Cherednik algebras and Calogero-Moser cells

63 Pages·2016·0.354 MB·English

by  Cédric Bonnafé, Raphael Rouquier

CÉDRIC BONNAFÉ
RAPHAËL ROUQUIER
CHEREDNIK ALGEBRAS AND
CALOGERO-MOSER CELLS
CÉDRIC BONNAFÉ
InstitutdeMathématiquesetdeModélisationdeMontpellier(CNRS:UMR5149),
UniversitéMontpellier2,CaseCourrier051,PlaceEugèneBataillon,34095
MONTPELLIERCedex,FRANCE.
E-mail: [email protected]
RAPHAËL ROUQUIER
UCLAMathematicsDepartmentLosAngeles,CA90095-1555,USA.
E-mail: [email protected]
The second author was partially supported by the NSF (grant DMS-1161999) and
byagrantfromtheSimonsFoundation(#376202).
October26,2016
CHEREDNIK ALGEBRAS AND
CALOGERO-MOSER CELLS
CÉDRIC BONNAFÉ, RAPHAËL ROUQUIER

CONTENTS
PartI. Reflectiongroups................................................. 7
1. Notations.............................................................. 9
1.1. Integers............................................................ 9
1.2. Gradings........................................................... 9
1.3. Modules........................................................... 10
2. Reflectiongroups...................................................... 11
2.1. Determinant,roots,coroots.......................................... 11
2.2. Invariants.......................................................... 12
2.3. Hyperplanesandparabolicsubgroups............................... 13
2.4. Irreduciblecharacters............................................... 14
2.5. Hilbertseries....................................................... 15
2.6. Coxetergroups..................................................... 17
PartII. Cherednikalgebras............................................... 19
3. GenericCherednikalgebra............................................. 21
3.1. Structure........................................................... 21
3.2. Gradings........................................................... 27
3.3. Eulerelement....................................................... 29
3.4. Sphericalalgebra................................................... 29
3.5. SomeautomorphismsofH(cid:101).......................................... 31
3.6. SpecialfeaturesofCoxetergroups................................... 33
4. Cherednikalgebrasatt =0............................................. 35
4.1. Generalities........................................................ 35
4.2. Center............................................................. 37
4.3. Localization........................................................ 39
4.4. Complements...................................................... 41
6 C.BONNAFÉ&R.ROUQUIER
4.5. SpecialfeaturesofCoxetergroups................................... 44
Appendices.............................................................. 45
A. Filtrations............................................................. 47
A.1. Filteredmodules................................................... 47
A.2. Filteredalgebras................................................... 47
A.3. Filteredmodulesoverfilteredalgebras............................... 48
A.4. Symmetricalgebras................................................ 50
A.5. Weylalgebras...................................................... 51
B. Invariantrings........................................................ 53
B.1. Moritaequivalence................................................. 53
B.2. Geometricsetting................................................... 54
Bibliography............................................................. 57
PART I
REFLECTION GROUPS

CHAPTER 1
NOTATIONS
1.1. Integers
Weput(cid:78)=(cid:90)(cid:190)0.
1.2. Gradings
1.2.A. Let k be a ring and X a set. We denote by kX =k(X) the free k-module with
basis X. We sometimes denote elements of kX as formal sums: (cid:80) α x, where
x∈X x
α ∈k.
x
1.2.B. Let Γ be a monoid. We denote by kΓ (or k[Γ]) the monoid algebra of Γ over
k. ItsbasisofelementsofΓisdenotedby{tγ}γ∈Γ.
A Γ-graded k-module is a k-module L with a decomposition L = (cid:76)γ∈ΓLγ (that is
the same as a comodule over the coalgebra kΓ). Given γ ∈ Γ, we denote by L〈γ 〉
0 0
the Γ-graded k-module given by (L〈γ0〉)γ = Lγγ0. We denote by k-freeΓ the additive
category of Γ-graded k-modules L such that Lγ is a free k-module of finite rank for
allγ∈Γ. Given L∈k-freeΓ,weput
(cid:88)
dimΓk(L)= rankk(Lγ)tγ∈(cid:90)Γ.
γ∈Γ
We have defined an isomorphism of abelian groups dimΓ : K (k-freeΓ)−∼→(cid:90)Γ. This
k 0
construction provides a bijection from the set of isomorphism classes of objects of
k-freeΓ to (cid:78)Γ. Given P =(cid:80)γ∈Γpγtγ with pγ∈(cid:78), we define the Γ-graded k-module kP
by(kP)γ=kpγ. WehavedimΓ(kP)=P.
k
Wesaythatasubset E ofaΓ-gradedmodule L ishomogeneousifeveryelementof
E isasumofelementsin E ∩Lγ forvariouselementsγ∈Γ.
1.2.C. A graded k-module L is a (cid:90)-graded k-module. We put L+ =(cid:76)i>0Li. If Li =0
for i (cid:28)0 (for example, if L is (cid:78)-graded), then dim(cid:90)(L) is an element of the ring of
k
10
Laurent power series (cid:90)((t)): this is the Hilbert series of L. Similarly, if L = 0 for
i
i (cid:29)0,thendim(cid:90)(L)∈(cid:90)((t−1)).
k
When L has finite rank over k, we define the weight sequence of L as the unique
sequenceofintegersr1(cid:182)···(cid:182)rm suchthatdim(cid:90)k(L)=tr1+···+trm.
A bigraded k-module L is a ((cid:90)×(cid:90))-graded k-module. We put t=t(1,0) and u=t(0,1),
sothatdim(cid:90)×(cid:90)(L)=(cid:80) dim (L )tiuj for L∈k-free(cid:90)×(cid:90). When L is((cid:78)×(cid:78))-graded,we
k i,j k i,j
havedim(cid:78)×(cid:78)(L)∈(cid:90)[[t,u]].
k
1.2.D. Assume k is a commutative ring. There is a tensor product of Γ-graded k-
modulesgivenby(L⊗kL(cid:48))γ=(cid:76)γ(cid:48)γ(cid:48)(cid:48)=γLγ(cid:48)⊗kLγ(cid:48)(cid:48). Whenthefibersofthemultiplication
map Γ×Γ→Γ are finite, the multiplication in Γ provides (cid:90)Γ with a ring structure,
thetensorproductpreservesk-freeΓ,anddimΓ(L⊗ L(cid:48))=dimΓ(L)dimΓ(L(cid:48)).
k k k k
AΓ-gradedk-algebraisak-algebraA withaΓ-gradingsuchthatAγ·Aγ(cid:48) ⊂Aγγ(cid:48).
1.3. Modules
Let A be a ring. Given L a subset of A, we denote by < L > the two-sided ideal
of A generated by L. Given M an A-module, we denote by Rad(M) the intersection
of the maximal proper A-submodules of M. We denote by A-mod the category of
finitely generated A-modules and we putG (A)=K (A-mod), where K ((cid:67)) denotes
0 0 0
theGrothendieckgroupofanexactcategory(cid:67).
Given M ∈ A-mod, we denote by [M] (or simply [M]) its class in G (A). When
A 0
A is a graded ring and M is a finitely generated graded A-module, we denote by
[M]gr (or simply [M]gr) its class in the Grothendieck group of the category A-modgr
A
of finitely generated graded A-modules. Note that K (A-modgr) is a (cid:90)[t±1]-module,
0
witht[M]gr=[M〈−1〉]gr.
We denote by Irr(A) the set of isomorphism classes of simple A-modules. As-
sume A is a finite-dimensional algebra over the field k. We have an isomorphism
∼
(cid:90)Irr(A)−→G (A), M (cid:55)→[M]. If A is semisimple, we have a bilinear form 〈−,−〉 on
0 A
G (A)givenby〈[M],[N]〉=dim Hom (M,N). WhenA issplitsemisimple,Irr(A)pro-
0 k A
videsanorthonormalbasis.
Let W be a finite group and assume k is a field. We denote by Irr (W) (or simply
k
by Irr(W)) the set of irreducible characters of W over k. When |W|∈k×, there is a
∼
bijection Irrk(W)−→Irr(kW), χ (cid:55)→ Eχ. The group Hom(W,k×) of linear characters of
W withvaluesinkisdenotedbyW∧k (orW∧). WehaveanembeddingW∧⊂Irr(W),
and equality holds if and only if W is abelian and k contains all e-th roots of unity,
where e istheexponentofW.