Finite dimensional representations of symplectic reflection algebras associated to wreath products II

0.2 MB·English

by  Silvia Montarani

Finite dimensional representations of symplectic reflection
algebras associated to wreath products II
Silvia Montarani
February 1, 2008
5
0
0 1 Introduction and main result
2

n Inthis paperweextendsomeresultsofapreviouspaper([EM])aboutfinitedimensionalrepresen-
a tations of the wreath product symplectic reflection algebra H (Γ ) of rank N attached to the
J 1,k,c N
group ΓN = SN ⋉ΓN ([EG]), where Γ ⊂ SL(2,C) is a finite subgroup, and (k,c) ∈ C(S), where
1
C(S) is the spaceof(complex valued) classfunctions onthe setS ofsymplectic reflectionsofΓ .
1 N
As in the previous paper, we use the results obtained in [CBH] for the rank 1 case and the

] homologicalpropertiesofH1,0,c(ΓN)toobtainirreduciblerepresentationsforsmallvaluesofk 6=0.
T
Specifically, let’s denote by B := H (Γ) the rank one symplectic reflection algebra attached to
1,c
R the pair (Γ, c) (in the rank 1 case there is no parameter k). For N~ =(N ,N ,...,N ) a partition
1 2 r
.
h of N, we consider W = W1 ⊗···⊗Wr, an irreducible representation of SN~ := SN1 ×···×SNr
t such that the Young diagram of W is a rectangle for any i and {Y }, i = 1,...,r, a collection of
a i i
m irreducible non-isomorphic representations of B with Ext1B(Yi,Yj) = 0 for any i 6= j. We denote
byY the tensorproductY⊗N1⊗···⊗Y⊗Nr. ThenM′ =W⊗Y isinanaturalwayanirreducible
[ 1 r
representation of the algebra S ♯B⊗N ⊂ S ♯B⊗N = H (Γ ). We show that the induced
N~ N 1,0,c N
1 representation M =IndSN♯B⊗NM′ of H (Γ) can be uniquely deformed along a linear subspace
v SN~♯B⊗N 1,0,c
6 of codimension r in C(S).
5
1
1.1 Symplectic reflection algebras of wreath product type
1
0
Before stating our main result in more detail, we recall the definition of the wreath product
5
0 symplectic reflection algebra H1,k,c(ΓN). Let L be a 2-dimensional complex vector space with a
/ symplectic form ω , and consider the space V =L⊕N, endowedwith the induced symplectic form
h L
t ωV =ωL⊕N. LetΓ be afinite subgroupofSp(L),andletSN be the symmetricgroupactingonV
a by permuting the factors. The group Γ :=S ⋉ΓN ⊂Sp(V) acts naturally on V. In the sequel
m N N
we will write γ ∈ Γ for any element γ ∈ Γ seen as an element in the i-th factor Γ of Γ . The
i N N
:
v symplecticreflectionsinΓN aretheelementsssuchthatrk(Id−s)|V =2. ΓN actsbyconjugation
i on the set S of its symplectic reflections. It is easy to see that there are symplectic reflections of
X
two types in Γ :
N
r
a (S) the elements s γ γ −1 where i,j ∈[1,N], s is the transposition (ij)∈S , and γ ∈Γ,
ij i j ij N
(Γ) the elements γ , for i∈[1,N] and γ ∈Γ\{1}.
i
Elements of type (S) are all in the same conjugacy class, while elements of type (Γ) form one
conjugacy class for any conjugacy class of γ in Γ. Thus elements f ∈C[S] can be written as pairs
(k,c),wherek isanumber(thevalueoff onelementsoftype(S)),andcisaconjugationinvariant
function on Γ\{1} (encoding the values of f on elements of type (Γ)).
Foranys∈S wewrite ω forthe bilinearformonV thatcoincideswith ω onIm(Id−s)and
s V
has Ker(Id−s) as radical. Denote by TV the tensor algebra of V.
1
Definition 1.1. For any f = (k,c) ∈ C[S], the symplectic reflection algebra H (Γ ) is the
1,k,c N
quotient
(Γ ♯TV)/h[u,v]−κ(u,v)i
N u,v∈V
where
κ:V ⊗V −→C[Γ ]:(u,v)7→ω(u,v)·1+ f ·ω (u,v)·s
N s s
s∈S
X
with f = f(s), and h...i is the two-sided ideal in the smash product Γ ♯TV generated by the
s N
elements [u,v]−κ(u,v) for u,v ∈V.
Following[GG], we will now givea more explicitdefinition of the algebraH (Γ ) by gener-
1,k,c N
ators and relations that we will need in the sequel. It is clear that choosing a symplectic basis x,
y for L we can consider Γ as a subgroup of SL(2,C). We will denote by x , y the corresponding
i i
vectorsinthe i-thL-factorofV =L⊕N andby c the valueofthe functionc onanelementγ ∈Γ.
γ
Lemma 1.2. ([GG]) The algebra H (Γ ) is the quotient of Γ ♯TV by the following relations:
1,k,c N N
(R1) For any i∈[1,N]:
k
[x ,y ]=1+ s γ γ−1+ c γ .
i i 2 ij i j γ i
Xj6=iγX∈Γ γ∈XΓ\{1}
(R2) For any u,v∈L and i6=j:
k
[u ,v ]=− ω (γu,v)s γ γ−1.
i j 2 L ij i j
γ∈Γ
X
2
Inthe caseN =1,there isno parameterk (there arenosymplectic reflectionsoftype (S)) and
H (Γ)=CΓ♯Chx,yi/(xy−yx=Λ) (1)
1,c
where
Λ=Λ(c)=1+ c γ ∈Z(C[Γ]) (2)
γ
γ∈XΓ\{1}
is the central element corresponding to c.
1.2 The main result
Let {Y } be a collection of non-isomorphic irreducible representations of the algebra B =H (Γ)
i 1,c
forsomec1andletY =Y1⊗N1⊗···⊗Yr⊗Nr. LetW =W1⊗···⊗Wr beanirreduciblerepresentation
ofS =S ×···×S ⊂S ,whereS isthesubgroupofpermutationsmovingonlytheindices
N~ N1 Nr N Ni
{ i−1N +1,..., i N }. There is a natural action of the algebra S ♯B⊗N on the vector
j=1 j j=1 j N~
space M′ :=W ⊗Y. Namely, eachcopy of H (Γ) acts in the correspondingY , while S acts on
P P 1,c i N~
W and simultaneously permutes the factors in the product Y. We will denote this representation
by M′. Since the algebra H (Γ ) is naturally isomorphic to S ♯B⊗N ⊃S ♯B⊗N we can form
c 1,0,c N N N~
the induced module M = IndSN♯B⊗NM′. The main theorem tells us when such a representation
c SN~♯B⊗N c
can be deformed to nonzero values of k.
1Suchrepresentations existonlyforspecialc,asforgenericcthealgebraH1,c(Γ)issimple;see[CBH].
2
If the Young diagram of W is a rectangle of height l and width m = N/l (the trivial
i i i i
representation corresponds to the horizontal strip of height 1) let H be the hyperplane in
Yi,mi,li
C(S) consisting of all pairs (k,c) satisfying the equation
k
dimY + |Γ|(m −l )+ c χ (γ)=0, (3)
i 2 i i γ Yi
γ∈XΓ\{1}
where χ is the character of Y .
Yi i
Let X =X(Y,W) be the moduli space of irreducible representations of H (Γ ) isomorphic
1,k,c N
toM asΓ -modules(where(k,c)areallowedtovary). Thisisaquasi-affinealgebraicvariety: itis
N
thequotientofthequasi-affinevarietyX(Y,W)ofextensionsoftheΓ -moduleM toanirreducible
N
H (Γ )-module by a free action of the reductive group G of basis changes in M compatible
1,k,c N
with Γ modulo scalars. Let f :X →eC(S) be the morphism which sends a representationto the
N
corresponding values of (k,c).
The main result of this paper is the following theorem that implies Theorem 3.1 of [EM] as a
particular case.
Theorem1.3. SupposeW hasrectangularYoungdiagramforanyi=1,...,r,andExt1(Y ,Y )=
i B i j
0 for any i6=j. Then:
(i) For any c the representation M of H (Γ ) can be formally deformed to a representa-
0 c0 1,0,c0 N
r
tionofH (Γ )alongthecodimension r linearsubspace H , butnotinotherdirections.
1,k,c N Yi,mi,li
i=1
\
This deformation is unique.
r
(ii) The morphism f maps X to H and is ´etale at M for all c . Its restriction to
Yi,mi,li c0 0
i=1
\
the formal neighborhood of M is the deformation from (i).
c0
r
(iii) There exists a nonempty Zariski open subset U of H such that for (k,c) ∈ U,
Yi,mi,li
i=1
\
the algebra H (Γ ) admits a finite dimensional irreducible representation isomorphic to M as
1,k,c N
a Γ -module.
N
The proof of this theorem occupies the remaining sections of the paper.
2 Main tools for the proof of Theorem 1.3
In this section we list some results we’ll need to prove our main theorem.
2.1 Representations of S with rectangular Young diagram
N
Wewillneedthefollowingstandardresultsfromtherepresentationtheoryofthesymmetricgroup.
The proofs are well known (for a quick reference see the previous article [EM]). Denote by h the
reflection representation of S . For a Young diagram µ we denote by π the corresponding
N µ
irreducible representation of S .
N
Lemma 2.1. (i) Hom (h⊗π ,π )=Cm−1, where m is the number of corners of the Young
SN µ µ
diagram µ. In particular Hom (h⊗π ,π )=0 if and only if µ is a rectangle.
SN µ µ
(ii) The element C =s +s +···+s acts by a scalar in π if and only if µ is a rectangle.
12 13 1n µ
In this case the scalar corresponding to C is the content c(µ) of the Young diagram µ i.e.
the sum of the signed distances of the cells from the diagonal.
2
3
2.2 Irreducible representations of H (Γ)
1,c
Werecallsomeresultsof[CBH]abouttheirreduciblerepresentationsoftherank1algebraH (Γ).
1,c
In [CBH], the classification of the finite-dimensional irreducible representations of H (Γ) is ob-
1,c
tained by establishing a Morita equivalence of this algebra with a deformationof the path algebra
of the double Q of the quiver Q, associated to the group Γ in the McKay correspondence. This
algebra is called deformed preprojective algebra, and we will denote it by Πλ(Q). Here λ ∈ Cν,
whereν isthenumberofnonisomorphicirreduciblerepresentationsofΓ,isaparameterrelatedto
ourparametercinthefollowingway. Ifwedenoteby{V },i=1,...,ν thecollectionofirreducible
i
non isomorphic representations of Γ, with V corresponding to the trivial representation, and by
1
Λ =1+ c γ the central element of C[Γ] that appears in the definition of H (Γ) given
γ∈Γ\{1} γ 1,c
at the end of Section 1.1 (see equation (2)), then we have:
P
λ =Tr| Λ.
i Vi
Thus, in particular, the representations of H (Γ) can be seen as representations of the path
1,c
algebra of a certain quiver, so that we can attach a dimension vector to each of them. We recall
that one can associate an affine root system to the McKay graph Q. The roots of such system
can be distinguished into real roots and imaginary roots. The real roots are divided into positive
and negative roots, and are the images of the coordinate vectors of Zν under sequences of some
suitably defined reflections. The imaginary roots, instead, are all the non-zero integer multiples
of the vector δ, with δ = dimV . If ”·” is the standard scalar product in Cν, let’s denote by R
i i λ
the set of real roots α such that λ·α=0, and by Σ the unique basis of R consisting of positive
λ λ
roots. We have the following result:
Theorem 2.2 ([CBH], Theorem 7.4). If λ·δ 6= 0, then Πλ(Q) has only finitely many finite-
dimensional simple modules up to isomorphisms, and they are in one-to-one correspondence with
the set Σ . The correspondence is the one assigning to each module its dimension vector.
λ
2
We recall that λ·δ = Tr| Λ, where R is the regular representation of Γ. Thus the theorem
R
holdsinthecaseTr| Λ6=0. LookingatourdefinitionofΛwededucethatthisisexactlyourcase.
R
2.3 A standard proposition in deformation theory
In this sectionwe describe the cohomologicalstrategywe will use to proveTheorem1.3. Let A be
anassociativealgebraoverC. In what follows,for eachA-bimodule E, we write Hn(A,E) for the
n-thHochschildcohomologygroupofAwithcoefficientsinE. WeremarkthatHi(A,E)coincides
with the vector space Exti (A,E), where Ao is the opposite algebra of A.
A⊗Ao
Let A be a flat formal deformation of A over the formal neighborhood of zero in a finite
U
dimensional vector space U with coordinates t ,...,t . This means that A is an algebra over
1 n U
C[[U]] = C[[t ,...,t ]] which is topologically free as a C[[U]]-module (i.e., A is isomorphic as a
1 n U
C[[U]]-module to A[[U]]), together with a fixed isomorphism of algebras A /JA ∼= A, where J
U U
is the maximal ideal in C[[U]]. Given such a deformation, we have a naturallinear map φ:U −→
H2(A,A).
Explicitly, we can think of A as A[[t ,...,t ]] equipped with a new C[[t ,...,t ]]-linear (and
U 1 n 1 n
continuous) associative product defined by:
a∗b= c (a,b) tpj a,b∈A
p1,...,pn j
p1X,...,pn Yj
where c :A×A−→A are C-bilinear functions and c (a,b)=ab, for any a,b∈A.
p1,...,pn 0,...,0
4
Imposing the associativity condition on ∗, one obtains that c must be Hochschild 2-
0,...,1j,...0
cocyclesforeachj. Themapφisgivenbytheassignment(t ,··· ,t )→ t [c ]forany
1 N j j 0,...,1j,...0
(t ,··· ,t )∈U, where [C] stands for the cohomology class of a cocycle C.
1 n
P
Now let M be a representation of A. In general it does not deform to a representation of
A . However we have the following standard proposition. Let η : U → H2(A,EndM) be the
U
composition of φ with the natural map ψ :H2(A,A)−→H2(A,EndM).
Proposition 2.3 ([EM]). Assume that η is surjective with kernel K, and H1(A,EndM) = 0.
Then:
(i) There exists a unique smooth formal subscheme S of the formal neighborhood of the origin
in U, with tangent space K at the origin, such that M deforms to a representation of the algebra
AS :=AU⊗ˆC[[U]]C[S] (where ⊗ˆ is the completed tensor product).
(ii) The deformation of M over S is unique.
2
We will show in the next sections that the conditions of Proposition 2.3 hold in our case.
2.4 Homological properties of the algebra H (Γ).
1,c
In order to show that we can make use of Proposition 2.3 we will need the following definitions
and results.
Definition 2.4 ( [VB1, VB2, EO]). An algebra A is defined to be in the class VB(d) if it is
of finite Hochschild dimension (i.e. there exists n ∈ N s.t. Hi(A,E) = 0 for any i > n and any
A-bimodule E) and H∗(A,A⊗Ao) is concentrated in degree d, where it is isomorphic to A as an
A-bimodule.
The nice properties of the class VB(d) are clarified by the following result by Van den Bergh.
Theorem2.5([VB1,VB2]). IfA∈VB(d)thenforanyA-bimoduleE,theHochschildhomology
H (A,E) is isomorphic to the Hochschild cohomology Hd−i(A,E).
i
2
Remark. The isomorphismof Theorem 2.5 depends on the choice of a bimodule isomorphism
φ:Hd(A,A⊗Ao)→∼ A and is unique for any such choice.
Now let B = H (Γ) and let Y be any irreducible representation of B. The following propo-
1,c i
sition summarizes the homological properties of this algebra that we will use in the proof of our
main theorem. All the proofs can be found in the previous article ([EM]).
Proposition 2.6.
(i) The algebra B belongs to the class VB(2);
(ii) H2(B,EndY )=H (B,EndY )=C;
i 0 i
(iii) H1(B,EndY )=0.
i
2
5
3 Proof of Theorem 1.3
3.1 Homological properties of A = H (Γ ).
1,0,c N
We are now ready to prove Theorem 1.3. Let A denote the algebra H (Γ ). The algebra
1,0,c0 N
A has a flat deformation over U = C(S), which is given by the algebra H1,k,c0+c′(ΓN). The
fact that this deformation is flat follows from the so called Poincar´e-Birkhoff-Witt Theorem for
symplectic reflection algebras (see [EG], Theorem 1.3). Our job is to compute the cohomology
groups H2(A,EndM), H1(A,EndM) and the map η :U −→H2(A,EndM).
Proposition 3.1. If W =W ⊗···⊗W is an irreducible representation of S such that W has
1 N N~ i
rectangular Young diagram for any i and Y =Y⊗N1⊗···⊗Y⊗Nr is an irreducible representation
1 r
of B⊗N with Y ≇Y and Ext1(Y ,Y )=Ext1 (B,Hom(Y ,Y ))=0 for any i6=j then
i j B i j B⊗Bo i j
H2(A,EndM)= H2(B,EndY )=Cr.
i
i
M
Proof. The second equality follows from Proposition 2.6, (ii). Let us prove the first equality.
We have:
H∗(A,EndM)=Ext∗ (A,EndM)=
A⊗Ao
=Ext∗ (S ♯B⊗N,EndM)=
SN♯B⊗N⊗SN♯Bo⊗N N
=Ext∗ (S ♯B⊗N,EndM).
SN×SN♯(B⊗N⊗Bo⊗N) N
Now, the S ×S ♯(B⊗N ⊗Bo⊗N)-module S ♯B⊗N is induced from the module B⊗N over the
N N N
subalgebra S ♯ B⊗N ⊗Bo⊗N , in which S acts simultaneously permuting the factors of B⊗N
N N
and Bo⊗N (note that S ♯(B⊗N ⊗Bo⊗N) is indeed a subalgebraof S ×S ♯(B⊗N ⊗Bo⊗N) as it
(cid:0) N (cid:1) N N
canbe identifiedwiththe subalgebraD♯(B⊗N⊗Bo⊗N)whereD ={(σ,σ), σ ∈S }⊂S ×S ).
N N N
Applying the Shapiro Lemma, we get:
Ext∗ (S ♯B⊗N,EndM)=
SN×SN♯(B⊗N⊗Bo⊗N) N
=Ext∗ (B⊗N,EndM)=
SN♯(B⊗N⊗Bo⊗N)
= Ext∗ (B⊗N,EndM) SN.
B⊗N⊗Bo⊗N
We observenow thatthe suba(cid:0)lgebraB⊗N is stable under th(cid:1)e inner automorphismsinduced by
the elements σ ∈S ⊂A. Thus the induced A-module M can be written as:
N
M =σ M′⊕σ M′⊕···⊕σ M′ (4)
1 2 n
where n = N! and {σ ,...,σ } is a set of representatives for the left cosets of S in S .
Each directNsu1m!...mNra!nd σM′1can itsenlf be written as: N~ N
l
σ M′ =σ W ⊗σ Y =σW ⊗Y ⊗···⊗Y .
l l l l ασl−1(1) ασl−1(N)
The actionofanelementσ(b ⊗···⊗b )onavector(w⊗y ⊗···⊗y ) ∈σ M′ is thefollowing:
1 N 1 N l l
σ(b ⊗···⊗b )(w⊗y ⊗···⊗y ) =
1 N 1 N l
= σ′w⊗b y ⊗···⊗b y ∈σ M′ (5)
(σ′σh)−1(1) (σ′σh)−1(1) (σ′σh)−1(N) (σ′σh)−1(N) h h
(cid:0) (cid:1)
6
where σ′ ∈ S , σ ∈ {σ ,...,σ } are the only elements satisfying σσ = σ σ′. Thus the latter
N~ h 1 n l h
module equals:
SN
Ext∗ B⊗N,End σ M′ =
B⊗N⊗Bo⊗N l
!!!
l
M
SN
= Ext∗ B⊗N, Hom(σ M′,σ M′)
 B⊗N⊗Bo⊗N  l h 
l,h
M
  
Since the actionof B⊗N ⊗Bo⊗N does not permute the direct factors in σ M′ and is trivial on
l
l
W in the module M′ we can rewrite the last term as: M
SN
Ext∗ B⊗N,Hom(σ M′,σ M′) =
 B⊗N⊗Bo⊗N l h 
l,h
M (cid:0) (cid:1)
 
SN
Ext∗ B⊗N,Hom(σ W,σ W)⊗Hom(σ Y,σ Y) =
 B⊗N⊗Bo⊗N l h l h 
l,h
M (cid:0) (cid:1)
 
SN
= Hom(σ W,σ W)⊗Ext∗ B⊗N,Hom(σ Y,σ Y) =
 l h B⊗N⊗Bo⊗N l h 
l,h
M (cid:0) (cid:1)
 
SN
N
= Hom(σ W,σ W)⊗Ext∗ B⊗N, Hom(Y ,Y ) .
l,h l h B⊗N⊗Bo⊗N i=1 ασl−1(i) ασh−1(i) !
M O
 
We want now to apply the Ku¨nneth formula in degree 2. Proposition2.6, (iii) and our conditions
on the Y s guarantee Ext1(Y ,Y ) = 0. Moreover, since for any i 6= j Y , Y are non-isomorphic
i B i j i j
irreducible representations of B we have Ext0(Y ,Y )=Hom (Y ,Y )=0. We get:
B i j B i j
SN
 
Hom(σ W,σ W)⊗ Ext2 B,Hom(Y ,Y )
 Ml,h l h Mi B⊗Bo(cid:18) ασl−1(i) ασh−1(i) (cid:19)
 
 (σ−1(i)σ−1(i))∈S ∀i 
 l h N~ 
 
where (σ−1(i)σ−1(i)) denotes the transposition moving the corresponding indices. Now we have
l h
(σ−1(i)σ−1(i)) ∈ S , ∀i if and only if σ = σ σ with σ ∈ S . But σ ,σ belong to different left
l h N~ l h N~ l h
cosets of S . Thus we can rewrite the last term as:
N~
SN
EndσW ⊗ Ext2 (B,EndY ) =
l l i B⊗Bo ασl−1(i) !
M M
r SN~
= EndW ⊗ Ext2 (B,EndY ) ⊕Ni =
B⊗Bo i
!
i=1
M (cid:0) (cid:1)
r r SN~
= EndW ⊗ Ext2 (B,EndY ) ⊕Ni .
  j B⊗Bo i 
i=1 j=1
M O (cid:0) (cid:1)
   
7
Now as an SNi-module, Ext2B⊗Bo(B,EndYi) ⊕Ni = Ext2B⊗Bo(B,EndYi)⊗CNi, where SNi
acts only on CNi permuting the factors and CNi = C⊕hi, where C is the trivial representation
(cid:0) (cid:1)
and hi is the reflection representation of SNi. So we have:
r
Ext2 (B,EndY )⊗ EndW ⊗···⊗(CNi ⊗EndW )···⊗EndW SN1×···×SNr =
B⊗Bo i 1 i r
i=1
M (cid:0) (cid:1)
r
= Ext2B⊗Bo(B,EndYi)⊗ EndSN1 W1⊗···⊗(C⊗EndWi⊕hi⊗EndWi)SNi ⊗···⊗EndSNr Wr =
Mi=1 (cid:16) (cid:17)
r
= Ext2B⊗Bo(B,EndYi)⊗ EndSNi Wi⊕HomSNi(hi⊗Wi,Wi) =
i=1
M (cid:0) (cid:1)
r
= Ext2 (B,EndY )=Cr
B⊗Bo i
i=1
M
as HomSNi(hi⊗Wi,Wi)=0, ∀i by Lemma 2.1 part (i). 2
Corollary 3.2. The map η :U −→H2(A,EndM) is surjective.
Proof.LetU ⊂U be the subspaceofvectors(0,c′). It issufficientto showthatthe restriction
0
of η to U is surjective. But this restriction is a composition of three natural maps:
0
U →H2(B,B)→H2(A,A)→H2(A,EndM).
0
Herethefirstmapη :U →H2(B,B)isinducedbythedeformationofBalongU ,thesecondmap
0 0 0
ξ : H2(B,B) → H2(A,A) comes from the Ku¨nneth formula, and the third map ψ : H2(A,A) →
H2(A,EndM) is induced by the homomorphism A→EndM.
Now,byProposition3.1,themapψ◦ξcoincideswiththemapψ :H2(B,B)→ r H2(B,EndY )
0 i=1 i
induced by the homomorphism φ:B → r EndY . Let K =Ker(ψ ) and U′ =η−1(K ). We
have to show that codimU′ ≥ r. But, byi=P1ropositiion 2.3,0the repres0entation0YLad0mits0a first
order deformation along U′0, for any i=1L,...,r. So on U′ we have Tr (Λ)=0, wihere Λ is as in
0 0 Yi
equation(2) (see equation (1)). Since the representationY correspondto the basis Σ of the root
i λ
system R (see Theorem 2.2), their characters are linearly independent, hence so are these linear
λ
equations. Thus the codimension of U′ is ≥r and η| is surjective, as desired.
0 U0
2
Proposition 3.3. H1(A,EndM)=0.
Proof.ArguingasintheproofofProposition3.1,wegetthatH1(A,EndM)= r H1(B,EndY ),
i=1 i
which is zero. This proves the proposition.
L 2
We have thus proved the following result.
Proposition3.4. IftheYoungdiagramcorrespondingtoW isarectangleforanyi,Ext (Y ,Y )=
i B i j
0 for any i6=j and M′ =W⊗Y =W1⊗···⊗Wr⊗Y1⊗N1⊗···⊗Yr⊗Nr then there exists a unique
smooth codimension r formal subscheme S of the formal neighborhood of the origin in U such
that the representation M =IndSN♯B⊗NM′ of H (Γ ) formally deforms to a representation of
SN~♯B⊗N 1,0,c0 N
H1,k,c0+c′(ΓN) along S (i.e., abusing the language, for (k,c′)∈S). Furthermore, the deformation
of M over S is unique.
Proof. Corollary 3.2 and Proposition 3.3 show that our case satisfies all the hypotheses of
Proposition 2.3. Moreover, from H2(A,EndM)= Cr we deduce dimKerη = dimU −r, and the
Proposition follows.
2
8
3.2 The subscheme S
Now we would like to find the subscheme S of Proposition 3.4. We will do this computing some
appropriate trace conditions for the deformation of the module M. To this end, we will use the
definition of H (Γ ) given at the end of Section 1.1 ( see Lemma 1.2). We recall that from
1,k,c N
the resultsof[CBH],thatwesummarizedinSection2.2,the irreduciblerepresentationsofH (Γ)
1,c
are in one-to-one correspondence with the basis Σ of the root system R , under the assignment
λ λ
sending each simple module to its dimension vector (see Theorem 2.2).
Since {Y } is a collection of non-isomorphic irreducible representations the dimension vectors
i
attached to these modules are distinct roots of such a basis, so in particular they are linearly
independent. Thus for any complex numbers z ,...,z there exists a central element Z of C[Γ]
1 r
such that Tr (Z)=z . Fix now Z(1) such that Tr (Z(1))=1−δ and consider the element:
Yi i Yi 1i
P =1⊗···⊗1⊗Z(1)⊗···⊗Z(1)∈H (Γ ).
1 t,k,c N
N1 N−N1
Such element commutes with|x ,{yz. C}ons|ider now{zthe rela}tion (R1) for i=1 and multiply it by
1 1
P on the right. The left hand side becomes [x ,y P ], thus it has trace zero. To compute the
1 1 1 1
traceoftherighthandsideoperatoritisconvenienttouseagainthe decompositionoftheinduced
module M given in the previous section (cfr. (4), (5)). The trace of the left hand side reduces to
the sum of the three terms:
Tr P =a(dimY )N1 dimW (6)
M 1 1 j
j
Y
k N1 k N1
Tr s γ γ−1 P = aTr s γ γ−1 dimW (7)
2 M 1j 1 j  1 2 W1⊗Y1⊗N1  1j 1 j  j
j=2γ∈Γ j=2γ∈Γ j6=1
XX XX Y
   
Tr c γ P =a c Tr (γ) (dimY )N1−1 dimW (8)
M γ 1 1  γ Y1  1 j
γ∈XΓ\{1} γ∈XΓ\{1} Yj
   
where a = (N−N1)! is the number of elements in the set of representatives σ ∈ {σ ,...,σ } such
N2!...Nr! l 1 n
that σ−1({1,2,...,N })={1,2,...,N }. But now we claim that:
l 1 1
Tr (s γ γ −1)=Tr (s )dimY N1−1 ∀j ≤N . (9)
W1⊗Y1⊗N1 1j 1 j W1 1j 1 1
To obtain (9), we observe that, for j ≤ N1, s1jγ1γj−1 is conjugate in ΓN1 = SN1 ⋉ΓN1 to s1j
and that the character of S on W ⊗Y⊗N1 is simply the product of the characters on W and
N1 1 1 1
Y⊗N1. An easy computation gives Tr s =dimY N1−1, hence the formula.
1 Y⊗N1 ij 1
1
By Lemma 2.1, we havethat for any transpositionσ ∈S , Tr (σ)= dimW1 c(µ), where
N1 W1 N1(N1−1)/2
c(µ) is the content of the Young diagram µ attached to W . In particular, if µ is a rectangular
1
diagram of size l ×m with l m =N , it can be easily computed that:
1 1 1 1 1
N (m −l )
1 1 1
c(µ)= ,
2
so we have:
(m −l )dimW
Tr| (s γ γ −1)= 1 1 1dimY N1−1. (10)
W1⊗Y1⊗N1 1j 1 j N1−1 1
Substituting in (7), summing up (6),(7),(8) and simplifying we get the equation:
k
dimY + |Γ|(m −l )+ c χ (γ)=0 (11)
1 2 1 1 γ Y1
γ∈XΓ\{1}
9
that is exactly the equation for the hyperplane H . Analogously we can define Z(i), P and
Y1,m1,l1 i
obtain the equations of the hyperplane H for i = 2,...,r. We get exactly r independent
Yi,mi,li
necessary linear conditions.
This shows that (0,c )+S ⊂ r H . But since the two subschemes have the same
0 i=1 Yi,mi,li
dimensionwehavethatS istheformalneighborhoodofzeroin r H −(0,c )andTheorem
T i=1 Yi,mi,li 0
1.3, (i) is proved. Parts (ii), (iii) follow exactly as in [EM].
T
Acknowledgments. I am very grateful to Pavel Etingof for many comments and useful
discussions.
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