Bel1]. We provide here a different proof, which uses the computation of Calogero-Moser cellular characters.
5.A. Families. — To any irreducible characterχ, Gordon [Gor] associates a simpleHc
-moduleLc(χ) (we follow the convention of [BoRo, Proposition 9.1.3]). We denote by
Ωχc : Z → C the morphism defined by the following property: if z ∈ Z, then Ωcχ(z ) is
the scalar by which z acts on Lc(χ) (by Schur’s Lemma). We say that two irreducible
characters χ and χ′ belong to the same Calogero-Moser c-family if Ωcχ = Ωχc′ (see [Gor]
or [BoRo, Lemma 9.2.3]: note that Calogero-Moser families are called Calogero-Moser blocksin [Gor]). We give here a different proof of a theorem of Bellamy [Bel1]:
Theorem 5.1 (Bellamy). — Let c ∈ C and let χ and χ′ be two irreducible characters of W.
Thenχ andχ′lies in the same Calogero-Moser families if and only ifΩcχ(eu) = Ωcχ′(eu).
Conse-quently, the Calogero-Moser families are given by Table 5.2.
Proof. — By [BoRo, Proposition 7.3.2], the values ofΩχc(eu)are given as follows:
(a) Ifd = 2e− 1is odd, then
Ω1cW(eu) = d a , Ωǫc(eu) =−d a , Ωχck(eu) = 0 if1 ¶ k ¶ e− 1. (b) Ifd = 2e is even, then Ωc1W(eu) = e (a + b ), Ωcǫ(eu) =−e (a + b ), Ωcǫ s(eu) = e (a− b ), Ωcǫ t(eu) = e (b− a ), Ωcχ k(eu) = 0 if1 ¶ k ¶ e− 1.
But two irreducible characters occuring in the same Calogero-Moserc-cellular character
necessarily belong to the same Calogero-Moserc-family [BoRo, Proposition 11.4.2]. So
the Theorem follows from (a), (b) and Table 4.14.
Remark 5.3. — Whenevera,b∈ R, the Kazhdan-Lusztig families for the dihedral groups
are easily computable (see for instance [Lus]) and a comparison with Table 5.2: this is [BoRo, Conjecture FAM] for dihedral groups. Note that this was already proved by Bellamy [Bel1].
5.B. Cuspidal families. — Recall that the algebrasZ andZc are endowed with a
Pois-son bracket{, }. This Poisson structure has been used by Bellamy [Bel2] to define the
notion of cuspidal Calogero-Moser families. IfF is a Calogero-Moser c-family, we set
mcF= Ker(Ωχc)⊂ Zc (note thatΩχc factorizes through the projectionZ ։ Zc). The
Calogero-Moserc-family F is called cuspidal if{mFc , mFc } ⊂ mcF. They have been determined for
most of the Coxeter groups by Bellamy and Thiel [BeTh]. In our case, we recall here their result, as well as a proof for the sake of completeness.
Proposition 5.4. — The list of cuspidal Calogero-Moser families is given by Table 5.5. The
following properties hold:
(a) A Calogero-Moser familyF is cuspidal if and only if|F | ¾ 2andχ1∈ F (and thenχk∈ F
Parameters d = 2e (even) d = 2e− 1(odd) a = b = 0 Irr(W ) Irr(W ) a6= b = 0 {1W,ǫt}, {ǫs,ǫ}, {χ1, . . .,χe−1} ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = 06= b {1W,ǫs}, {ǫt,ǫ}, {χ1, . . .,χe−1} ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = b6= 0 {1W}, {ǫ}, {ǫs,ǫt,χ1, . . .,χe−1} {1W}, {ǫ}, {χ1, . . .,χe−1} a =−b 6= 0 {ǫs}, {ǫt}, {1W,ǫ, χ1, . . .,χe−1} ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a b (a2− b2)6= 0 {1W}, {ǫs}, {ǫt}, {ǫ}, {χ1, . . .,χe−1} ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛
TABLE5.2. Calogero-Moser families of W
(b) There is at most one cuspidal family. Ifd ¾ 5, there is always exactly one cuspidal family.
Proof. — The main (easy) observation is that{q ,Q } = eu. This implies that, ifχ belongs
to a cuspidal families, thenΩcχ(eu) = 0. Since it follows from Table 5.2 that the
Calogero-Moser c-families are determined by the values of Ωcχ(eu), this implies that there is at
most one cuspidal Calogero-Moser c-family, and that it must contain χ1 (and χk, for
1 ¶ k< d /2).
Also, since a Calogero-Moser c-family of cardinality 1cannot be cuspidal [Gor], this
shows the “only if” part of (a). It remains to prove the “if” part of (a). So assume that
χ1∈ F and that |F | ¾ 2. By Bellamy theory [Bel2], there exists a non-trivial parabolic subgroupW′ofW and a cuspidal Calogero-Moserc′-familyF′of W′(here,c′denotes the restriction ofc toRef(W′)) which are associated withF. Again, by [Bel2],|F | = |F′|.
We must show thatW = W′. So assume thatW′6= W. Then|W′| = 2and so|F′| ¶ 2and c′must be equal to0. This forces|F | = 2anda b = 0(andc 6= 0). This can only occur in
typeG2: but the explicit computation of the Poisson bracket in typeG2shows thatF is necessarily cuspidal.
Parameters d = 2e (even) d = 2e− 1(odd) a = b = 0 Irr(W ) Irr(W ),e ¾ 2 a6= b = 0 {χ1, . . .,χe−1},e ¾ 3 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = 06= b {χ1, . . .,χe−1},e ¾ 3 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a = b6= 0 {ǫs,ǫt,χ1, . . .,χe−1},e ¾ 2 {χ1, . . .,χe−1},e ¾ 3 a =−b 6= 0 {1W,ǫ, χ1, . . .,χe−1},e ¾ 2 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛ a b (a2− b2)6= 0 {χ 1, . . .,χe−1},e ¾ 3 ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛❛
TABLE5.5. Calogero-Moser cuspidal families of W
IfF is a cuspidal Calogero-Moserc-family, then the Poisson bracket{, }stabilizes the
maximal idealmcF and so it induces a Lie bracket [, ] on the cotangent space Liec(F ) =
mcF/(mFc )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 §8, using explicit computations).
Theorem 5.6. — LetF be a cuspidal Calogero-Moserc-family ofW. Then:
(a) Ifd = 4(i.e. ifW is of typeB2) anda = b6= 0, thenLiec(F ) ≃ sl3(C)is a simple Lie algebra of typeA2.
(b) Ifd = 6(i.e. ifW is of typeG2) anda b (a2− b2)6= 0, thenLiec(F ) ≃ sp4(C)is a simple Lie algebra of typeB2.
6. Calogero-Moser cells
Letc ∈ C. The main theme of [BoRo] is a construction of partitions ofW into
Calogero-Moser left, right and two-sidedc-cells, using a Galois closureM of the field extension
Calogero-Moser cells are defined [BoRo, Definition 6.1.1] as orbits of particular sub-groups ofG. Our aim in this section is to prove [BoRo, Conjectures L and LR] whenever d is odd. We first start by trying to determine the Galois groupG.
It is proved in [BoRo, §5.1.C] that there is an embedding
G,−→ SW
(here,SW denotes the symmetric group on the setW, and we identifyG with its image)
such that
ι(W × W ) ⊂ G ,
whereι : W × W −→ SW denotes the morphism obtained by lettingW× W act by left and
right translations ((x , y )· z = x z y−1). LetA
W denote the alternating group onW.
Theorem 6.1. — Ifd is odd, thenG = SW.
Proof. — We first prove an easy lemma about finite permutation groups.
Lemma 6.2. — LetΓ be a subgroup ofSW. We assume that:
(1) d is odd;
(2) Γ containsι(W × W );
(3) Γ is primitive. ThenΓ = SW.
Proof of Lemma 6.2. — Sinced is odd, the action of σ = ι(c , c )onW is a
cycle of lengthd (it fixes〈c 〉and acts by a cycle onW\ 〈c 〉by (1.1)).
More-over,ι(c , 1)andι(1, c )belong to the centralizer ofι(c , c )inΓ, soCΓ(c )6= 〈c 〉.
SinceΓ is primitive, it follows from [DiMo, Exercise 7.4.12] thatΓ = SW or
AW.
But note that the action ofι(s , 1)is a product ofd transpositions, so it is
an odd permutation (becausedis odd). Therefore,Γ 6= AW and the Lemma
is proved.
Assume that d is odd. By the description of the Calogero-Moserc-families given in
Table 5.2, it follows from [BoRo, Theorem 10.2.7] that there exists a subgroup I of G
which have three orbits for its action onW, of respective lengths1, 1and2d− 2. Since
ι(W × W )is transitive onW,G is also transitive and we may assume that one of the two
orbits of length1is the singleton{1}. Let∆W denotes the diagonal inW × W. Its action
onW is by conjugacy: it has only one fixed point (because the center ofW is trivial). This
proves that the subgroup〈I , ι(∆W )〉acts transitively onW \ {1}. SoG is2-transitive and,
in particular, primitive. The Theorem now follows from Lemma 6.2 above.
Proof. — Assume thatcs ∈ Rfor alls ∈ Ref(W ). The computation of Calogero-Moserc
-families andc-cellular characters shows that, if we choose randomly two prime ideals as
in [BoRo, Chapter 15], then the associated Calogero-Moser two-sided or leftc-cells have
the same sizes as the Kazhdan-Lusztig two-sided or leftc-cells respectively (see [BoRo,
Chapters 10 and 11]). Since the Galois groupG coincides with SW, we can manage to
change the prime ideals so that Calogero-Moser and Kazhdan-Lusztigc-cells coincide.
Remark 6.4. — Let
SBW ={σ ∈ SW | ∀ w ∈ W , σ(w0w ) = w0σ(w )}
and SDW =SBW∩ AW.
Note that, in our case,SB
W (respectivelySDW) is a Weyl group of typeBd (respectivelyDd)
and thatSDW is a normal subgroup ofSWB of index2.
Assume here, and only here, that d is even. It then follows from [BoRo,
Proposi-tion 5.5.2] thatG ⊂ SB
W. We would bet a few euros (but not more) thatG = SDW. This has
been checked ford = 4in [BoRo, Theorem 19.6.1] and it will be checked in 8.12 whenever d = 6. Let us just prove a few general facts.
First, let W = W¯ /Z(W ) (it is a dihedral group of order d) and let ¯ι : ¯W × ¯W → SW¯
denote the morphism induced by the action by left and right translations. LetG¯ denote
the image ofG in SW¯ (indeedG ⊂ SWB and there is a natural morphism SWB → SW¯).
Then, ifa = b6= 0, the Calogero-Moser two-sidedc-cells have cardinalities1,1and2d− 2
(by [BoRo, Theorem 10.2.7]) so it follows from the definition of Calogero-Moser cells that there exists a subgroupI1ofG whose orbits have cardinalities1,1and2d− 2. Therefore, the image I¯1 of I1 in G¯ have orbits of cardinalities 1 andd − 1. Consequently, G¯ is 2
-transitive. Similarly, takingc such thata b (a2− b2)6= 0, we get that there is a subgroupI¯ 2 ofG¯ whose orbits have cardinalities1,1andd− 2. Therefore,
(♦) G¯ is3-transitive.
On the other hand,
(♥) G¯ contains¯ι( ¯W× ¯W ).
As a consequence, we get
(♠) Ifd/2is odd, thenG = S¯ W¯. Indeed, this follows from Lemma 6.2.
7. Fixed points
The Z-grading on the C-algebra H (defined in Remark 3.4) induces an action of the
groupC×onHas follows [BoRo, §3.5.A]. Ifξ∈ C×then: • Ify ∈ V, then ξy =ξ−1y.
• Ifx∈ V∗, thenξx =ξx.
• Ifw∈ W, then ξw = w.
So the centerZ inherits an action of C×, which may be viewed as a C×-action on the
Calogero-Moser spaceZ, which stabilizes all the fibersZc (forc∈ C).
Now, if m ∈ N∗, we denote by µ
m the group of m-th root of unity in C×. In [BoRo,
Conjecture FIX], R. Rouquier and the author conjecture that all the irreducible compo-nents of the fixed point varietyZµm are isomorphic to the Calogero-Moser space of some
other complex reflection groups (here,Zµm is endowed with its reduced structure). This conjecture will be checked ford ∈ {3, 4, 6}and anym in Section 8.
Theorem 7.1. — Assume thatd is odd. Then
Zµd
≃ {(a , u, v, e ) ∈ C4| (e − d a )(e + d a )ed−2= u v}.
Remark 7.2. — By [BoRo, Theorem 18.2.4], the above Theorem shows thatZµd
max,c is iso-morphic to the Calogero-Moser space associated with the cyclic group of order d and
some parameters, so it proves [BoRo, Conjecture FIX] in this case.
Proof. — The case whered = 1is not interesting, so we assume thatd ¾ 3. LetIdenote
the ideal ofZ generated by{ζz− z | z ∈ Z }. Then the algebra of regular functions onZµd isZ/pI. We will describeZ/I, and this will prove thatI =pIin this case. Therefore,
I =〈q ,Q , a1, a2, . . ., ad−1〉,
and so Z/I is generated by the images of A, r, R andeu. In the quotientZ/I, all the
equations ofZ-degree which is not divisible byd are automatically fulfiled, so it only
remains the equations(Zi ,d−i)(which is bi-homogeneous of bi-degree(d , d )). Also, note that(Z0i ,d−i)implies that
eud= r R mod〈I, A〉.
The only bi-homogeneous monomials inA,r,R andeuof bi-degree(d , d )arer Rand the eukAd−k (for0 ¶ k ¶ d). Therefore, the above equation implies that there exist complex
numbersλ0,. . . ,λd−1such that
(∗) eud+λd−1Aeud−1+··· + λ1Ad−1eu+λ0Ad≡ r R mod I. On the other hand, it follows from [BoRo, Corollary 9.4.4] that
(eu− d A)(eu + d A)eu2d−2= Y
χ∈Irr(W )
(eu− Ωχ(eu))χ(1)2≡ 0 mod 〈q ,Q , r, R 〉.
So(∗)implies that the polynomialtd+λd−1Atd−1+··· + λ1Ad−1t+λ0Ad divides(t− d A)(t + d A)t2d−2inC[A][t].
Since all the C×-fixed points belong to Zµd, this implies that t− d A andt+ d A both
dividetd+λd−1Atd−1+··· + λ1Ad−1t+λ0Ad. Therefore,
td+λd−1Atd−1+··· + λ1Ad−1t+λ0Ad= (t− d A)(t + d A)td−2, and so there remains only one relations in the quotientZ/I, namely
(eu− d A)(eu + d A)eud−2≡ r R mod I,
8. Examples
We are interested here in the cases where d ∈ {3, 4, 6}. This are the Weyl groups of
rank2 (of typeA2, B2 orG2). For each of these cases, we give a complete presentation of the centreZ ofH (using the algorithms developed in [BoTh]). We use these explicit
computations to check some of the facts that have been stated earlier in this paper. Most of the computations are done usingMAGMA[Mag].
8.A. The typeA2. — We work here under the following hypothesis:
We assume in this subsection, and only in this subsection, thatd = 3. In other words,W is a Weyl group of typeA2.
Using MAGMA and [BoTh], we can compute effectively the generators of the C[C ]
-algebraZ and we obtain the following presentation forZ (note thatC[C ] = C[A]because
A = B):
Proposition 8.1. — TheC[A]-algebraZ admits the following presentation:
Generators: q , r,Q , R , eu, a1, a2 Relations: eu a1= q a2+ r Q eu a2= Q a1+ q R a2 1= 4q2Q + r a2− q eu2+ 9A2q a1a2= 4qQ eu + r R− eu3+ 9A2eu a2 2= 4qQ 2+ R a 1−Q eu2+ 9A2Q The minimal polynomial ofeuis given by
t6− (6qQ + 9A2) t4− r R t3+ 9(q2Q2+ 2A2qQ ) t2
+3q r Q R t + q3R2+ r2Q3− 4q3Q3− 9A2q2Q2.
(8.2)
We conclude by proving [BoRo, Conjecture FIX] in this case (about the variety Zµm).
Note that the only interesting case is wherem divides the order of an element ofW. So m∈ {1, 2, 3}. The casem = 1is stupid while the casem = 3is treated in Theorem 7.1:
Proposition 8.3. — TheC[A]-algebraC[Zµ2]admits the following presentation:
Generators: q ,Q , eu Relations: q (eu2− 9A2− 4qQ ) = 0, eu(eu2− 9A2− 4qQ ) = 0, Q (eu2− 9A2− 4qQ ) = 0, In particular, ifa6= 0, then the varietyZµ2
c has two irreducible components:
(1) A component of dimension2defined by the equationeu2− 9a2− 4qQ = 0(which contains the pointsz1andzǫ).
