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Cédric Bonnafé, Ruslan Maksimau
To cite this version:
Cédric Bonnafé, Ruslan Maksimau. Fixed points in smooth Calogero-Moser spaces. Annales de l’Institut Fourier, Association des Annales de l’Institut Fourier, In press. �hal-01728667v3�
by
CÉDRIC BONNAFÉ & RUSLANMAKSIMAU
Abstract. —We prove that every irreducible component of the fixed point variety under the action ofµdin a smooth Calogero-Moser space is isomorphic to a Calogero-Moser space
asso-ciated with another reflection group.
Consider a finite subgroup W ⊂ GLC(V ) of automorphisms of a finite dimensional complex vector spaceV, generated by reflections and let c be a complex valued
func-tion on the set of conjugacy classes of reflecfunc-tions ofW. To the triple (V , W , c ), Etingof
and Ginzburg [10] have associated an affine varietyZc(V , W ), the Calogero-Moser space,
which is defined as the spectrum of the center of the rational Cherednik algebraHc “at
t = 0”(see Section 1 for a precise definition). The algebraHc carries aZ-grading (i.e., a
C×-action), which induces aC×-action on the Calogero-Moser spaceZc(V , W ).
We define a reflection subquotient of(V , W )to be a pair(V′, W′)whereV′is a subspace
ofV,W′⊂ GLC(V′)is generated by reflections and there exists a subgroupH ofW which stabilizesV′and whose image inGL
C(V′)is exactlyW′.
Note that any element in GLC(V ) normalizing W acts on Zc(V , W ). In [4,
Conjec-ture FIX], Rouquier and the first author proposed the following conjecConjec-ture:
Conjecture F.Letσ ∈ GLC(V )be an element of finite order and normalizingW and letX be an irreducible component ofZc(V , W )σ(endowed with its reduced
scheme structure). Then there exists a reflection subquotient (V′, W′)ofW and a complex valued functionc′on the set of conjugacy classes of reflections inW′
such thatX ≃ Zc′(V′, W′), as varieties endowed with aC×-action.
Note that the conjecture stated in [4, Conjecture FIX] does not give an explicit descrip-tion ofc′in terms ofc: it is just mentioned that the mapc 7→ c′ should be linear. Note
also that, as stated, it might be a little bit optimistic: maybe one should replaceX by its
normalization (as it is not clear whetherX is normal or not). A special case of an element
σnormalizingW is whenσ is a root of unity, viewed as the corresponding homothety
onV. The aim of this paper is to prove the following result:
Theorem.Conjecture F holds ifZc(V , W )is smooth andσis a root of unity.
The proof is by case-by-case analysis, as the triples (V , W , c ) such that Zc(V , W ) is
smooth are classified (see the works of Etingof-Ginzburg [10], Gordon [12], Bellamy [1]). The classification and the conjecture can be easily reduced to the case whereW acts
ir-reducibly onV/VW and in this case, the smoothness of Z
c(V , W ) implies that W is of
typeG (l , 1, n)or G4 in Shephard-Todd classification. The case ofG4 can be handled by
computer calculations (see Section 5), while the infinite family case will be handled by using an isomorphism betweenZc(V , W )and a quiver variety.
Commentary. The motivation for Conjecture F comes from the modular representation theory of finite reductive groups and conjectures of Broué-Malle-Michel about the en-domorphism algebra of some Deligne-Lusztig variety (see [7, §1.A] and [8, Conj. 5.7]). Experimentally, there is an astonishing analogy between the combinatorics involved in Broué-Malle-Michel conjectures and the one involved in the geometry ofZc(V , W )µd.
The results of this paper confirm this analogy in the particular case of the general linear groupG = GLn(Fq)(whose Weyl groupW is the symmetric groupSn). Let us be more
precise. Letℓbe a prime number not dividingq and letd denote the order ofq modulo ℓ. Then (see §4):
– Theℓ-blocks of unipotent characters ofG are parametrized by the possibled-cores
of partitions ofn, as well as the irreducible components ofZ1(Cn, Sn)µd.
– Unipotent characters lying in the blockBγparametrized by thed-coreγare in
bijec-tion with the irreducible characters ofG (d , 1, r )(wherer = (n − |γ|)/d). On the other
side, the irreducible componentXγofZ1(Cn, Sn)µd parametrized byγis isomorphic
toZc′(Cr,G (d , 1, r ))and so itsC×-fixed points are also parametrized byIrrG (d , 1, r ).
– The conjectures of Broué-Malle-Michel say that the unipotent characters lying inBγ
are exactly the ones appearing in the cohomology of some Deligne-Lusztig variety
Xγ, and that the endomorphism algebra of the cohomology ofXγ is isomorphic to
some Hecke algebraH (G (d , 1, r ), c′′)for some explicitly given parameterc′′. Exam-ple 4.19 show thatc′′= c′(!).
1. Notation and main result
All along this note, we will abbreviate ⊗C as ⊗. By an algebraic variety, we mean a reduced scheme of finite type overC.
Set-up. We fix in this paper a C-vector spaceV of finite dimension n
and a finite subgroupW ofGLC(V ). We set
Ref(W ) = {s ∈ W | dimCVs= n − 1} and we assume that
W = 〈Ref(W )〉.
We also fix an elementσ ∈ GLC(V )of finite order normalizingW.
We setǫ : W → C×,w 7→ det(w ). Ifs ∈ Ref(W ), we denote byα∨s andαs two elements of
V andV∗ respectively such thatVs= Ker(αs)andV∗s = Ker(α∨s), whereα∨s is viewed as a
1.A. Rational Cherednik algebra att = 0. — All along this note, we fix a function c :
Ref(W ) → Cwhich is invariant under conjugacy. We define theC-algebra Hc to be the
quotient of the algebraT(V ⊕V∗) ⋊ W (the semi-direct product of the tensor algebraT(V ⊕
V∗)with the groupW) by the relations
(Hc) [x , x′] = [y , y′] = 0, [y , x ] = X s ∈Ref(W ) (ǫ(s ) − 1)cs 〈y , αs〉〈α∨s, x 〉 〈α∨s,αs〉 s ,
for allx ,x′∈ V∗,y,y′∈ V. The algebraH
c is called the rational Cherednik algebra att = 0.
The first commutation relations imply that we have morphisms of algebrasC[V ] → Hc
andC[V∗]→ H
c. Recall [10, Corollary 4.4] that we have an isomorphism of C-vector
spaces
(1.1) C[V ] ⊗ CW ⊗ C[V∗]−→ H∼ c
induced by multiplication (this is the so-called PBW-decomposition).
We denote byZc the center ofHc: it is well-known [10, Lemma 3.5] thatZc is an integral
domain, which is integrally closed and containsC[V ]W andC[V∗]W as subalgebras (so it
containsP = C[V ]W⊗ C[V∗]W), and which is a freeP-module of rank|W |. We denote by
Zc the algebraic variety whose ring of regular functionsC[Zc]isZc: this is the
Calogero-Moser spaceassociated with the datum(V , W , c ). If necessary, we will writeZc(V , W )for Zc.
Example 1.2. — Let l ¾ 1, let ζ be a primitive l-th root of unity and let µl = 〈ζ〉. We assume in this example thatn = 1andW = 〈t 〉 ≃ µl, wheret (v ) =ζv for allv ∈ V. Then
Ref(W ) = {t , t2, . . ., tl −1}and we set for simplification
kj = 1 l l −1 X i =1 ζ−i ( j −1)cti
(0 ¶ j ¶ l − 1). Note thatk0+ k1+· · · kl −1= 0. Then
Zc ≃ {(x , y , e ) ∈ C3|
l −1
Y
i =0
(e − l ki) = x y }
(see for instance [4, Theorem 18.2.4]). In particular,Zc is smooth if and only if
Y
0 ¶ i< j ¶ l −1
(ki− kj)6= 0.
Note also that the inclusionP ⊂ Zc corresponds to the morphism of algebraic varieties Zc → C2,(x , y , e ) 7→ (x , y ).
TheC×-action is given by the formula
ξ · (x , y , e ) = (ξdx ,ξ−dy , e ),
1.B. Action of the normalizer. — The normalizerNGLC(V )(W )ofW inGLC(V )acts onV naturally, onV∗ by the contragredient action and on W by conjugation. This endows
T(V ⊕ V∗) ⋊ W with an action ofNGLC(V )(W )and it is easily checked that the bunch of
rela-tions(Hc)is stable under this action. SoHc inherits an action ofNGLC(V )(W ). In particular, its centreZc also inherits such an action, and this defines an action ofNGLC(V )(W )on the
Calogero-Moser spaceZc.
Theorem 1.3. — Assume that Zc is smooth and that σ is a root of unity, and let X be an
irreducible component ofZσ
c. Then there exists a reflection subquotient(V′, W′)of(V , W )and a
complex-valued mapc′on the set of conjugacy classes of reflections ofW′such that
X≃ Zc′(V′, W′),
as varieties endowed with aC×-action.
Remark 1.4. — Note that ifX is reduced to a point, then the theorem is easy because in
this case,X≃ Z0(0, 1).
This Theorem will be proved in Sections 3 and 5.
1.C. Filtration of the group algebra. — If w ∈ W, we set cod(w ) = codimC(Vw). We define a filtrationF•(CW )of the group algebra ofW as follows: let
Fi(CW ) = M cod(w ) ¶ i Cw. Then CIdV =F0(CW ) ⊂ F1(CW ) ⊂ · · · ⊂ Fn(CW ) = CW = Fn +1(CW ) = · · ·
is a filtration ofCW. For any subalgebraAofCW, we setFi(A) = A ∩ Fi(CW ), so that
CIdV =F0(A) ⊂ F1(A) ⊂ · · · ⊂ Fn(A) = A = Fn +1(A) = · · ·
is also a filtration ofA.
Now, assume that Zc(V , W )is smooth, letσ be a root of unity and letX be an
irre-ducible component ofZc(V , W )σ. According to Theorem 1.3, there exists a reflection
sub-quotient(V′, W′)of(V , W ), a complex-valued mapc′on the set of conjugacy classes of
re-flections ofW′and aC×-equivariant isomorphismi : Zc′(V′, W′)
∼
−→ X. We will viewias
a closed immersioni : Zc′(V′, W′),→ Zc(V , W ). By [12, Corollary 5.8], the smoothness of Zc(V , W )implies that there exists a bijection betweenIrr(W )andZc(V , W )C×. As the fixed
point variety of a finite group in a smooth complex algebraic variety is still smooth, this means thatXis smooth, and soZc′(V′, W′)is smooth. Applying again [12, Corollary 5.8],
we get another bijection betweenIrr(W′)andZc′(V′, W′)C ×
. AsXC×⊂ Z
c(V , W )C
×
, this gives an injective mapiX : Irr(W′),→ Irr(W ), depending onX and the choice ofi. This allows to define a surjective morphism of algebras
iX∗ : Z(CW ) −։ Z(CW′)
as follows: ifχ ∈ Irr(W ), let eW
χ denote the corresponding primitive central idempotent
ofCW and set
iX∗ (eχW) = ¨
eχW′′ ifχ′∈ Irr(W′)is such thatiX(χ′) =χ,
Using results of Shan and the first author [5], Theorem 1.3 has the following consequence:
Corollary 1.5. — Ifi ¾ 0, theniX∗ (Fi(Z(CW ))) ⊂ Fi(Z(CW′)).
Proof. — IfY is a complex algebraic variety endowed with aC×-action, we denote by HCi×(Y ) the i-th group of equivariant cohomology, with coefficients in C. Let ħh be an
indeterminate and identifyHC×(pt)withC[ħh ]as usual, withħh homogeneous of degree2.
SinceZc(V , W )is smooth, it follows from [5, Theorem A] that
¨ H2i +1C× (Zc(V , W )) = 0 ifi ¾ 0, H2• C×≃ ReesF(Z(CW )) asC[ħh ]-algebras, whereReesF(Z(CW )) = L
i ¾ 0ħhiFi(Z(CW )) ⊂ C[ħh ]⊗CZ(CW )is the Rees algebra associated with the filtrationF•(Z(CW )). Using the analogous result forZc′(V′, W′)and the
functo-riality of equivariant cohomology, we get a morphism of algebrasi∗: ReesF(Z(CW )) −→
ReesF(Z(CW′)).
Now, we will use not only [5, Theorem A] but also its proof: the proof goes by re-striction to the fixed point subvarietyZc(V , W )C× and identifyingH2•
C×(Zc(V , W )C
×
)with C[ħh ] ⊗ Z(CW )using [12, Corollary 5.8]. As the same strategy holds for Zc′(V′, W′), the
functoriality of equivariant cohomology implies that the mapi∗ fits into a commutative
diagram ReesF(Z(CW )) i ∗ // _ ReesF(Z(CW′)) _ C[ħh ] ⊗CZ(CW ) IdC[ħh]⊗CiX∗ //C[ħh ] ⊗ CZ(CW′). This shows the corollary.
In Remark 4.18 and Theorem 4.21, we will describe combinatorially the injective map
iX : Irr(W′),→ Irr(W )wheneverW is of typeG (l , 1, n).
2. Preliminaries on quiver varieties
2.A. Quiver varieties. — LetQl denote the cyclic quiver withl vertices, defined as
fol-lows:
• Vertices:i ∈ Z/l Z;
• Arrows: yi: i −→ i + 1,i ∈ Z/l Z.
We denote byQl the double quiver ofQl that is, the quiver obtained fromQl by adding
0 1 2 −1 y−1 y0 y1 x1 x0 x−1
Fig. 1. The quiverQl
Now, ifd = (di)i ∈Z/l Zis a family of elements ofN, we denote byGL(d )the direct product
GL(d ) = Y
i ∈Z/l Z
GLdi(C),
by∆C×the image of C× inGL(d )through the diagonal embedding∆ : C×,→ GL(d )and
we set
PGL(d ) = GL(d )/∆C×.
The groupPGL(d )acts on the varietyRep(Ql, d )of representations ofQl in the family of
vector spaces(Cdi)
i ∈Z/l Z. The orbits are the isomorphism classes of representations ofQl
of dimension vectord. We denote by
µd: Rep(Ql, d ) −→
L
i ∈Z/l ZMatdi(C)
(Xi, Yi)i ∈Z/l Z 7−→ (XiYi− Yi −1Xi −1)i ∈Z/l Z
the corresponding moment map. Finally, if θ = (θi)i ∈Z/l Z is a family of complex
num-bers, we denote byIθ(d )the family(θiIdCdi)i ∈Z/l Zand by Oθ(d )the closed subvariety of
Matd0(C)consisting of matrices of rank ¶1and trace−
P
i ∈Z/l Zθidi. Finally, we set
Yθ(d ) =µ−1d (Iθ(d ) + Oθ(d )) and Xθ(d ) = Yθ(d )//PGL(d ).
We will usually denote an element(Xi, Yi)i ∈Z/l Z ofYθ(d )by (X , Y ), where X = (Xi)i ∈Z/l Z
andY = (Yi)i ∈Z/l Z. Note thatYθ(d )is endowed with aC×-action: ifξ ∈ C×, we set
ξ · (X , Y ) = (ξ−1X ,ξY ).
This action commutes with the action ofGL(d )and the moment map is constant onC×
-orbits, so it induces aC×-action onXθ(d ).
Remark 2.1. — We extend the definition of Xθ(d ) to the case where d ∈ ZZ/l Z by the
2.B. Action of the affine Weyl group. — Ifl ¾ 2, letWlaffdenote the affine Weyl group
of type A˜l −1. It is the Coxeter group with associated Coxeter system(Waff
l ,Slaff), where
Slaff={si | i ∈ Z/l Z}and the Coxeter graph is given by
s0
s1
s2
s−1
Fig. 2.Coxeter graph of(Wlaff,Slaff)
We extend this notation to the case wherel = 1by settingW1aff= 1.
Consider the Lie algebra gl = sll(C)and its affine version bgl = bsll(C) = sll(C)[t , t−1]⊕
C1 ⊕ C∂. Let h ⊂ gl be the Cartan subalgebra formed by the diagonal matrices and set bh = h ⊕ C1 ⊕ C∂.
The C-vector space bh∗ has a basis(α0, α1, . . ., αl −1, Λ0), where α0, α1, . . ., αl −1 are the
simple roots ofbgl andΛ0 is such thatΛ0 annihilates h and∂ andΛ0(1) = 1. Denote by
RlaffandRl the affine and the non-affine root lattices respectively (i.e.,Rlaffis theZ-lattice
generated byα0, α1, . . ., αl −1andRl is the sublattice generated byα1, . . ., αl −1.)
Following [19], we define two actions ofWlaff: a non-linear one onZZ/l Z, and a linear
one onCZ/l Z. If l = 1, there is nothing to define so we may assume thatl ¾ 2. If d =
(di)i ∈Z/l Z∈ ZZ/l Zand if j ∈ Z/l Z, we setsj(d ) = (di′)i ∈Z/l Z, where
di′= ¨
di ifi 6= j,
δj 0+ di +1+ di −1− di ifi = j.
Remark 2.2. — We can identifyZZ/l Zwith the root latticeRaff
l by
d 7→ X
i ∈Z/l Z
diαi.
Beware, the action considered here is not the usual action of Wlaff on the root lattice.
When we have w (d ) = d′ with respect to the action defined above, this corresponds to w (Λ0− d ) = Λ0− d′for the usual action ofWlaffonbh∗.
Ifθ = (θi)i ∈Z/l Z∈ CZ/l Z, we setsj(θ ) = (θi′)i ∈Z/l Z, where θi′= θi ifi 6∈ { j − 1, j , j + 1}, θj+θi ifi ∈ { j − 1, j + 1}, −θi ifi = j.
It is readily seen that these definitions on generators extend to an action of the whole groupWlaff. We also define a pairingZZ/l Z× CZ/l Z→ C,(d ,θ ) 7→ d · θ, where
d ·θ = X
i ∈Z/l Z
diθi.
Then
(2.3) sj(d ) · sj(θ ) = (d · θ ) − δj 0θ0. It is proved in [19, Corollary 3.6] that
(2.4) Xs
j(θ )(sj(d )) ≃ Xθ(d ) ifθj 6= 0.
Note that this isomorphism takes into account the convention of Remark 2.1.
The isomorphism above motivates to consider the following equivalence relation on the setZZ/l Z× CZ/l Z. Let∼be the transitive closure of
(d ,θ ) ∼ (si(d ), si(θ )), θi6= 0.
The isomorphism (2.4) implies that if (d ,θ ) ∼ (d′,θ′), then we have an isomorphism of algebraic varietiesXθ(d ) ≃ Xθ′(d′).
Recall that the affine Weyl group has another presentation. We have Wlaff= Wl ⋉Rl,
whereWl = Sl is the non-affine Weyl group, that is, the subgroup generated by s1,. . . ,
sl −1. For eachα ∈ Rl, denote bytα the image of αinWlaff. Each element ofWlaffcan be
written in a unique way in the formw · tα, where w ∈ Wl andα ∈ Rl. Letδl denote the
constant familyδl = (1)i ∈Z/l Z∈ ZZ/l Z.
Recall also from [10, §11] the following result, which follows from the fact thatXθ(nδl)
is isomorphic to some Calogero-Moser space (we will use this fact later, and make this statement more precise):
Lemma 2.5. — Ifn ¾ 0, thenXθ(nδl)is normal and irreducible of dimension2n.
Definition 2.6. — We say that the pair(d ,θ )is smooth if it is equivalent to a pair of the form
(nδl,θ′)such thatn ¾ 0andXθ′(nδl)is smooth. In particular, in this case, the varietyXθ(d )is
smooth and non-empty.
In the following lemma we identifyZZ/l ZwithRlaff.
Lemma 2.7. — Assumeα ∈ Rl andd ∈ ZZ/l Z. Then we havetα(d ) ≡ d −α mod Zδl.
Proof. — This statement is a partial case of [15, (6.5.2)] (see also Remark 2.2).
Lemma 2.8. — For eachd ∈ ZZ/l Z, there exists a uniquen ∈ Zsuch thatd andnδ
l are in the
Proof. — Let us prove the existence ofn. Setα = d − d0δl. Thenαis clearly an element of
Rl. Then by Lemma 2.7, the elementtα(d )is of the formnδl.
Let us prove the uniqueness of n. Assume thatn1δl andn2δl are in the sameWlaff
-orbit. Sincen1δl isWl-stable, theWlaff-orbit of n1δl coincides with theRl-orbit ofn1δl.
So there existsα ∈ Rl such that tα(n1δl) = n2δl. By Lemma 2.7, this forces α = 0 and
n1= n2.
Consider theZ-linear map
Rlaff→ CZ/l Z, d 7→ d ,
given by
(αr)i= 2δi ,r− δi ,r +1− δi ,r −1.
The kernel of this map isZδl. Set
Σ(θ ) =
X
i ∈Z/l Z
θi.
Lemma 2.9. — For eachα ∈ Rl andθ ∈ CZ/l Z, we havetα(θ ) = θ + Σ(θ )α.
Proof. — TheWlaff-action onCZ/l Zdefined above coincides with the (usual) action ofWlaff
on the dual of the span ofα0,α1, . . .,αl −1inbh∗. The statement follows from [15, (6.5.2)].
We denote by(ZZ/l Z)
+the set ofd ∈ ZZ/l Zwhich belong to the orbit of somenδl, with
n ¾ 0. This set has a more precise combinatorial description. For example, it will follow
from Proposition 4.4 that an elementd ∈ ZZ/l Zis in(ZZ/l Z)
+if and only ifd is a residue of some Young diagram. In particular, the set(ZZ/l Z)
+is contained inNZ/l Z.
Lemma 2.10. — Assume that d ∈ ZZ/l Z is such that there exists a simple representation in
Yθ(d ). Then
(a) the varietyXθ(d )is normal and irreducible,
(b) we haved ∈ (ZZ/l Z)
+.
Proof. — Letn ∈ Zandw ∈ Wlaffbe such thatnδl = w (d ). Let us show that (2.4) implies an
isomorphismXθ(d ) ≃ Xw (θ )(nδl). (In particular, we must haven ¾ 0.) The only difficulty
that we have is caused by the conditionθi6= 0in (2.4). But [9, Lemma 7.2] implies that we
can find a sequence of reflections such thatsi1si2. . . sir(d ) = nδl and such that we can apply
2.C. Smoothness. — The following result is proved in [13, Lemma 4.3 and its proof] (which is based on [17, Theorem 1.2] and [9, Theorem 1.2]):
Theorem 2.11. — Letn ¾ 0. Then the following are equivalent:
(1) The varietyXθ(nδl)is smooth.
(2) Every element ofYθ(nδl)defines a simple representation of the quiverQl.
(3) The familyθ satisfies
Σ(θ ) Y 1 ¶ i ¶ j ¶ l −1 −(n −1) ¶ k ¶ n −1 (θi+θi +1+· · · + θj) + k Σ(θ ) 6= 0.
If these equivalent conditions hold, then PGL(nδl) acts freely on Yθ(nδl), and so, as a set,
Xθ(nδl)identifies withGL(nδl)-orbits inYθ(nδl).
2.D. Fixed points. — For every m ¾ 1, the group of m-th roots of unity µm acts on
Xθ(nδl) (this is the restriction of the C×-action defined in §2.A). We aim to compute
the fixed point varietyXθ(nδl)µm wheneverXθ(nδl) is smooth. Note first thatµl acts
trivially on Xθ(nδl): indeed, if ζ ∈ µl and if (X , Y ) ∈ Yθ(nδl), then ζ · (X , Y ) = g0(X , Y ),
whereg0= (ζiIdCn)i ∈Z/l Z∈ GL(nδl). This means that, in order to describeXθ(nδl)µm, we
may replaceµm by〈µm, µl〉or, in other words, we may assume thatl dividesm.
Hypothesis.We fix in this subsection a non-zero natural numberk and we setm = k l.
Ifθ = (θi)i ∈Z/l Z∈ CZ/l Z, we defineθ [k ]to be the element(θ [k ]j)j ∈Z/m Zsuch that
θ [k ]j =θi
if j ≡ i mod l. Roughly speaking,θ [k ]is the concatenation ofk copies ofθ.
We denote byZZ/m Z[nδl]the set ofd = (dj)j ∈Z/m Z∈ ZZ/m Zsuch that
n = X
j ∈Z/m Z j ≡i mod l
dj
for alli ∈ Z/l Z. Set also E (k , l , n) = ZZ/m Z[nδ
l]∩ (ZZ/m Z)+. If d ∈ E (k , l , n)and(X , Y ) ∈ Rep(Qm, d ), we set ik(d )(X , Y ) = (X′, Y′), whereXi′=L j ∈Z/m Z j ≡i mod l Xj and Yi′= L j ∈Z/m Z j ≡i mod l
Yj. By the definition of E (k , l , n), we have
Xi′,Yi′∈ Matn(C). In other words,(X′, Y′)∈ Rep(Ql, nδl): it is clear thatik(d ): Rep(Qm, d ),→
Rep(Ql, nδl)is a closed immersion.
Proof. — Write ik(d )(X , Y ) = (X′, Y′)and let i ∈ Z/l Z. Write p0 for the rank one matrix X0Y0− Y−1X−1− θ [k ]0IdCd0. Then Xi′Yi′− Yi −1′ Xi −1′ = M j ∈Z/m Z j ≡i mod l (XjYj− Yj −1Xj −1) = M j ∈Z/m Z j ≡i mod l θ [k ]jIdCd j +δ0ip0 = θiIdCn+δ0ip0, as desired.
It is also clear that, if(X , Y )and( eX , eY )are two elements ofYθ [k ](d )which are conjugate
underGL(d ), thenik(d )(X , Y )andik(d )( eX , eY )are conjugate underGL(nδl). This means that
ik(d )induces a morphism of algebraic varieties still denoted by ik(d ): Xθ [k ](d ) −→ Xθ(nδl).
Ford ∈ E (k , l , n), set
Xd= ik(d )(Xθ [k ](d )) ⊂ Xθ(nδl).
For the moment, we allow the possibility that the varietiesXθ [k ](d )andXd are empty.
Set
E (k , l , n)6=∅={d ∈ E (k , l , n); Xd6= ∅}.
We will show in Corollary 4.13 that we haveE (k , l , n)6=∅=E (k , l , n).
The next result describes the fixed point varietyXθ(nδl)µm: it is the main step towards
the proof of Theorem 1.3.
Theorem 2.13. — Assume thatXθ(nδl)is smooth. Then:
(a) Xθ(nδl)µm is smooth. (b) We have Xθ(nδl)µm= a d ∈E (k ,l ,n )6=∅ Xd
andXd are exactly the irreducible components ofXθ(nδl)µm.
(c) Ifd ∈ E (k , l , n)6=∅, thenik(d ): Xθ [k ](d ) → Xd is an isomorphism.
Proof. — (a) follows from the fact already mentioned in the proof of Corollary 1.5 that the fixed point variety under the action of a finite group on a smooth variety is again smooth.
We will now prove (b). Letξ be a primitivem-th root of unity and let ζ = ξk (it is a
primitive l-th root of unity). Letg0 denote the element (ζiId
Cn)i ∈Z/l Z in GL(nδl). This
element satisfies
(∗) g0(X , Y ) = (ζ−1X ,ζY )
for all(X , Y ) ∈ Yθ(nδl). We will use the following variety:
wherepX ,Y = ([Xi, Yi]−θi)i ∈Z/l Z: it is0except on the0-th component, where it is equal to a
rank one matrix of trace−nΣ(θ )(recall from Theorem 2.11 thatΣ(θ ) 6= 0becauseXθ(nδl)
is assumed to be smooth).
We denote by p′ : Z → Yθ(nδl) the projection on the first two factors, by p : Z →
Xθ(nδl)the map induced byp′and byq : Z → GL(nδl)the projection on the third factor.
Note thatPGL(nδl)acts onZ by conjugacy on the three factors, and so the mapsp′and
qare equivariant for this action.
First, it is clear that p (Z ) ⊂ Xθ(nδl)µm. Conversely, if (X , Y ) ∈ Yθ(nδl)is a
represen-tative of an element of Xθ(nδl)µm, then there exists g ∈ PGL(nδl) such that g(X , Y ) =
(ξ−1X ,ξY ). In particular, gpX ,Y = pX ,Y. Sog stabilizes the image ofpX ,Y, which is of
di-mension1: this means thatg acts by a non-zero scalarωonIm(pX ,Y). Settingg′=ω−1g, we see that(X , Y , g ) ∈ Z. So we have proved the following fact:
Fact 1.p (Z ) = Xθ(nδl)µm.
The next fact follows from the freeness of the action ofPGL(nδl)onYθ(nδl)(see
The-orem 2.11).
Fact 2.If(X , Y , g )and(X , Y , g′)belong toZ, theng = g′andgk= g0. Indeed, the hypothesis implies that g(X , Y ) =g′(X , Y )andgk
(X , Y ) =g0(X , Y ). So there
exists two non-zero complex numbersαandβ such thatg′=αg andgk =β g0. But the conditiong pX ,Y = g′pX ,Y = g0pX ,Y forcesα = β = 1becausepX ,Y 6= 0.
Now, letC denote the set of elementsg ∈ GL(nδl)such thatgk= g0. Sinceg0is central inGL(nδl),C is a disjoint union of finitely many semisimple conjugacy classes (which
are the irreducible components ofC). We fixg ∈ C such thatq−1(g ) 6= ∅and we denote
byCg its conjugacy class inGL(nδl). We will describeq−1(g ). For this, letE =
L
i ∈Z/l ZCn
and letEj = Eξ
− jg
(for j ∈ Z/mZ). In other words, Ej is theξj-eigenspace of g in E: it
is contained in thei-th component ofE, wherei is the element ofZ/l Zsuch that j ≡ i mod l. Letdj = dimC(Ej)and letd = (dj)j ∈Z/m Z.
If(X , Y , g ) ∈ Z, thenX sendsEj toEj −1(we denote byXj : Ej → Ej −1the induced map)
while Y sends Ej to Ej +1 (we denote byYj : Ej → Ej +1 the induced map). By choosing
a basis of everyEj, the family(Xj, Yj)j ∈Z/m Z defines a representation of the quiverQm,
of dimension vectord. This means that we have defined a mapq−1(g ) −→ Rep(Q
m, d ),
which is clearly a closed immersion, and whose image isYθ [k ](d )(because Im(pX ,Y) is
contained inE0).
Note that by construction we have d ∈ ZZ/m Z[nδl]. Moreover, by Lemma 2.10(b) and
Theorem 2.11 we also haved ∈ (ZZ/m Z)
+. This means thatd is an element ofE (k , l , n) =
ZZ/m Z[nδl]∩ (ZZ/m Z)+. Moreover, it is clear that the set of elementsd that can appear as above (from someg ∈ C such thatq−1(g ) 6= ∅) isE (k , l , n)6=∅.
This shows the following fact.
Fact 3.With the above notation,q−1(g ) ≃ Yθ [k ](d ). In particular,q−1(g )is irreducible.
Fact 3 and its proof imply the following. Fact 4.We havep (q−1(Cg)) =Xd.
Denote byπthe projectionπ: Yθ(nδl)→ Xθ(nδl). Fact 2 implies that the mapp′: Z →
π−1(Xθ(nδl)µm)is bijective. Moreover, sinceπ−1(Xθ(nδl)µm)is smooth (the groupPGL(nδl)
acts freely onYθ(nδl)), this bijection is an isomorphism of algebraic varieties. Let
be its inverse.
Consider the morphismτ: πe −1(Xθ(nδl)µm)→ GL(nδl)given byτ = q ◦ (pe ′)−1. Since we
haveπ−1(Xθ(nδl)µm)//GL(nδl) =Xθ(nδl)µm, we get a morphismτ: Xθ(nδl)µm→ GL(nδl)//GL(nδl),
where the groupGL(nδl)acts on itself by conjugation.
Moreover, since we clearly haveτ−1(Cg) = p (q−1(Cg)), Fact 4 implies the following.
Fact 5.We haveτ−1(Cg) =Xd. In particularXd is closed.
This completes the proof of (b). Now we prove (c). The groupGL(nδl)acts transitively
onCg. So we get an isomorphism of varieties
q−1(g )//CGL(nδl)(g )
∼
−→ q−1(Cg)//GL(nδl).
Now, ifd = (dimC(Eξ
− jg
))j ∈Z/m Z, q−1(g )identifies with Yθ [k ](d )andCGL(nδl)(g )identifies
withGL(d ). Glueing this with Facts 3 and 4, we get an isomorphism of varieties
Xθ [k ](d )
∼
−→ Xd,
which is the mapik(d ). This proves (c).
Corollary 2.14. — Ifk> n, then we haveXθ(nδl)µm=Xθ(nδl)C
×
.
Proof. — Taked ∈ E (k , l , n)6=∅. Since k > n, some component of d must be zero. This implies thatC×acts trivially onXθ [k ](d )because the action ofC×is induced by elements
ofGL(d ).
Sinceik(d )isC×-equivariant and Xθ(nδl)µm=
a
d ∈E (k ,l ,n )6=∅
ik(d )(Xθ [k ](d ))
(by Theorem 2.13(b)), theC×-action on the varietyXθ(nδl)µm is also trivial.
3. Proof of Theorem 1.3
First, the explicit description ofZc whenevern = 1(see Example 1.2) allows to prove
easily Theorem 1.3 in this case. Also, let us decomposeV as V = VW⊕ V1⊕ · · · ⊕ Vr,
where theVi are the non-trivial irreducible components ofV as aCW-module. ThenW
decomposes asW = W1×· · ·×Wr, where(Vi, Wi)is then a reflection subquotient (subgroup)
ofW. Letci denote the restriction ofc toWi. Then
Zc(V , W ) ≃ VW× V∗W × Zc
1(V1, W1)× · · · × Zcr(Vr, Wr)
so that the proof of Theorem 1.3 is easily reduced to the case whereW acts irreducibly on V/VW.
In this case, the smoothness ofZc(V , W )implies thatW is of typeG4 orG (l , 1, n)for
somel ¾ 1. The case of typeG4will be handled by computer calculations: see Section 5
Hypothesis.From now on, and until the end of this section, we assume thatn ¾ 2, thatV = Cn and thatW = G (l , 1, n). We also assume thatσ
is anm-th root of unity, wherem = k l for somek ¾ 1.
Recall that G (l , 1, n) is the group of monomial matrices with coefficients in µl (the group ofl-th root of unity inC×) and recall thatn ¾ 2.
3.A. Quiver varieties vs Calogero-Moser spaces. — We fix a primitivel-th root of unity ζ. We denote bys the permutation matrix corresponding to the transposition(1, 2) and
we set
t = diag(ζ, 1, . . ., 1) ∈ W .
Thens,t, t2,. . . ,tl −1 is a set of representatives of conjugacy classes of reflections ofW.
We set for simplification
a = cs and kj = 1 l l −1 X i =1 ζ−i ( j −1)cti forj ∈ Z/l Z. Then (3.1) k0+· · · + kl −1= 0 and cti= X j ∈Z/l Z ζi ( j −1)kj
for1 ¶ i ¶ l − 1. Finally, ifi ∈ Z/l Z, we set
(3.2) θi=
¨
k−i− k1−i ifi 6= 0,
−a + k0− k1 ifi = 0. andθ = (θi)i ∈Z/l Z.
The following result is proved in [13, Theorem 3.10]. (Note that ourki is related with
Gordon’sHi viaHi= k−i− k1−i.)
Theorem 3.3. — With the above notation, there is aC×-equivariant isomorphism of varieties
Zc−→ X∼ θ(nδl).
The theorem above is also true forn = 1. But in this case, the varietyZc has no
param-etera. On the other hand, the varietyXθ(δl)is independent ofθ0. Putting Theorems 2.11 and 3.3 together, one gets:
Corollary 3.4. — The varietyZc is smooth if and only if
a Y
0 ¶ i 6= j ¶ l −1 0 ¶ r ¶ n −1
Proof of Theorem 1.3. — Assume now thatZc is smooth and letX be an irreducible com-ponent ofZµm
c . Using the isomorphism of Theorem 3.3, we see that
Zµm
c
∼
−→ Xθ(nδl)µm.
So, by Theorem 2.13, there existsd ∈ E (k , l , n)6=∅such thatX ≃ Xθ [k ](d ). By Lemma 2.8,
there exists r ¾ 0 and w ∈ Wmaff such that w (d ) = rδm. So it follows from the
isomor-phism (2.4) that
X ≃ Xw (θ [k ])(rδm)
(see also Lemma 2.10 and its proof). Using Theorem 3.3 in the other way, we see that there exists a complex valued functionc′ on the set of conjugacy classes of reflections
ofW′= G (m, 1, r )such thatXw (θ [k ])(rδm)≃ Zc′(W′). Moreover, as the action ofWmaff on
CZ/m Zis linear, this implies that the mapc 7→ c′is linear.
This proves almost every statement of Theorem 1.3, except thatW′can be realized as
a reflection subquotient of(V , W ). For this, we need the following fact.
Fact.We havek r ¶ n.
Indeed, the elementd − rδm is in theWmaff-orbit of0. Then Proposition 4.4 implies that
d − rδm is a residue of somem-core. In particular, each coordinates ofd − rδm must be
positive. Since the sum of the coordinates ofd isnl, this givesnl − r k l ¾ 0. This implies k r ¶ n.
Now, using the fact proved above, letx denotes the matrix x = diag(ζ, 1, . . ., 1
| {z }
kterms
)· M(1,2,...,k )∈ GLk(C),
whereM(1,2,...,k )is the permutation matrix associated with the cycle(1, 2, . . ., k ). Now, letg be the matrix
g = diag(x , . . ., x
| {z }
rterms
, IdCn −r k)∈ G (l , 1, n).
LetV′denote theξ-eigenspace ofg, whereξis a primitivem-th root of unity such that ξk =ζ. ThenC
W(g )acts onV′, V′ is of dimensionr and, if we denote byK the kernel
of this action, thenCW(g )/K ≃ G (m, 1, r ). So(V′,G (m, 1, r ))is a reflection subquotient of
(V , W ).
Remark 3.5. — The formula (3.2) above yields a bijection betweenCZ/l Z and the set of (l +1)-tuples of complex numbers(a , k0, . . ., kl −1)such thatk0+k1+. . .+kl −1= 0. Moreover,
theWlaff-action onCZ/l Z(see Section 2.B) translates to the action on the(l + 1)-tuples in
the following way
sr(a , k0, . . ., kl −r, kl −r +1, . . ., kl −1) = (a , k0, . . ., kl −r +1, kl −r, . . ., kl −1), forr = 1, 2, . . ., l − 1
and
s0(a , k0, k1, k2, . . ., kl −1) = (a , k1+ a , k0− a , k2, . . ., kl −1).
In particular, we see that the subgroupWl ofWlaffacts by permutation of the parameters
k0,k1,. . . ,kl −1.
The next result follows from [2], but we provide here a different proof (which works only in typeG (d , 1, n)).
Corollary 3.6. — A permutation of parameters k0, k1, . . ., kl −1 does not change the
Calogero-Moser spaceZc (up to an isomorphism of algebraic varieties).
Proof. — The subgroupWl ofWlaff stabilizesδl. Then by (2.4), for eachw ∈ Wl,n ∈ Z>0
andθ ∈ CZ/l Z, we haveX
θ(nδl)≃ Xw (θ )(nδl). This proves the statement by Theorem 3.3
and Remark 3.5
4. Combinatorics
In this subsection, we aim to make the statement of Theorem 1.3 more precise in the case whereW = G (l , 1, n): we wish to describe precisely the map c 7→ c′as well as the
mapiX∗ : Irr(W′),→ Irr(W )in terms of the combinatorics of partitions, cores,l-quotients,
etc.
4.A. Partitions and cores. — Let l and n be positive integers. A partition is a tuple λ = (λ1,λ2, . . .,λr)of positive integers (with no fixed length) such that λ1¾λ2¾· · · ¾ λr.
Set|λ| =Pi =1r λi. If|λ| = n, we say thatλis a partition ofn.
Denote by P (resp. P [n]) be the set of all partitions (resp. the set of all partitions of
n). By convention,P [0]contains one (empty) partition. We will identify partitions with
Young diagrams. The partitionλcorresponds to a Young diagram withr lines such that
theith line containsλiboxes. For example the partition(4, 2, 1)corresponds to the Young
diagram
Letb be a box of a Young diagram in the liner and columns. Thel-residue of the box b is the numbers − r modulol. (We also say that the integers − r is the∞-residue of the
boxb). Then we obtain a map
Resl: P → Z
Z/l Z, λ 7→ Res l(λ),
such that for eachi ∈ Z/l Zthe number of boxes with residuei inλis(Resl(λ))i. (Similarly,
we obtain a mapRes∞: P → ZZ.)
Example 4.1. — For the partitionλ = (4, 2, 1)andl = 3the residues of the boxes are
0 1 2 0 2 0 1
In this case we haveResl(λ) = (3, 2, 2)because there are three boxes with residue 0, two
boxes with residue1and two boxes with residue2.
We say that a box of a Young diagram is removable if it has no box on the right and on the bottom. Forλ, µ ∈ P, we writeµ ¶ λif the Young diagram ofµcan be obtained from
Definition 4.2. — We say that the partitionλis anl-core if there is no partitionµ ¶ λsuch that the Young diagram ofµdiffers from the Young diagram ofλbyl boxes withl differentl-residues.
See [3] for more details about the combinatorics of l-cores. LetCl ⊂ P be the set of
l-cores. SetCl[n] = P [n] ∩ Cl.
If a partitionλis not anl-core, then we can get a smaller Young diagram from its Young
diagram by removing l boxes with differentl-residues. We can repeat this operation
again and again until we get anl-core. It is well-known that the l-core that we get is
independent of the choice of the boxes that we remove. Then we get an application
Corel: P → Cl.
Ifµ = Corel(λ), we will say that the partitionµis thel-coreof the partitionλ.
Example 4.3. — The partition(4, 2, 1)from the previous example is not a3-core because
it is possible to remove three bottom boxes. We get
0 1 2 0
But this is still not a3-core because we can remove three more boxes and we get 0
This shows that the partition(1)is the3-core of the partition(4, 2, 1).
As mentioned in Lemma 2.8, for eachd ∈ ZZ/l Z, theWlaff-orbit ofd contains an element
of the formnδl for a uniquen ∈ Z.
Proposition 4.4. — Letd be an element ofZZ/l Z. The following statements are equivalent.
(a) d is of the formd = Resl(λ)for somel-coreλ,
(b) d is in theWlaff-orbit of0.
Proof. — Consider the Wlaff-action on Cl as in [3, Sec. 3]. By construction, the map Resl: Cl → ZZ/l Z isWlaff-invariant. Moreover, the residue of the empty partition is zero.
The stabilizer of the empty partition inWlaff isWl and the stabilizer of0 ∈ ZZ/l ZinWlaff
is alsoWl. This shows that the mapResl yields a bijection between the set ofl-cores and
theWlaff-orbit of0inZZ/l Z.
Denote byπthe obvious surjectionπ: ZZ/l Z→ ZZ/l Z/Zδ l.
Proposition 4.5. — The chain of maps
Cl Resl //ZZ/l Z π //ZZ/l Z/Zδ l
yields a bijection betweenCl andZZ/l Z/Zδl.
Proof. — We have seen in the proof of Proposition 4.4 thatResl yields a bijection between
Cl and theWlaff-orbit of0 ∈ ZZ/l Z. This proves the statement because the restriction ofπ
For d = (di)i ∈Z/l Z∈ ZZ/l Z we set|d | =
P
i ∈Z/l Zdi. Set ZZ/l Z[n] = {d ∈ ZZ/l Z; |d | = n}.
Denote byCl[≡ n]the subset ofCl that contains onlyl-cores with the number of boxes
congruent ton modulo l. Proposition 4.5 yields the following bijection ǫ : ZZ/l Z[n]↔∼
Cl[≡ n].
Lemma 4.6. — Assumed ∈ ZZ/l Z[n]. Thend is in(ZZ/l Z)
+if and only if|ǫ(d )| ¶ n.
Proof. — By construction,ǫ(d )is the uniquel-coreλ ∈ Cl[≡ n] such thatResl(λ)is
con-gruent tod moduloδl. Write d = Resl(λ) + r δl. Since by Proposition 4.4, the element
Resl(λ)is in theWlaff-orbit of0, thend is in theW
aff
l -orbit ofrδl. Then, by definition, the
elementd is in(ZZ/l Z)
+if and only ifr ¾ 0. Now, let us count the sum of the coordinates in the sides of the equalityd = Resl(λ) + r δl. We obtainn = |λ| + r l. This shows that we
haver ¾ 0if and only if|λ| ¶ n.
Letνbe anl-core. Denote byPl ,νthe set of partitions with thel-coreν. In particular
we denote byPl ,∅the set of partitions with triviall-core. LetPl be the set ofl-partitions
(i.e., the set ofl-tuples of partitions). Ifλis a partition, we denote byβl(λ)itsl-quotient
(see for example [18, Sec. 2.2]).
Lemma 4.7. — Ifνis anl-core, thenβl restricts to a bijectionβl ,ν: Pl ,ν
∼
↔ Pl.
4.B. Description ofE (k , l , n)in terms of cores. — Letk be a positive integer. Setm = k l.
Set also(Cm)l ,ν=Pl ,ν∩ Cm.
Recall that we have a bijectionǫ : ZZ/m Z[nl ]↔ C∼
m[≡ nl ]. Assumed ∈ ZZ/m Z[nl ].
Lemma 4.8. — We haved ∈ ZZ/m Z[nδ
l]if and only if thel-core ofǫ(d )is trivial, where the set
ZZ/m Z[nδl]is as in Section 2.D.
Proof. — By definition,ǫ(d )is the uniquem-coreλ ∈ Cm[≡ nl ]such thatResm(λ)is equal
tod moduloδm. On the other hand, by Proposition 4.4, the partitionλis anl-core if and
only ifResl(λ)is a multiple ofδl. By commutativity of the following diagram,
P Resm {{①①①① ①①①① Resl "" ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ZZ/m Z //ZZ/l Z
this condition is clearly equivalent tod ∈ ZZ/m Z[nδl].
Recall that the set E (k , l , n) was defined as the intersection E (k , l , n) = ZZ/m Z[nδ l]∩
(ZZ/m Z)+. Combining Lemmas 4.6 and 4.8, we obtain the following.
Lemma 4.9. — The bijectionǫ : ZZ/m Z[nl ] ≃ Cm[≡ nl ]restricts to a bijection
Forλ = (λ0, . . .,λl −1)∈ Pl we set|λ| = |λ0| + · · · + |λl −1|. Set alsoPl[n] = {λ ∈ Pl; |λ| = n}.
We say that an element ofPl[n]is anl-partition ofn.
Now recall the bijection βl ,ν: Pl ,ν
∼
↔ Pl obtained by restriction fromβ
l as in Lemma
4.7. By construction, it has the following properties.
Lemma 4.10. — (a) For eachλ ∈ Pl ,∅, we haveλ ∈ Cm if and only ifβl ,∅(λ) ∈ (Ck)l.
(b) For eachλ ∈ Pl ,ν, we have|λ| = l · |βl ,ν(λ)| + |ν|.
Part (a) of the lemma above shows that βl ,∅ can be restricted to a bijection(Cm)l ,∅ ≃
(Ck)l. Moreover, part (b) shows that new bijection restricts to a bijection Cm[≡ nl ] ∩ Cm[¶ nl ] ∩ (Cm)l ,∅≃ (Ck)l[≡ n, ¶ n],
where
(Ck)l[≡ n, ¶ n] = {λ ∈ (Ck)l; |λ| ¶ nand|λ| ≡ n mod k }.
Combining this with Lemma 4.9, we get a bijection
δ: E (k , l , n) → (Ck)l[≡ n, ¶ n].
4.C. Parametrization of theµm-fixed points. — We have proved in Theorem 2.13 that the subsetE (k , l , n)6=∅ofE (k , l , n)parametrizes the irreducible components ofXθ(nδl)µm.
(But we will show in Corollary 4.13 that we haveE (k , l , n)6=∅=E (k , l , n).) Then the bijec-tionδabove gives another parametrization of the irreducible components ofXθ(nδl)µm
in terms of(Ck)l[≡ n, ¶ n]. For eachλ ∈ (Ck)l[≡ n, ¶ n], we setXλ=Xδ−1(λ).
Recall the map Corek: P → Ck. It yields a map Corek: Pl → (Ck)l (component by
component). Note that the set(Ck)l[≡ n, ¶ n] considered above is nothing else but the image ofPl[n]byCorek. In other words, the set(Ck)l[≡ n, ¶ n]is the same thing as the
set ofk-cores ofl-partitions ofn.
This new parametrization has the following nice property.
Lemma 4.11. — Fix positive integers k1 and k2 such that k1 divides k2. Set m1 = k1l and
m2= k2l. Fixλ1∈ (Ck1) l[≡ n, ¶ n]andλ 2∈ (Ck2) l[≡ n, ¶ n]. Then we haveX λ2 ⊂ Xλ1 if and only ifCorek1(λ2) =λ1.
Proof. — Consider the map
p : ZZ/m2Z→ ZZ/m1Z, p (d )
i=
X
j ≡i mod m1
dj.
Letd1∈ E (k1, l , n) andd2∈ E (k2, l , n)be such thatδ(d1) =λ1 and δ(d2) =λ2. We have
Xd1 =Xλ
only ifp (d2) = d1. Then the statement follows from the commutativity of the following diagram E (k2, l , n) p // E (k1, l , n) Cm2[≡ nl ] ∩ Cm2[¶ nl ] ∩ (Cm2)l ,∅ Corem1 // Cm1[≡ nl ] ∩ Cm1[¶ nl ] ∩ (Cm1)l ,∅ (Ck2)l[≡ n, ¶ n] Corek1 //(C k1) l[≡ n, ¶ n],
where the vertical maps are the bijections discussed above.
Proposition 4.12. — TheC×-fixed points inXθ(nδl)are parametrized by the setPl[n]. This
parametrizationXθ(nδl)C
×
={pµ; µ ∈ Pl[n]}can be chosen in such a way that for eachk ∈ Z,
eachλ ∈ (Ck)l[≡ n, ¶ n]and eachµ ∈ Pl[n]we havepµ∈ Xλif and only ifλis thek-core ofµ.
Proof. — Assumek> n. Then by Corollary 2.14, we haveXθ(nδl)µm=Xθ(nδl)C
×
. More-over, the setXθ(nδl)C
×
is finite. We already know that Xθ(nδl)µm is in bijection with a
subset of (Ck)l[≡ n, ¶ n]. Moreover, k > n also implies that we have (C
k)l[≡ n, ¶ n] =
Pl[n]. This shows that X
θ(nδl)µm =Xθ(nδl)C
×
is in bijection with a subset of Pl[n].
Moreover, by Lemma 4.11, this bijection is independent of the choice of k. But it is
well-known (see [12]) thatXθ(nδl)µm andPl[n]have the same cardinalities. This shows
that the bijection above is a bijection betweenXθ(nδl)µm andPl[n](not just a subset of
Pl[n]).
The second statement follows from Lemma 4.11.
Corollary 4.13. — For each λ ∈ (Ck)l[≡ n, ¶ n], we have Xλ 6= ∅. In particular, we have
E (k , l , n) = E (k , l , n)6=∅.
Proof. — Assumeλ ∈ (Ck)l[≡ n, ¶ n]. Then there existsµ ∈ Pl[n]such thatCorek(µ) = λ.
ThenXλis not empty because we havepµ∈ Xλby Proposition 4.12.
4.D. C×-fixed points. — In the previous section we have constructed a parametrization
of the C×-fixed points in Xθ(nδl) by Pl[n]. Another parametrization of the C×-fixed
points inXθ(nδl)by the same set is done by Gordon [12]. This parametrization is given
in terms of baby Verma modules.
It is not obvious at all how to compare these two parametrizations. But this can be deduced from the main theorem in [21].
Proposition 4.14. — The parametrization of theC×-fixed points inXθ(nδl)given in
Proposi-tion 4.12 differs from Gordon’s parametrizaProposi-tion by the twist
Pl[n] → Pl[n], λ 7→ λ♭.
Proof. — We have constructed in Lemma 4.9 a parametrization of the irreducible com-ponents ofXθ(nδl)µm by the setCm[≡ nl ] ∩ Cm[¶ nl ] ∩ (Cm)l ,∅. Similarly to the proof of
Proposition 4.12, this yields a bijection betweenXθ(nδl)C
×
andP [nl ] ∩ Pl ,∅.
Indeed, ifk> n, then we haveXθ(nδl)C
×
=Xθ(nδl)µm by Corollary 2.14 and
Cm[≡ nl ] ∩ Cm[¶ nl ] ∩ (Cm)l ,∅=P [nl ] ∩ Pl ,∅.
Moreover, an argument similar to the proof of Proposition 4.12 shows that the obtained bijectionXθ(nδl)C
×
≃ P [nl ] ∩ Pl ,∅ is independent ofk.
On the other hand, [21, Theorem 1.1 a)] also constructs a bijection betweenXθ(nδl)C
×
andP [nl ] ∩ Pl ,∅. It is clear from the construction that this bijection coincides with ours.
Theorem [21, Theorem 1.2] claims that the bijection
P [nl ] ∩ Pl ,∅→ Pl[n], λ 7→ (βl ,∅(λ))♭
identifies the parametrization of theC×-fixed points by the setP [nl ]∩Pl ,∅with Gordon’s
parametrization of theC×-fixed points by the setPl[n].
On the other hand, the parametrization of the C×-fixed points by P [nl ] ∩ Pl ,∅
com-posed with the l-quotient bijection βl ,∅ is nothing else but the parametrization of the
C×-fixed points byPl[n]from Proposition 4.12.
Remark 4.15. — The parametrization of the points of Xθ(nδl)C
×
given in Proposition 4.12 is constructed as a parametrization of the irreducible components ofXθ(nδl)µm for
a very bigm. This parametrization can be seen in an "m-independent way" if we replace
the quiverQm with a very bigmby an infinite quiver. This can be done in the following
way.
LetQ∞ be the quiver defined in the same way asQl with respect to the vertex setZ
instead ofZ/l Z. The dimension vectors of the quiverQ∞are inZZ. Set ZZ
fin={ bd ∈ Z Z
; bd has finitely many non-zero components}.
Consider the map
p : ZZfin→ ZZ/l Z, (p ( bd ))i= X j ≡i mod l b dj. For eachd ∈ Zb Z
fin, we have a linear mapi ( bd )
∞: Rep(Q∞, bd ) → Rep(Ql, p ( bd )), defined in the
same way is in Section 2.D. Now, for eachθ ∈ CZ/l Zwe consider the elementθ [∞] ∈ CZ given byθ [∞]i=θi mod l. Then we obtain aC×-invariant morphism of algebraic varieties
i∞( bd ): Xθ [∞]( bd ) → Xθ(p ( bd )). But the varietyXθ [∞]( bd )is obviouslyC×-stable because theC×
action is induced by elements ofGL( bd ). SinceXθ [∞]( bd )is connected, it is a singleton (if it
is not empty). Set
E (∞, l , d ) = { bd ∈ ZZfin; p ( bd ) = d , Xθ [∞]( bd ) 6= ∅}.
Now, we assume d = nδl, n ¾ 0 and that Xθ(d ) is smooth. The construction of
the parametrization of theC×-fixed points inXθ(nδl)given in Proposition 4.12 implies
that each C×-fixed point pµ, µ ∈ Pl[n], in Xθ(nδl)is of the form pdbfor a unique d ∈b
E (∞, l , nδl). Moreover, thisdbis given byd = Resb ∞(βl ,∅−1(µ)).
Lemma 4.16. — Letd ∈ ZZ/l Zbe such that the pair(d ,θ )is smooth (see Definition 2.6). Then
by (2.4), the variety Xθ(d ) is smooth and non-empty. Let ν be the l-core such that we have
d = nδl + Resl(ν)(see Proposition 4.4). Then we have the following.
(a) TheC×-fixed points inXθ(d )are exactlypdb,d ∈ E (∞, l , d )b (without repetition). (b) We haveE (∞, l , d ) = {Res∞(λ); λ ∈ Pl ,ν[|d |]}.
Proof. — The cased = nδl follows from Remark 4.15. But the proof in the general case
is completely the same. It is just important to know that under the given assumption on
(d ,θ ), each representation inXθ(d )is simple.
The lemma above yields parametrizations of Xθ(d )C× by E (∞, l , d ) and byPl ,ν[|d |].
Then the l-quotient bijection βl ,ν: Pl ,ν[|d |] → Pl[n] (see Lemma 4.10(b)) yields also a
parametrization ofXθ(d )C× byPl[n]. Forµ ∈ Pl[n]we denote byp
µ the corresponding
C×-fixed point ofXθ(d )(i.e., we havepµ= pdbifd = Resb ∞(βl ,ν−1(µ)). We can writepµdinstead
ofpµto stress that it is a point inXθ(d )C
×
.
The groupWlaffacts onPl[n]by permutation of components. More precisely, for each
i ∈ Z/l Z, the elementsi exchanges the componentsµi andµi +1 ofµ = (µ0, . . .,µl −1). It is
clear that the kernel of the canonical surjectionWlaff→ Wl acts trivially.
Lemma 4.17. — Let(d ,θ )be a smooth pair (see Definition 2.6) inZZ/l Z× CZ/l Z. Assume that
θi 6= 0. Then for every µ ∈ Pl[n], the bijection Xθ(d ) ≃ Xsi(θ )(si(d )) given by(2.4) sends the
C×-pointpd
µ ∈ Xθ(d )topsi
(d )
si(µ) ∈ Xsi(θ )(si(d )).
Proof. — There is an action ofWlaffonZZ
fingiven by (si( bd ))j= δj 0+ bdj +1+ bdj −1− bdj ifi ≡ j mod l , b dj ifi 6≡ j mod l .
It is clear that this action lifts theWlaff-actions onZZ/l Zwith respect to the mapp : ZZ
fin→
ZZ/l Z.
Letνandν′be thel-cores such that we haved = Resl(ν)+nδl andsi(d ) = Resl(ν′)+nδl.
For eachd ∈ E (∞, l , d )b , the following diagram is commutative Xθ [∞]( bd ) // i∞( bd ) Xsi(θ )[∞](si( bd )) i(si( bd )) ∞ Xθ(d ) //X si(θ )(si(d )).
