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Cédric Bonnafé, Peng Shan
To cite this version:
Cédric Bonnafé, Peng Shan. On the cohomology of Calogero-Moser spaces. International Mathematics Research Notices, Oxford University Press (OUP), 2020, 2020, pp.1091-1111. �10.1093/imrn/rny036�. �hal-01579435v5�
by
CÉDRIC BONNAFÉ & PENG SHAN
Abstract. —We compute the equivariant cohomology of smooth Calogero-Moser spaces and
some associated symplectic resolutions of symplectic quotient singularities.
1. Notation and main results
Throughout this note, we will abbreviate⊗C as⊗. By an algebraic variety, we mean a
reduced scheme of finite type overC.
1.A. Reflection group. — LetV be aC-vector space of finite dimensionn and letW be
a finite subgroup ofGLC(V ). We set
Ref(W ) = {s ∈ W | dimCVs= n − 1}
and we assume that
W = 〈Ref(W )〉.
We setǫ : W → C×,w 7→ det(w ).
If s ∈ Ref(W ), we denote byα∨s andαs two elements ofV andV∗, respectively, such
thatVs= Ker(αs)andV∗s= Ker(α∨s), whereα∨s is viewed as a linear form onV∗.
Ifw ∈ W, we set
cod(w ) = codimC(Vw)
and 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 = CF0(A) ⊂ F1(A) ⊂ · · · ⊂ Fn(A) = A = Fn +1(A) = · · ·
The first author is partly supported by the ANR (Project No ANR-16-CE40-0010-01 GeRepMod). The second author is partly supported by the ANR (Project No ANR-13-8501-001-01 VARGEN)..
is also a filtration ofA. Letħh be a formal variable, and write
Rees•F(A) =M
i ¾0
ħ
hiFi(A) ⊂ C[ħh ] ⊗ A (the Rees algebra),
gr•F(A) =M
i ¾0
Fi(A)/Fi −1(A).
Recall thatgr•F(A) ≃ Rees•F(A)/ħh Rees•F(A).
1.B. Rational Cherednik algebra at t = 0. — Throughout this note, we fix a function
c : Ref(W ) → C which is invariant under conjugacy. We define the rational Cherednik
algebraHc 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. Here〈 , 〉 : V ×V∗→ Cis the standard pairing. The first
commu-tation relations imply that we have morphisms of algebrasC[V ] → Hc andC[V∗]→ H
c.
Recall [5, Theorem 1.3] that we have an isomorphism ofC-vector spaces
(1.1) C[V ] ⊗ CW ⊗ C[V∗]−→ H∼ c
induced by multiplication (this is the so-called PBW-decomposition).
We denote by Zc the center of Hc: it is well-known [5] that Zc is an integral domain,
which is integrally closed and containsC[V ]W andC[V∗]W as subalgebras (so it contains
P = C[V ]W ⊗ C[V∗]W), and that it is a free P-module of rank|W |. We denote byZ c the
affine algebraic variety whose ring of regular functionsC[Zc]isZc: this is the Calogero-Moser spaceassociated with the datum(V , W , c ).
Using the PBW-decomposition, we define aC-linear mapΩcH: Hc −→ CW by
ΩcH( f w g ) = f (0)g (0)w
for allf ∈ C[V ],g ∈ C[V∗]andw ∈ CW. This map isW-equivariant for the action on both
sides by conjugation, so it induces a well-definedC-linear map
Ωc: Zc −→ Z(CW ).
Recall from [4, Corollary 4.2.11] thatΩc is a morphism of algebras, and that
(1.2) Zc is smooth if and only ifΩc is surjective.
The “only if” part is essentially due to Gordon [7, Corollary 5.8] (see also [4, Proposi-tion 9.6.6 and (16.1.2)]) while the “if” part follows from the work of Bellamy, Schedler and Thiel [3, Corollary 1.4].
1.C. Grading. — The algebra T(V ⊕ V∗) ⋊ W can be Z-graded in such a way that the
generators have the following degrees
deg(y ) = −1 ify ∈ V, deg(x ) = 1 ifx ∈ V∗, deg(w ) = 0 ifw ∈ W.
This descends to a Z-grading on Hc, because the defining relations (Hc) are
Z-graded. So the Calogero-Moser spaceZc inherits a regularC×-action. Note also that
by definitionP = C[V ]W ⊗ C[V∗]W is clearly a graded subalgebra ofZ c.
1.D. Main results. — For a complex algebraic variety X (equipped with its classical topology), we denote byHi(X ) itsi-th singular cohomology group with coefficients in
C. IfX carries a regular action of a torusT, we denote byHi
T(X )itsi-th T-equivariant
cohomology group (still with coefficients in C). Note thatH2•(X ) = L
i ¾ 0H2i(X ) is a
gradedC-algebra andH2•T(X ) =Li ¾ 0H2iT(X )is a gradedH2•T(pt)-algebra. The following
result [5, Theorem 1.8] describes the algebra structure on the cohomology of Zc (with coefficients inC):
Theorem 1.3 (Etingof-Ginzburg). — Assume thatZc is smooth. Then:
(a) H2i +1(Z
c) = 0for alli.
(b) There is an isomorphism of gradedC-algebrasH2•(Zc)≃ gr•
F(Z(CW )).
In this note, we prove an equivariant version of this statement (we identify H∗C×(pt) withC[ħh ]in the usual way):
Theorem A. —Assume thatZc is smooth. Then:
(a) H2i +1C× (Zc) = 0for alli.
(b) There is an isomorphism of gradedC[ħh ]-algebrasH2•C×(Zc)≃ Rees•F(Z(CW )).
Note that Theorem A(a) just follows from the statement (a) of Etingof-Ginzburg Theo-rem by Proposition 2.4(a) below. As a partial consequence of TheoTheo-rem A, we also obtain the following application to the equivariant cohomology of some symplectic resolutions. Theorem B. —Assume that the symplectic quotient singularity(V ×V∗)/W admits a symplectic resolutionπ : X −→ (V × V∗)/W. Recall that theC×-action on(V × V∗)/W lifts (uniquely) toX (see[12, Theorem 1.3(ii)]). Then:
(a) H2i +1C× (X ) = 0for alli.
(b) There is an isomorphism of gradedC[ħh ]-algebrasH2•C×(X )≃ Rees•F(Z(CW )).
Note that forW = Sn acting onCn, Theorem B describes the equivariant cohomology
of the Hilbert scheme ofn points inC2: this was already proved by Vasserot [14]. In [6,
Conjecture 1.3], Ginzburg-Kaledin proposed a conjecture for the equivariant cohomology of a symplectic resolution of a symplectic quotient singularity E/G, where E is a finite
dimensional symplectic vector space andG is a finite subgroup ofSp(E ). However, their conjecture cannot hold as stated, because they considered the C×-action by dilatation,
which is contractible. Theorem B shows that the correct equivariant cohomological re-alization of the Rees algebra is provided by the symplecticC×-action, which exists only
whenG stabilizes a Lagrangian subspace ofE.
Remark 1.4. — Recall from the works of Etingof-Ginzburg [5], Ginzburg-Kaledin [6], Gordon [7] and Bellamy [1] that the existence of a symplectic resolution of(V × V∗)/W is
equivalent to the existence of a parameterc such thatZc is smooth, and that it can only occur if all the irreducible components ofW are of typeG (d , 1, n)(for somed,n ¾ 1) or G4in Shephard-Todd classification.
1.E. Conjectures. — In [4, Chapter 16, Conjectures COH and ECOH], Rouquier and the first author proposed the following conjecture which aims to generalize the above Etingof-Ginzburg Theorem 1.3 into two directions: it includes singular Calogero-Moser spaces and it extends to equivariant cohomology.
Conjecture 1.5. — With the above notation, we have:
(1) H2i +1(Zc) = 0for alli.
(2) We have an isomorphism of gradedC-algebrasH2•(Z
c)≃ grF(Im(Ωc)).
(3) We have an isomorphism of gradedC[ħh ]-algebrasHC2•×(Zc)≃ ReesF(Im(Ωc)).
By (1.2), when Zc is smooth, the image of Ωc coincide with the center of CW. So
Theorem A proves this conjecture for smoothZc.
We will also prove in Example 3.6 the following result:
Proposition 1.6. — IfdimC(V ) = 1, then Conjecture 1.5 holds.
1.F. Structure of the paper. — The paper is organized as follows. The proof of The-orem A relies on classical theThe-orems on restriction to fixed points in cohomology and K-theory. In Section 2, we first recall basic properties on equivariant cohomology and equivariant K-theory and restriction to fixed points. Section 3 explains how these general principles can be applied to Calogero-Moser spaces. Theorem A will be proved in Sec-tion 4. The cyclic group case (ProposiSec-tion 1.6) will be handled in Example 3.6. The proof of Theorem B will be given in Section 5.
2. Equivariant cohomology,K-theory and fixed points
2.A. Equivariant cohomology. — LetX be a complex algebraic variety equipped with a regular action of a torusT. Recall that the equivariant cohomology ofX is defined by
HT•(X ) = H•(ET×BTX),
whereET→ BTis a universalT-bundle. The pullback of the structural morphismx : X →
ptyields a ring homomorphismH•T(pt) → HT•(X ), which makesH•T(X )a gradedH•T(pt)
-algebra.
Denote byX (T)the character lattice ofT. Then for eachχ ∈ X (T), denote byCχ the one
dimensionalT-module of characterχ, the first Chern classcχ of the line bundleET×TCχ
onBTis an element inH2(BT). Identify the vector spaceC⊗ZX (T)with the dualt∗ of the
Lie algebratof Tvia χ 7→ d χ. Then the assignmentχ 7→ cχ yields an isomorphism of
2.B. EquivariantK-theory. — We denote byKT(X ) the Grothendieck ring of the
cate-gory ofT-equivariant vector bundles onX. Note that aT-equivariant vector bundle on a point is the same as a finite dimensionalT-module. We have a canonical isomorphism
KT(pt) = Z[X (T)]which sends the class of aT-moduleM to
(2.1) dimT(M ) = X
χ∈X (T)
dimC(Mχ)χ,
whereMχ is theχ-weight space inM.
Let Hˆ2•
T (X )be the completion of HT2•(X ) with respect to the ideal
L
i>0H2iT(X ). The
equivariant Chern characterprovides a ring homomorphism
chX : KT(X )−→ ˆH2•T(X )
with the following properties. First, whenX is a pointpt, we have (2.2) chpt: KT(pt) = Z[X (T)] −→ Hˆ2•T (pt) = ˆS(t
∗)
χ 7−→ exp(dχ).
Next the Chern character commutes with pullback. More precisely, ifY is another variety with a regular action of the same torusTand ifϕ : X → Y is aT-equivariant morphism, then the following diagram commutes
(2.3) KT(Y ) ϕ∗ // chY KT(X ) chX ˆ H2•T(Y ) ϕˆ ∗ //Hˆ2• T (X ).
Hereϕ∗ denotes both the pullback map inK-theory or in equivariant cohomology, and
ˆ
ϕ∗is the map induced after completion by the pullback map in equivariant cohomology.
In particular, by applying the above diagram to Y = pt and the structural morphism
X→ pt, we may viewchX as a morphism of algebras overKT(pt), with theKT(pt)-algebra
structure onHˆ2•
T(X )provided by the embedding (2.2).
2.C. Fixed points. — We denote byXT the (reduced) variety consisting of fixed points ofTin X. LetiX : XT,−→ X be the natural closed immersion. SinceTacts trivially on XT, we haveH•T(XT) = HT•(pt) ⊗ H•(XT)asHT•(pt)-algebras. Recall that theT-action onX is called equivariantly formal if the Leray-Serre spectral sequence
Ep q2 = Hp(BT; Hq(X )) =⇒ Hp +qT (X )
for the fibrationET×TX → BT degenerates atE2. We have the following standard result
on equivariant cohomology (see for instance [10, Proposition 2.1]):
Proposition 2.4. — Assume thatH2i +1(X ) = 0for alli. ThenX is equivariantly formal, and:
(a) There is an isomorphism of gradedHT•(pt)-modulesHT•(X )≃ H•
T(pt) ⊗ H
•(X ). In particular
H2i +1T (X ) = 0for alli.
(b) The pullback mapiX∗ : H
•
T(X )→ H
•
T(X
T)is injective.
(c) Let m be the unique graded maximal ideal of H•T(pt). Then we have an isomorphism of algebrasH•(X )≃ H•T(X )/mHT•(X ).
In particular, this shows that Conjectures 1.5(1) and (3) imply Conjecture 1.5(2).
Example 2.5 (Blowing-up). — LetY be an affine variety with aT-action and letC be a T-stable closed subvariety (not necessarily reduced) ofY. LetX be the blowing-up ofY alongC. Writeπ : X → Y for the natural morphism, and we equipX with the unique T-action such thatπis equivariant. We assume thatXTis finite. We write
H2•T (XT) = M
x ∈XT
S(t∗)ex,
whereex ∈ H0T(XT)is the primitive idempotent associated with x (i.e., the fundamental
class ofx).
ThenD=π∗(C )is aT-stable effective Cartier divisor, and we denote by[ D ]the class inKT(X )of its associated line bundle (which isT-equivariant). We denote bych1X([ D ])∈
HT2(X )its firstT-equivariant Chern class. We want to computeiX∗ (ch
1
T(D)).
First, letI be the ideal ofC[Y ]associated withC. As it isT-stable, we can find a family of T-homogeneous generators(ai)1 ¶ i ¶ k of I. We denote by λi ∈ X (T) theT-weight of
ai. The choice of this family of generators induces a T-equivariant closed immersion
X,→ Y × Pk −1(C). We denote byXi the affine chart corresponding to “ai6= 0”. Ifx ∈ XT,
we denote byi(x ) ∈ {1, 2, . . ., k }an element such thatx ∈ Xi(x ). Then
(2.6) iX∗ (ch 1 T(D)) =−ħh X x ∈DT (dλi(x ))ex.
Indeed, we just need to compute the local equation ofDaroundx ∈ XT, and this can be done inXi(x ). ButD∩ Xi is principal for alli, defined by
D∩ Xi={(y , ξ) ∈ Y × Pk −1(C)| (y , ξ) ∈ Xi andai(y ) = 0}.
So D∩ Xi is defined by a T-homogeneous equation of degree λi ∈ X (T), and so (2.6)
follows.
3. Localization and Calogero-Moser spaces
In this section, we apply the previous discussions toX = Zc andT = C×. Denote by
q : C×→ C×the identity map. ThenX (C×) = qZ, and we have
KC×(pt) = Z[q , q−1], H2•
C×(pt) = C[ħh ],
withħh = cq, following the notation of Section 2.A. SoHˆ2•C×(pt) = C[[ħh ]]and the Chern map in this case is given by
chpt: Z[q , q−1],→ C[[ħh ]], q 7→ exp(ħh ).
Note also that a finite C×-module is nothing but a finite dimensional Z-graded vector
space M = Li ∈ZMi such that C× acts on Mi by the character qi. The identification
KC×(pt) = Z[q , q−1]sends the class ofM to its graded dimension (or Hilbert series):
dimgr(M ) =X
i ∈Z
3.A. Fixed points. — For χ ∈ Irr(W ), we denote by ωχ : Z(CW ) → C the associated
morphism of algebras, that is,ωχ(z ) =χ(z )/χ(1) is the scalar through which z acts on
the irreducibleCW-module with the character χ. We denote byeW
χ the unique
primi-tive idempotent ofZ(CW )such that ωχ(eχW) = 1. IfE is a subset of Irr(W ), then we set
eEW =Pχ∈EeχW.
Now, consider the algebra homomorphism
Ωχc =ωχ◦ Ωc : Zc −։ C.
Its kernel is a maximal ideal ofZc: we denote by zχ the corresponding closed point in
Zc. It follows from [4, Lemma 10.2.3 and (14.2.2)] thatzχ∈ ZC×
c and that the map
(3.1) z : Irr(W ) −→ ZC
×
χ 7−→ zχ
is surjective. The fibers of this map are called the Calogero-Moser c-families. They were
first consider by Gordon [7] and Gordon-Martino [9]: see also for instance [4, §9.2]. Let
CMc(W )be the set of Calogero-Moserc-families. ForE ∈ CMc(W ), we denote byzE∈ ZC
× its image under the mapz. On the other hand, by [4, (16.1.2)] we have
(3.2) Im(Ωc) = M
E ∈CMc(W )
CeEW.
Hence we get an isomorphism ofC-algebras
H2•(ZcC×)≃ Im(Ωc), ezE 7→ e
W
E ,
which extends to an isomorphism ofC[ħh ]-algebras H2•C×(ZC ×
c )≃ C[ħh ] ⊗ Im(Ωc).For
sim-plification, we setic = iZc : ZC
×
c ,→ Z and, under the above identification, we view the
pullback mapic∗ as a morphism of algebras
ic∗: H2•C×(Zc)−→ C[ħh ] ⊗ Im(Ωc).
So, by Proposition 2.4, Conjecture 1.5 is implied by the following one:
Conjecture 3.3. — With the above notation, we have:
(1) H2i +1(Zc) = 0for alli.
(2) Im(ic∗) = ReesF(Im(Ωc)).
Remark 3.4. — Set
FiH(Im(Ωc)) ={z ∈ Im(Ωc)| ħhiz ∈ Im(ic∗)}.
Then, by construction and Proposition 2.4,F•H(Im(Ωc))is the filtration ofIm(Ωc)satisfying H2•C×(Zc)≃ Im(i∗c) = ReesFH(Im(Ωc))
and H2•(Zc)≃ grFH(Im(Ωc)).
So showing Conjecture 3.3 is then equivalent to showing that the filtrationsF•(Im(Ωc))
andFH
• (Im(Ωc))coincide.
Note also that, sinceZc is an affine variety of dimension2n, we haveHi(Zc) = 0for
i> 2n, so this shows that
On the other hand, sinceZc is connected, we have
(♯) F0(Im(Ωc)) =F0H(Im(Ωc)) = C.
These two particular cases will be used below.
The following result, based on the previous remark, can be viewed as a reduction step for the proof of Conjecture 3.3:
Proposition 3.5. — Assume thatH2i +1(Z
c) = 0and
dimC(H2i(Zc)) = dimC(Fi(Im(Ωc))/Fi −1(Im(Ωc)))
for alli. Then Conjecture 3.3 holds if and only ifReesF(Im(Ωc))⊂ Im(ic∗).
Proof. — Assume thatH2i +1(Zc) = 0and
dimC(H2i(Zc)) = dimC(Fi(Im(Ωc))/Fi −1(Im(Ωc)))
for alli. We keep the notation of Remark 3.4. It then follows from this remark and the
hypothesis that
dimC(Fi(Im(Ωc))/Fi −1(Im(Ωc))) = dimC(H2i(Zc)) = dim(FiH(Im(Ω c))
/Fi −1H (Im(Ωc)))
for alli. So, by induction, we get thatdimC(Fi(Im(Ωc))) = dim(FiH(Im(Ωc)))for alli (by
the equality (♯) of Remark 3.4). This shows that ReesF(Im(Ωc)) = Im(ic∗) if and only if
ReesF(Im(Ωc))⊂ Im(ic∗), as desired.
Example 3.6. — Assume in this example, and only in this example, thatdimC(V ) = 1. It
is proved in [4, Theorem 18.5.8] that in this caseH2i +1(Zc) = 0and dimC(H2i(Zc)) = dimC(Fi(Im(Ωc))/Fi −1(Im(Ωc)))
for alli. SinceZc is affine of dimension2, we haveHi(Z
c) = 0ifi 6∈ {0, 2}. So it follows
from the equalities (♭) and (♯) of Remark 3.4 that Conjecture 3.3 holds in this case (this
proves Proposition 1.6).
3.B. Chern map. — IfE is a Calogero-Moser family, we denote bymE ⊂ Zc the ideal of
functions vanishing atzE∈ ZC × c . We also set Im(Ωc)Z= M E ∈CMc(W ) ZeEW.
We make the natural identificationKC×(ZC ×
c ) = Z[q , q−1]⊗ZIm(Ωc)Z. Through these
identi-fications, the Chern mapchZC×
c just becomes the natural inclusionZ[q , q
−1]⊗
ZIm(Ωc)Z,→
C[[ħh ]]⊗Im(Ωc). Moreover, ifP is aZ-graded finitely generated projectiveZc-module, then
the commutativity of the diagram (2.3) just says that (3.7) ic∗(chZc([ P ])) =
X
E ∈CMc(W )
dimgr(P/mEP )eEW ⊂ C[[ħh ]] ⊗ Im(Ω
4. Proof of Theorem A
Hypothesis and notation. We assume in this section, and only in this sec-tion, thatZc is smooth.
IfM =Li ∈ZMi andN =Li ∈ZNi are two finite-dimensional gradedCW-modules, we
set
〈M , N 〉grW = X
i , j ∈Z
〈Mi, Nj〉W qi + j,
where 〈E , F 〉W = dim HomCW(E , F ) for any finite dimensional CW-modules E and F.
We extend this notation to the case whereM or N is a graded virtual character (i.e. an
element ofZ[q , q−1]⊗ZZIrr(W )). Finally, we denote byC[V ]co(W ) the coinvariant algebra,
that is, the quotient of the algebraC[V ]by the ideal generated by the elementsf ∈ C[V ]W
such that f (0) = 0. Then C[V ]co(W ) is a gradedCW-module, which is isomorphic to the
regular representationCW when one forgets the grading.
4.A. Localization inK-theory. — Recall from (1.2) that the smoothness of Zc implies thatIm(Ωc) = Z(CW ), so thatz : Irr(W ) → ZcC× is bijective. Lete = e1W = (1/|W |)Pw ∈Ww.
The smoothness ofZc also implies that the functor Hc-mod −→ Zc-mod
M 7−→ e M = e Hc⊗HcM
is an equivalence of categories [5, Theorem 1.7]. If E is a finite dimensional Z-graded CW-module, the Hc-module Hc ⊗CW E is finitely generated, Z-graded and projective.
Therefore,e Hc⊗CWE is a finitely generated graded projectiveZc-module, which can be
viewed as aC×-equivariant vector bundle onZc. For simplification, we set
chc(E ) = ic∗(chZc(e Hc⊗CW E )) ∈ C[[ħh ]] ⊗ Im(Ωc).
Proposition 4.1. — Assume that Zc is smooth. Let E be a finite dimensional graded CW -module. Then chc(E ) = X χ∈Irr(W ) 〈χ, C[V ]co(W )⊗ E 〉gr W 〈χ, C[V ]co(W )〉gr W eχW.
Proof. — As the formula is additive, we may, and we will, assume thatE is an irreducible
CW-module, concentrated in degree0.
Now, let χ ∈ Irr(W ): we denote bymχ the maximal ideal ofZc corresponding to the
fixed pointzχ. We setp = mχ∩ P: it does not depend on χ (it is the maximal ideal of
P = C[V/W × V∗/W ]of functions which vanishes at(0, 0); see for instance [4, (14.2.2)]).
By (3.7),
chc(E ) =
X
χ∈Irr(W )
dimgr((e Hc⊗CWE )/mχ(e Hc⊗CWE ))eχW.
Now, letZ¯c= Zc/pZc andH¯c= Hc/pHc. We setm¯χ= mχ/pZc. ThenH¯c is a finite
dimen-sionalC-algebra (called the restricted rational Cherednik algebra) and again the bimodule
( ¯Zc/ ¯mχ)⊗Z¯ce ¯Hc is a simple rightH¯c-module (which will be denoted byLc(χ): it is iso-morphic to the shift by somerχ ∈ Zof the quotient of the baby Verma module denoted
byL(χ)in [7]).
SinceCW is semisimple, theCW-moduleE is flat, and so
(e Hc⊗CW E )/mχ(e Hc⊗CWE ) ≃ (e Hc/mχ(e Hc))⊗CW E
≃ (e ¯Hc/ ¯mχ(e ¯Hc))⊗CW E .
≃ Lc(χ) ⊗CWE .
But the graded dimension ofLc(χ) ⊗CWE is known wheneverZc is smooth and is given
by the expected formula (see [1, Lemma 3.3 and its proof]), up to a shift in grading:
chc(E ) = X χ∈Irr(W ) qrχ′〈χ, C[V ] co(W )⊗ E 〉gr W 〈χ, C[V ]co(W )〉gr W eχW,
whererχ′ ∈ Zdoes not depend onE. Now, ifCdenotes the trivial CW-module
concen-trated in degree0, thenchc(C) = 1, which shows thatrχ′= 0for allχ, as desired.
Let KC×(CW ) denote the Grothendieck group of the category of finite dimensional gradedCW-modules. If E is a finite dimensional graded CW-module, we denote by
[ E ] its class inKC×(CW ). We still denote by chc : KC×(CW ) → C[[ħh ]] ⊗ Z(CW )the map defined by
chc([ E ]) = chc(E ).
Now, letW′ be a parabolic subgroup ofW and setV′= VW′ andr = codim
C(V′). We
identify the dualV′∗ofV′withV∗W′and note that
(4.2) V∗= V′∗⊕ (V′)⊥.
We denote by∧(V′)⊥the element ofK
C×(CW′)defined by
∧(V′)⊥=X
i ¾ 0
(−1)i[∧i(V′)⊥].
Recall also that there existnalgebraically independent homogeneous polynomials f1,. . . ,
fn inC[V ]W such thatC[V ]W = C[ f
1, . . ., fn], and we denote bydi the degree of fi.
Corollary 4.3. — Assume that Zc is smooth. Let E′ be a finite dimensional graded CW′ -module. Then chc IndWW′(∧(V ′)⊥ ⊗ [ E′])=(1 − q d1)· · · (1 − qdn) (1 − q )n −r X χ∈Irr(W ) 〈χ, IndWW′E′〉 gr W 〈χ, C[V ]co(W )〉gr W eχW.
Proof. — The groupW′acts trivially onV′so it acts trivially on∧iV′∗for alli. Therefore,
(1 − q )n −rchc IndWW′(∧(V
′)⊥⊗ [ E′])
= chc(IndWW′ ∧V
′∗⊗ ∧(V′)⊥⊗ [ E′]) .
But∧V′∗⊗ ∧(V′)⊥= ResWW′(∧V∗)by (4.2), so, by Frobenius formula,
(1 − q )n −rchc IndWW′(∧(V
′)⊥
⊗ [ E′])= chc ∧V∗⊗ IndWW′E
′ .
So it follows from Proposition 4.1 that (1 − q )n −rchc IndWW′(∧(V ′)⊥ ⊗ [ E′]) = X χ∈Irr(W ) 1 〈χ, C[V ]co(W )〉gr W 〈χ, C[V ]co(W )⊗ (∧V∗)⊗ IndWW′(E ′)〉gr W e W χ .
But, if w ∈ W, the Molien’s formula implies that the graded trace ofw onC[V ]co(W ) is
equal to
(1 − qd1)· · · (1 − qdn)
det(1 − w−1q ) ,
while its graded trace on∧V∗ is equal todet(1 − w−1q ). So the class ofC[V ]co(W )⊗ ∧V∗
is equal to (1 − qd1)· · · (1 − qdn) times the class of the trivial module, and the corollary follows.
Proof of Theorem A. — Assume thatZc is smooth. This implies in particular that Conjec-ture 1.5(1) and (2) hold (Etingof-Ginzburg Theorem 1.3), and so the hypotheses of Propo-sition 3.5 are satisfied. Note also that Im(Ωc) = Z(CW ) by (1.2). It is then sufficient to
prove thatħhiFi(Z(CW )) ⊂ Im(ic∗)for alli.
Let us introduce some notation. IfG is a finite group andH is a subgroup, we define a
C-linear mapTrGH : Z(CH ) −→ Z(CG )by TrGH(z ) = X g ∈[G/H ] gz = 1 |H | X g ∈G gz
(here,[G/H ]denotes a set of representatives of elements ofG/H andgz = g z g−1). It is
easy to check that
(4.4) TrGH(eηH) =η(1) |H | X γ∈Irr(G ) |G | γ(1)〈γ, Ind G H(η)〉Ge G γ
for allη ∈ Irr(H ). Also, ifh ∈ H andΣH(h ) ∈ Z(CH )denotes the sum of the conjugates of
h inH, then
(4.5) TrGH(ΣH(h )) =
|CG(h )|
|CH(h )|
ΣG(h ),
whereCG(h )andCH(h )denote the centralizers ofh inG andH respectively. LetPr(W )
denote the set of parabolic subgroups W′ of W such thatcodimC(VW
′ ) = r. It follows from (4.5) that (4.6) Fr(Z(CW )) = X W′∈P r(W ) TrWW′(Z(CW ′)).
Therefore, by Proposition 3.5, it is sufficient to prove that
(Æ) ħhrTrWW′(eW ′ χ′ )∈ Im(i ∗ c)for allW ′∈ P
r(W )and allχ′∈ Irr(W′).
But it turns out that the coefficient ofħhr in χ′(1)
|W′|chc Ind
W W′(∧(V
is equal, according to Corollary 4.3, to (−1)rχ ′(1) |W′| X χ∈Irr(W ) |W | 〈χ, IndWW′χ′〉W χ(1) e W χ .
Indeed, sinceq = exp(ħh ), this follows from the fact that the polynomial 〈χ, C[V ]co(W )〉gr
W
takes the valueχ(1)wheneverq = 1(i.e. ħh = 0) and from the fact that
(1 − qd1)· · · (1 − qdn)
(1 − q )n −r ≡ (−1) rd
1· · · dnhħr mod ħhr +1
and that|W | = d1· · · dn. Note that by definition χ
′(1)
|W′|chc IndWW′(∧(V′)⊥⊗ χ′)
belongs to
C[[ħh ]] ⊗C[ħh]Im(ic∗), so each of its homogeneous components belong toIm(ic∗). By (4.4), this
proves that(Æ)holds, and the proof of Theorem A is complete.
5. Proof of Theorem B
Hypothesis and notation. We assume in this section, and only in this sec-tion, that the symplectic quotient singularityZ0= (V ×V∗)/W admits a sym-plectic resolutionX→ Z0.
Recall [12, Theorem 1.3(ii)] that the C×-action on Z0 lifts uniquely toX. As it is ex-plained in Remark 1.4, the existence of a symplectic resolution ofZ0implies that all the irreducible components ofW are of typeG (d , 1, n)orG4. Since the proof of Theorem B
can be easily reduced to the irreducible case, we will separate the proof in two cases. Proof of Theorem B forW = G (d , 1, n). — Assume here thatW = G (d , 1, n). Let S1 be the group of complex numbers of modulus1. In this case, it follows from [8] that X is dif-feomorphic to some smoothZc, and that the diffeomorphism might be chosen to be S1-equivariant. As theS1-equivariant cohomology is canonically isomorphic to theC×
-equivariant cohomology, this proves Theorem B in this case.
Proof of Theorem B forW = G4. — Assume here thatW = G4. It is possible (probable?) that
againX isS1-diffeomorphic to some smoothZc, but we are unable to prove it (it is only known that they are diffeomorphic). So we will prove Theorem B in this case by brute force computations.
We fix a primitive third root of unityζand we assume thatV = C2and thatW = 〈s , t 〉,
where s = ζ 0 ζ2 1 and t =1 −ζ 2 0 ζ .
By the work of Bellamy [2], there are only two symplectic resolutions ofZ0= (V ×V∗)/W.
They have both been constructed by Lehn and Sorger [13]: one can be obtained from the other by exchanging the role ofV andV∗, so we will only prove Theorem B for one of
them. Let us describe it.
Let H = Vs and letH denote the image ofH × V∗ in Z0, with its reduced structure of closed subvariety. We denote byβ : Y → Z0 the blowing-up ofZ0along H and we
denote byα : X → Y the blowing-up ofY along its reduced singular locusS. Then [13]
is a symplectic resolution.
We now give more details, which all can be found in [13, §1]. First,C[Z0]is generated
by8homogeneous elements(zi)1 ¶ i ¶ 8whose degrees are given by the following table:
(5.1) z z1 z2 z3 z4 z5 z6 z7 z8
deg(z ) 0 4 −4 2 −2 −6 6 0
The defining ideal ofH in C[Z0]is generated by 6 homogeneous elements(bj)1 ¶ j ¶ 6
whose degrees are given by the following table:
(5.2) b b1 b2 b3 b4 b5 b6
deg(b ) 2 6 0 12 8 4
This defines aC×-equivariant closed immersionY ,→ Z0× P5(C). We denote byYi the affine chart defined by “bi 6= 0”. The equations of the zero fiber β−1(0) given in [13,
§1] show thatYC× ={p
2, p3, p4, p6}, wherepi is the unique element ofYC
×
i . We use the
notation of Example 2.5. By (2.6) and (5.2), we have
(♣) iY∗ (ch
1
Y([β
∗H ])) =−ħh (6e
p2+ 12ep4+ 4ep6).
Now,S is contained inY2∪ Y3, soαis an isomorphism in a neighborhood ofp4andp6.
So letq4=α−1(p4)andq6=α−1(p6). These are elements ofXC
× .
On the other hand, Y2 is a transversalA1-singularity, so the defining ideal of S ∩ Y2
inC[Y2]is generated by three homogeneous elementsa+,a◦ anda−(of degree6,0and
−6, by [13, §1]), and it is easily checked thatα−1(p2)C×={q+
2, q2−}whereq ±
2 is the unique
C×-fixed element in the affine chart defined by “a±6= 0”.
Also, Y3 is isomorphic to(h × h∗)/S3, whereh is the diagonal Cartan subalgebra of
sl3(C)and S3 is the symmetric group on 3 letters, viewed as the Weyl group of sl3(C).
LetHilb3(C2)denote the Hilbert scheme of3 points in C2, and let Hilb0
3(C2) denote the
(reduced) closed subscheme defined as the Hilbert scheme of three points inC2 whose
sum is equal to(0, 0). By [11, Proposition 2.6],
(5.3) X3≃ Hilb0
3(C 2).
It just might be noticed that the isomorphismY3≃ (h × h∗)/S3 becomesC×-equivariant
if one “doubles the degrees” in(h × h∗)/S3, that is, ifC×acts on h(respectivelyh∗) with
weight2(respectively−2). Recall thatC×-fixed points inHilb03(C2)are parametrized by
partitions of3: we denote byq3+,q3◦andq3−the fixed points inX3corresponding respec-tively to the partitions(3),(2, 1)and(1, 1, 1)of3, so that
(5.4) XC× 3 ={q + 3, q ◦ 3, q − 3}. Finally, we have (5.5) XC×={q2+, q2−, q3+, q3◦, q3−, q4, q6}.
Also,π∗(H ) =α∗(β∗(H ))is an effective Cartier divisor ofX, and it follows from (♣) and
the commutativity of the diagram (2.3) that
(♦) iX∗ (ch 1 X([π ∗H ])) =−ħh (6e q+ 2 + 6eq2−+ 12eq4+ 4eq6).
We now wish to compute iX∗ (ch
1
X([α
∗S ])). As the singular locus S is contained in
Y2∪ Y3, and containsp2andp3, there existsn2+,n2−,n3+,n3◦ andn3−inZsuch that
iX∗ (ch 1 X([α ∗S ])) =−ħh (n+ 2eq+ 2 + n − 2eq− 2 + n + 3eq+ 3 + n ◦ 3eq3◦+ n3−eq− 3).
Sincea+anda−have degree6and−6, it follows from (2.6) thatn2+= 6andn2−=−6.
As we can exchange the roles ofhandh∗in the description ofY3and its singular locus (becauseh ≃ h∗as anS3-module), this shows thatn3−=−n+
3 andn3◦=−n3◦. Son3◦= 0and
it remains to computen3+. So letU+be the open subset ofHilb03(C2)consisting of ideals J
of codimension3ofC[x , y ]such that the classes of1, x andx2form a basis ofC[x , y ]/ J.
Then we have an isomorphism J+: C4 ∼
−→ U+given by
(a , b , c , d ) 7−→ J+(a , b , c , d ) = 〈x3+ a x + b , y − c x2− d x −
2 3a c 〉.
The form of the generators of the ideal J+(a , b , c , d )is here to ensure that J+(a , b , c , d ) ∈
Hilb03(C2). The fixed point q3+ is the unique one inU+. Through this identification, the
action ofC×onC4is given by
ξ · (a , b , c , d ) = (ξ4a ,ξ6b ,ξ−6c ,ξ−4d )
(remember that we must “double the degrees” of the usual action). The equation ofα∗(S )
on this affine chart≃ C4 can then be computed explicitly and is given by4a3+ 27b2= 0,
so is of degree12. This shows thatn3+= 12. Finally,
(♥) iX∗ (ch 1 X([α ∗S ])) =−ħh (6e q2+− 6eq2−+ 12eq3+− 12eq − 3). Let us now conclude. First, recall from [6, Theorem 1.2] that
(5.6) ¨ H2i +1(X ) = 0 for alli, H2•(X )≃ grF(Z(CW )). By Proposition 2.4, we get (♠) dimCH2iC×(X ) = 1 ifi = 0, 3 ifi = 1, 7 ifi ¾ 2.
Now,W has seven irreducible characters1,ǫ,ǫ2,χ,χǫ,χǫ2andθ, whereχis the unique
irreducible character of degree2with rational values andθ is the unique one of degree
3. We denote by
Ψ : HC2•×(X
C×)−→ C[ħ∼ h ] ⊗ Z(CW )
the isomorphism ofC[ħh ]-algebras such that Ψ(eq4) = e W 1 , Ψ(eq6) = e W θ , Ψ(eq+ 2) = e W χǫ, Ψ(eq2−) = eχǫW2, Ψ(eq+ 3) = e W
ǫ2 , Ψ(eq3◦) = eχW and Ψ(eq3−) = eǫW.
By Proposition 2.4, HC2•×(X ) is isomorphic to its image by Ψ ◦ iX∗ in C[ħh ] ⊗ Z(CW ). But,
by (♦) and (♥), and after investigation of the character table ofW, this image contains
ħ h (6eχǫW− 6eχǫW2+ 12e W 1 + 4e W θ ) = ħh (4 + ΣW(s ) + ΣW(s2))
and ħh (12eǫW2 − 12e
W
So it containsh Σħ W(s ) and ħh ΣW(s2). Also, by (♠) and Proposition 2.4, it also contains
ħ
h2Z(CW ), so
Rees•F(Z(CW )) ⊂ Im(Ψ ◦ iX∗ ).
Using again (♠) and Proposition 2.4, a comparison of dimensions yields that Rees•F(Z(CW )) = Im(Ψ ◦ iX∗ ),
and the proof is complete.
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March 13, 2018
CÉDRIC BONNAFÉ, Institut Montpelliérain Alexander Grothendieck (CNRS: UMR 5149), Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34095 MONTPELLIER Cedex, FRANCE E-mail :cedric.bonnafe@umontpellier.fr
PENG SHAN, Yau Mathematical Sciences Center, Tsinghua University, 100086, BEIJING, CHINA
