From Jantzen to Andersen Filtration via Tilting Equivalence

0.23 MB·English

by  Johannes Kübel

FROM JANTZEN TO ANDERSEN FILTRATION VIA TILTING
EQUIVALENCE
1
1 JOHANNESKU¨BEL
0
2
Abstract. The space of homomorphisms between a projective object and a

n Verma module in category O inherits an induced filtration from the Jantzen
a filtration on the Verma module. On the other hand there is the Andersen
J filtration on the space of homomorphisms between a Verma module and a

0 tiltingmodule. Arkhipov’stiltingfunctor,acontravariantself-equivalenceofa
2 certainsubcategoryofO,whichmapsprojectivetotiltingmodulesinducesan
isomorphismofthesekindsofHom-spaces. Wewillshowthatthisequivalence

] evenidentifies bothfiltrations.
T
R
.
h
1. Introduction
t
a
m Let g⊃ b⊃h be a semisimple complex Lie algebra with a Borel and a Cartan.
In BGG-category O, prominent objects are indecomposable projective and tilting
[
modules.

3 Tilting modules were introduced in [3] as selfdual Verma flag modules and the
v indecomposable tilting modules are classified by their highest weight. Let ρ ∈ h∗
4
be the halfsum of positive roots relative to b and let T denote the ring of regular
9
functions on the line Cρ. T is a quotient of the universal enveloping algebra of h.
7
1 For every weight λ ∈ h∗ the quotient map b ։ h induces a b-module structure on
. C. We denote this b-module by C . The h-module structure on T also restricts
1 λ
1 to a b-module structure by the map b ։ h. Now we can form the Verma module
0 ∆(λ)=U(g)⊗ C ∈g-mod and the deformed Verma module
U(b) λ
1
∆ (λ)=U(g)⊗ (C ⊗T)∈g−mod−T
: T U(b) λ
v
where tensor products without any specification are to be understood over C. The
i
X T-module structure on ∆ (λ) is just multiplication from the right while the b-
T
r module structure on C ⊗T is the tensor product representation.
a λ
Taking the direct sum of the T-dualweight spaces and twisting the contragredient
g-module structure with a Chevalley automorphism leads to the deformed dual
Verma module ∇ (λ) ∈ g-mod-T. We will see that every invertible T-module
T
homomorphism on the λ-weight spaces extends to an injective homomorphism of
g-T-bimodules
can:∆ (λ)֒→∇ (λ)
T T
which forms a basis of the T-module Hom (∆ (λ),∇ (λ)). Since T can be
g−T T T
understood as a polynomial ring in one variable v we get the Jantzen filtration
on ∆(λ) by taking the images of can−1(∇ (λ)vi) for i = 0,1,2,3,... under the
T
Date:December5,2010.
Key words and phrases. representationtheory,categoryO.
I would like to thank Wolfgang Soergel and Peter Fiebig for their support and many helpful
discussions.
1
2 JOHANNESKU¨BEL
surjection∆ (λ)։∆(λ) induced by·⊗ C. Fora projectiveobjectP ∈O we get
T T
an induced filtration on Hom (P,∆(λ)).
g
Let T be the completion of T at the maximalideal of 0. So we canidentify T with
the ring of formal power series CJvK in one variable. In this article, we will also
b b
introducedeformedtiltingmodules whicharecertaing-T-bimodulescorresponding
totiltingmodulesinOafterspecializingwith·⊗bC. NowletK besuchadeformed
T b
tilting module and consider the composition pairing
Hom(∆b(λ),K)×Hom(K,∇b(λ))→Hom(∆b(λ),∇b(λ))∼=T
T T T T
where all Hom-spaces are meant to be homomorphisms of g-T-bimodulebs. We will
see that this is a nondegenerate pairing of free T-modules of finite rank and leads
b
to an injection
b
Hom(∆b(λ),K)֒→(Hom(K,∇b(λ)))∗
T T
where (·)∗ denotes the T-dual.
Taking the preimages of the T-submodules (Hom(K,∇b(λ)))∗ ·vi under this em-
b T
bedding and applying ·⊗bC to these preimages defines the Andersen filtration on
T b
Homg(∆(λ),K ⊗TbC).
In [9], Soergel introduces the tilting functor t which forms a contravariant self-
equivalence of the category of modules with a Verma flag, i.e. a filtration with
subquotients isomorphic to Verma modules. The functor t takes projective mod-
ules to tilting modules and sends a Verma module ∆(ν) to the Verma module
∆(−2ρ−ν). So t induces an isomorphism of vector spaces
∼
Hom (P,∆(λ))−→Hom (∆(−2ρ−λ),t(P))
g g
which we denote by t as well. In this paper we will prove that t even identifies
the filtration induced by the Jantzen filtration on the left side with the Andersen
filtration on the right side.
In [10], Soergel uses a hard Lefschetz argument to prove that the Andersen
filtration on Hom (∆(λ),K) for K a tilting module coincides with the grading
g
filtration induced from the graded version of O as described in [2]. Since this is
very similar to the result in [1] about the semisimplicity of the subquotients of the
Jantzenfiltration,the relationofboth filtrations mightgiveanalternativeproofof
this semisimplicity.
2. Preliminaries
ThissectioncontainssomeresultsaboutthedeformedcategoryOofasemisimple
complex Lie algebra g with Borel b and Cartan h, which one can also find in [4]
and [10]. By S we will denote the universal enveloping algebra of the Cartan h
which is equal to the ring of polynomial functions C[h∗]. Let T be a commutative,
associative,noetherian,unital,localS-algebrawithstructuremorphismτ :S →T.
We call T a local deformation algebra.
In this article we will mostly deal with the S-algebras R = S , the localisation
(0)
of S at the maximal ideal of 0 ∈ h∗, localisations R of R at a prime ideal p of
p
height 1 or the residue fields of these rings K =R /R p. To apply results of this
p p p
sectiontobothfiltrationswewillalsobe concernedwiththepowerseriesringCJvK
in one variable and the quotient field Q of S. All these rings are local deformation
algebras.
FROM JANTZEN TO ANDERSEN FILTRATION VIA TILTING EQUIVALENCE 3
2.1. Deformed category O. LetT be alocaldeformationalgebrawithstructure
morphism τ :S →T and let M ∈g-mod-T. For λ∈h∗ we set
M ={m∈M|hm=(λ+τ)(h)m ∀h∈h}
λ
where(λ+τ)(h) is meantto be anelementofT. We callthe T-submoduleM the
λ
deformed λ-weight space of M.
We denote by O the full subcategory of all bimodules M ∈ g-mod-T such
T
thatM = M andwiththepropertiesthatforeverym∈M theb-T-bimodule
λ
λL∈h∗
generatedbymisfinitelygeneratedasaT-moduleandthatM isfinitelygenerated
as a g-T-bimodule. For example, if we put T = C, O is just the usual BGG-
T
category O.
For λ∈h∗ we define the deformed Verma module
∆ (λ)=U(g)⊗ T
T U(b) λ
where T denotes the U(b)-T-bimodule T with b-structure given by the composi-
λ
λ+τ
tion U(b)→S −→T.
As in [10], we now introduce a functor
d=d :g⊗T−mod−→g⊗T−mod
τ
by letting dM ⊂ Hom (M,T)σ be the sum of all deformed weight spaces in the
T
spaceofhomomorphismsofT-modulesfromM toT withitsg-actiontwistedbyan
involutive automorphism σ : g→g with σ| =−id. We now set ∇ (λ)= d∆ (λ)
h T T
for λ∈h∗ and call it the deformed nabla module. As in [10], one shows d∇ (λ) ∼=
T
∆ (λ)andthattensoringwithafinitedimensionalrepresentationE ofgcommutes
T
with d up to the choice of an isomorphism dE ∼=E.
Proposition 2.1 ([10], Proposition 2.12.). (1) For all λ the restriction to the
deformed weight space of λ together with the two canonical identifications
∆ (λ) →∼ T and ∇ (λ) →∼ T induces an isomorphism
T λ T λ
∼
Hom (∆ (λ),∇ (λ))−→T
OT T T
(2) For λ6=µ in h∗ we have Hom (∆ (λ),∇ (µ))=0.
OT T T
(3) For all λ,µ∈h∗ we have Ext1 (∆ (λ),∇ (µ))=0.
OT T T
Corollary 2.2 ([10],Corollary2.13.). Let M,N ∈O . If M has a ∆ -flag and N
T T
a ∇ -flag, then the space of homomorphisms Hom (M,N) is a finitely generated
T OT
free T-module and for any homomorphism T → T′ of local deformation algebras
the obvious map defines an isomorphism
Hom (M,N)⊗ T′ −∼→Hom (M ⊗ T′,N ⊗ T′)
OT T OT′ T T
Proof. ThisfollowsfromProposition2.1byinductiononthelengthofthe∆ -and
T
∇ -flag. (cid:3)
T
If m⊂T is the unique maximal ideal in our local deformation algebra T we set
K=T/mT for its residue field.
4 JOHANNESKU¨BEL
Theorem 2.3 ([4], Propositions 2.1 and 2.6). (1) Thebasechange·⊗ Kgives
T
a bijection
simple isomorphism simple isomorphism
←→
(cid:26) classes of OT (cid:27) (cid:26) classes of OK (cid:27)
(2) The base change ·⊗ K gives a bijection
T
projective isomorphism projective isomorphism
←→
(cid:26) classes of OT (cid:27) (cid:26) classes of OK (cid:27)
The category OK is the direct summand of the category O over the Lie algebra
g⊗K consisting of all objects whose weights lie in the complex affine subspace
τ +h∗ = τ +HomC(h,C) ⊂ HomK(h⊗K,K) for τ the restriction to h of the map
thatmakesKtoaS-algebra. Sothesimple objectsofOK aswellasthe onesofOT
areparametrizedby their highestweightin h∗. Denote by L (λ) the simple object
T
with highest weight λ. We also use the usual partial order on h∗ to partially order
τ +h∗.
Theorem 2.4 ([4],Propositions2.4and2.7). Let T be a local deformation algebra
and K its residue field. Let L (λ) be a simple object in O .
T T
(1) There is a projective cover P (λ) of L (λ) in O . Every projective object
T T T
in O is isomorphic to a direct sum of projective covers.
T
(2) P (λ) has a Vermaflag, i.e. afinitefiltration with subquotients isomorphic
T
to Verma modules, and for the multiplicities we have the BGG-reciprocity
formula
(PT(λ):∆T(µ))=[∆K(µ):LK(λ)]
for all Verma modules ∆ (µ) in O .
T T
(3) Let T → T′ be a homomorphism of local deformation algebras and P pro-
jective in OT. Then P ⊗T T′ is projective in OT′ and the natural transfor-
mation
Hom (P,·)⊗ T′ −→Hom (P ⊗ T′,·⊗ T′)
OT T OT′ T T
is an isomorphism of functors from O to T′-mod.
T
SinceFiebigworksoveracomplexsymmetrizableKac-Moodyalgebra,hehasto
introduce truncated subcategories and has to set some finiteness assumptions. In
case of a finite dimensional semisimple Lie algebra we do not need these technical
tools.
2.2. Block decomposition. Let T again denote a local deformation algebra and
K its residue field.
Definition 2.5. Let ∼ be the equivalence relation on h∗ generated by λ ∼ µ if
T T
[∆K(λ):LK(µ)]6=0.
Definition 2.6. Let Λ ∈ h∗/ ∼ be an equivalence class. Let O be the full
T T,Λ
subcategory of O consisting of all modules M such that every highest weight of a
T
subquotient of M lies in Λ.
FROM JANTZEN TO ANDERSEN FILTRATION VIA TILTING EQUIVALENCE 5
Proposition 2.7 ([4], Proposition 2.8). The functor
O −→ O
T,Λ T
Λ∈hL∗/∼T
(MΛ)Λ∈h∗/∼T 7−→ MΛ
Λ∈hL∗/∼T
is an equivalence of categories.
The isomorphism above is called block decomposition.
Later we will be especially interested in the case T =R=S where S denotes
(0) (0)
the localisation of S at the maximal ideal generated by h, i.e. the maximal ideal
of 0∈h∗. Since ∼R=∼C, the block decomposition of OR corresponds to the block
decomposition of the BGG-category O over g.
Let τ : S → K be the induced map that makes K into a S-algebra. Restricting
to h and extending with K yields a K-linear map h⊗K → K which we will also
call τ. Let R ⊃ R+ be the root system with positive roots according to our data
g⊃b⊃h. For λ∈h∗K =HomK(h⊗K,K) and αˇ ∈h the dual root of a root α∈R
we set hλ,αˇi =λ(αˇ)∈K. Let W be the Weyl group of (g,h).
K
Definition 2.8. For R the root system of g and Λ∈h∗/∼ we define
T
R (Λ)={α∈R|hλ+τ,αˇi ∈Z⊂K for some λ∈Λ}
T K
and call it the integral roots corresponding to Λ. Let R+(Λ) denote the positive
T
roots in R (Λ) and set
T
W (Λ)=h{s ∈W|α∈R+(Λ)}i⊂W
T α T
We call it the integral Weyl group with respect to Λ.
From [4] Corollary 3.3 it follows that
Λ=W (Λ)·λ for any λ∈Λ
T
where we denote by · the ρ-shifted dot-action of the Weyl group.
Since most of our following constructions commute with base change, we are
particularly interested in the case when T = R is a localization of R at a prime
p
idealpofheightone. Applyingthefunctor·⊗ R willsplitthedeformedcategory
R p
O into generic and subgeneric blocks which is content of the next
T
Lemma 2.9 ([5], Lemma 3). Let Λ∈h∗/∼ and let p∈R be a prime ideal.
R
(1) If αˇ ∈/ p for all roots α ∈ R (Λ), then Λ splits under ∼ into generic
R Rp
equivalence classes.
(2) If p = Rαˇ for a root α ∈ R (Λ), then Λ splits under ∼ into subgeneric
R Rp
equivalence classes of the form {λ,s ·λ}.
α
We recall that we denote by P (λ) the projective cover of the simple object
T
L (λ). It is indecomposable and up to isomorphism uniquely determined. For an
T
equivalence class Λ∈h∗/∼ which contains λ and is generic, i.e. Λ={λ}, we get
T
P (λ) = ∆ (λ). If Λ = {λ,µ} and µ < λ, we have P (λ) = ∆ (λ) and there is a
T T T T
non-split short exact sequence in O
T
0→∆ (λ)→P (µ)→∆ (µ)→0
T T T
6 JOHANNESKU¨BEL
In this case, every endomorphism f :P (µ)→P (µ) maps ∆ (λ) to ∆ (λ) since
T T T T
λ>µ. So f induces a commutative diagram
0 −−−−→ ∆ (λ) −−−−→ P (µ) −−−−→ ∆ (µ) −−−−→ 0
T T T
fλ f fµ
    
    
y0 −−−−→ ∆Ty(λ) −−−−→ PTy(µ) −−−−→ ∆Ty(µ) −−−−→ y0
SinceendomorphismsofVermamodulescorrespondtoelementsofT,wegetamap
χ:End (P (µ)) −→ T ⊕T
OT T
f 7−→ (f ,f )
λ µ
For p=Rαˇ we define R :=R for the localization of R at the prime ideal p.
α p
Proposition 2.10 ([4], Corollary 3.5). Let Λ ∈ h∗/ ∼ . If Λ = {λ,µ} and
Rα
λ=s ·µ>µ, the map χ from above induces an isomorphism of R -modules
α α
End (P (µ))∼={(t ,t )∈R ⊕R |t ≡t mod αˇ}
ORα Rα λ µ α α λ µ
3. Tilting modules and tilting equivalence
In this chapter, T will be a localisation of R =S at a prime ideal p⊂R and
(0)
let K be its residue field. Let λ∈h∗ be such that ∆K(λ) is a simple object in OK.
Thus, we have ∆K(λ)∼=∇K(λ) and the canonical inclusion Can:∆T(λ)֒→∇T(λ)
becomes an isomorphism after applying ·⊗ K. So by Nakayama’s lemma, we
T
conclude that Can was bijective already.
3.1. Deformed tilting modules.
Definition 3.1. By K we denote the full subcategory of O which
T T
(1) includes the self-dual deformed Verma modules
(2) is stable under tensoring with finite dimensional g-modules
(3) is stable under forming direct sums and summands.
For T = S/Sh = C the category K is just the usual subcategory of tilting
T
modulesofthe categoryO overg. Ingeneral,KK is the categoryoftilting modules
of category O over the Lie algebra g⊗K whose weights live in the affine complex
subspaceτ+h∗ ⊂HomK(h⊗K,K),where τ :h⊗K→K comesfromthe mapthat
makes K into an S-algebra.
Proposition 3.2. The base change ·⊗ K gives a bijection
T
isomorphism classes isomorphism classes
←→
(cid:26) of KT (cid:27) (cid:26) of KK (cid:27)
Proof. For K,H ∈ K with K ⊗ K ∼= H ⊗ K we conclude K ∼= H from
T T T
Nakayama’s lemma applied to the weight spaces, since the weight spaces of tilt-
ing modules are finitely generated and free over T. This shows injectivity.
For surjectivity we only have to show that every indecomposable tilting module
in OK has an indecomposable preimage in KT. Since we are not working over a
complete local ring we cannot apply the idempotent lifting lemma as in the proof
of Proposition 3.4. in [10]. Rather, the proof works very similar to the proof of
Theorem 6 in [7].
Let K ∈ OK be an indecomposable tilting module. For the sake of simplicity, we
will assume the highest weight of K to be regular. The singular case is treated
FROM JANTZEN TO ANDERSEN FILTRATION VIA TILTING EQUIVALENCE 7
analogously.
If the highest weight λ of K is minimal in its equivalence class under ∼ then,
T
K ∼=∆K(λ)∼=∇K(λ) and we can take ∆T(λ) as a preimage of K in KT.
Now denote by λ the equivalence class of λ in h∗/ ∼ and let λ = w · µ with
T
w ∈W (λ) and µ minimal in W (λ)·λ. In addition, let w =s s ...s be a mini-
T T 1 2 n
mal expression of w with simple reflections s ∈W (λ¯). We denote by θ =θK the
i T i i
translationfunctor of OK through the si-wall. Then K is a direct summand of the
tilting module
M =θ1...θn∆K(µ)
and we get a decomposition M ∼= K ⊕K′. K′ decomposes into indecomposable
tilting modules with highest weights of the form w′ ·µ and l(w) > l(w′) where l
denotes the length of a Weyl group element. By using induction on the length of
w, we get a preimage K˜ ∈K of K′ together with a splitting inclusion
T
K˜ ⊗ K֒→M
T
Using Nakayama’s lemma, this induces a splitting lift
K˜ ֒→θT...θT∆ (µ)
1 n T
whereθT denotes the T-deformedwall-crossingfunctorofO correspondingto θK.
i T i
Finally, the cokernel of this inclusion is the indecomposable tilting module in K
T
we were looking for. (cid:3)
3.2. Tilting functor. Let M denote the subcategory of all modules in O ad-
T T
mittingaVermaflag. Werecallthemapτ :S →T whichmakesT intoaS-algebra.
We get a new S-algebra structure on T via τ ◦γ : S →T, where γ : S → S is the
isomorphism given by γ(h) = −h for all h ∈ h. We denote this new S-algebra by
T. Let S be the semiregular U(g)-bimodule of [9]. If N is a g⊗T-module which
2ρ
decomposes into weight spaces we get a T-module
N⋆ = Hom (N ,T)
T λ
λM∈h∗
Thenwe geta g⊗T-module via the g-action(Xf)(v)=−f(Xv)forall X ∈g,f ∈
N⋆ and v ∈N.
Now we get a functor
t′ :M −→Mopp
T T T
by setting t′ (M)=(S ⊗ M)⋆.
T 2ρ U(g)
For T a localisation of S at a prime ideal p which is stable under γ, for a residue
fieldofthis or forthe ringCJvK offormalpowerseriescomingfromaline Cλ⊂h∗,
∼
γ induces an isomorphism of S-algebras γ :T −→T which induces an equivalence
of categories
γ :M −→M
T T
Theorem 3.3 ([5], Section 2.6). The functor t = (S ⊗ ·)⋆ ◦γ induces an
T 2ρ U(g)
equivalence of categories
t :M −→Mopp
T T T
which respects block decomposition, makes short exact sequences to short exact se-
quences and sends a Verma module ∆ (λ) to the Verma module ∆(−2ρ−λ) for
T
any weight λ∈h∗.
8 JOHANNESKU¨BEL
Proposition 3.4. Let λ∈h∗. Then
t (P (λ))∼=K (−2ρ−λ)
T T T
whereP (λ)denotestheindecomposableprojectivecoverofL (λ)andK (µ)theup
T T T
to isomorphism unique indecomposable deformed tilting module with highest weight
µ∈h∗.
Proof. We set µ =−2ρ−λ. This proof is very similar to the proof of Proposition
3.1 of [8]. As we have already seen, the tilting module K (µ) can be described as
T
the up to isomorphism unique indecomposable module in O with the properties:
T
(1) K (µ) admits a Verma flag
T
(2) K (µ) has a ∇ -flag
T T
(3) K (µ) is free of rank one over T
T µ
(4) If γ is a weight of K (µ), we have γ ≤µ.
T
Since t is fully faithful, we already conclude the indecomposability of t (P (λ)).
T T T
Theorem3.3alsotells usthatt (P (λ))hasaVermaflag. ByBGG-reciprocitywe
T T
get a short exact sequence
N ֒→P (λ)։∆ (λ)
T T
where N has a Verma flag in which the occurring weights are strictly larger than
λ. By applying t we get a new short exact sequence
T
t (N)ևt (P (λ))←֓t (∆ (λ))
T T T T T
By induction onthe length of the Verma flag of N we conclude that the weights of
t (N) are strictly smaller than µ. So the weightspace (t (P (λ))) is free of rank
T T T µ
one, since t (∆ (λ)) ∼= ∆ (µ). Now, P (λ) being projective and t being fully
T T T T T
faithful, we get
Ext1 (∆ (δ),t (P (λ)))=0 ∀δ ∈h∗
OT T T T
Now we set D =t (P (λ)) and consider the diagramm
T T T
∆ (µ) ֒→ K (µ) ։ coker
T T
k
∆ (µ) ֒→ D ։ coker′
T T
Since cokerhasaVermaflagweconcludeExt1 (coker,D )=0usinginduction
OT T
on the length of a Verma flag of coker. So the restriction induces a surjection
Hom (K (µ),D ) ։ Hom (∆ (µ),D ) and we get a map α : K (µ) → D
OT T T OT T T T T
which induces the identity on ∆ (µ). For the same reason we also get a map
T
β : D → K (µ) with the same property, since coker′ has a Verma flag while
T T
K (µ) admits a nabla flag which implies Ext1 (coker′,K (µ))=0 by Propostion
2.1T. Applying the base change functor ·⊗ KO,Twe get twoTmaps
T
ϕ:=(β◦α)⊗idK :KK(µ)→KK(µ) andψ :=(α◦β)⊗idK :DT ⊗T K→DT ⊗T K
which induce the identity on ∆K(µ). We conclude that ϕ and ψ are not nilpotent
andsinceDT⊗TKandKK(µ)havefinitelengthandareindecomposable,itfollows
fromtheFittinglemmathatϕandψareisomorphisms. NowbyNakayama’slemma
applied to all weight spaces, we conclude K (µ)∼=D . (cid:3)
T T
Lemma 3.5. Let T → T′ be a homomorphism of S-algebras where also T′ is a
localisation of S at a prime ideal which is stable under γ, its residue field or the
FROM JANTZEN TO ANDERSEN FILTRATION VIA TILTING EQUIVALENCE 9
ring of formal power series CJvK. Let M,N ∈M and let M be projective in O .
T T
Then the diagram
Hom (M,N)⊗ T′ −−−−→ Hom (M ⊗ T′,N ⊗ T′)
OT T OT′ T T
tT⊗idT′ tT′
 
 
HomOT(tT(N)y,tT(M))⊗T T′ −−−−→ HomOT′(tT′(N ⊗TyT′),tT′(M ⊗T T′))
commutes,wherethehorizontalsarethebasechangeisomorphisms andtheverticals
are induced by the tilting functors tT resp. tT′.
Proof. All composition factors of the tilting functor commute with base change in
the sense of the lemma. (cid:3)
4. The Jantzen and Andersen filtrations
We fix a deformed tilting module K ∈ K and let λ ∈ h∗. The composition of
T
homomorphisms induces a T-bilinear pairing
Hom (∆ (λ),K)×Hom (K,∇ (λ)) −→ Hom (∆ (λ),∇ (λ))∼=T
OT T OT T OT T T
(ϕ,ψ) 7−→ ψ◦ϕ
For any T-module H we denote by H∗ the T-module Hom (H,T). As in [10]
T
Section4oneshowsthatforT alocalizationofS ataprimeidealporforT =CJvK
our pairing is nondegenerate and induces an injective map
E =Eλ(K):Hom (∆ (λ),K)−→(Hom (K,∇ (λ)))∗
T OT T OT T
of finitely generated free T-modules.
If we take T = CJvK the ring of formal power series around the origin on a line
Cδ ⊂ h∗ not contained in any hyperplane corresponding to a reflection of the
Weyl group, we get a filtration on Hom (∆ (λ),K) by taking the preimages of
OT T
(Hom (K,∇ (λ)))∗·vi for i=0,1,2,... under E.
OT T
Definition 4.1 ([10], Definition 4.2.). Given KC ∈ KC a tilting module of O and
K ∈KCJvK apreimageofKC underthefunctor·⊗CJvKC,which ispossiblebyPropo-
sition 3.2 with S → CJvK the restriction to a formal neighbourhood of the origin
in the line Cρ, then the image of the filtration defined above under specialization
·⊗CJvKC is called the Andersen filtration on Homg(∆(λ),KC).
The Jantzen filtration on a Verma module ∆(λ) induces a filtration on the vec-
tor space Hom (P,∆(λ)), where P is a projective object in O. Now consider
g
the embedding ∆CJvK(λ) ֒→ ∇CJvK(λ). Let PCJvK denote the up to isomorphism
unique projective object in OCJvK that maps to P under ·⊗CJvKC, which is possi-
ble by Theorem 2.3. Then we get the same filtration by taking the preimages of
HomOCJvK(PCJvK,∇CJvK(λ))·vi, i=0,1,2,..., under the induced inclusion
J =Jλ(P):Hom (P ,∆ (λ))−→Hom (P ,∇ (λ))
T OT T T OT T T
for T = CJvK and taking the images of these filtration layers under the map
Hom (P ,∆ (λ))։Hom (P,∆(λ)) induced by ·⊗ C.
OT T T g T
Forwhatfollows,wedefine µ′ =−2ρ−µandλ′ =−2ρ−λ. Toavoidambiguity,
we sometimes write (·)∗T, when we mean the T-dual of the T-module in brackets.
10 JOHANNESKU¨BEL
Theorem 4.2. Let λ,µ∈h∗. Denote by R=S the localization of S at 0. There
(0)
exists an isomorphism L=L (λ,µ) which makes the diagram
R
J
Hom (P (λ),∆ (µ)) −−−−→ Hom (P (λ),∇ (µ))
OR R R OR R R
(4.1)
t L
 
Hom (∆ y(µ′),K (λ′)) −−−E−→ (Hom (K (yλ′),∇ (µ′)))∗
OR R R OR R R
commutative. Here J = Jµ(P (λ)) and E = Eµ′(K (λ′)) denote the inclusions
R R R R
defined above and t=t denotes the isomorphism induced by the tilting functor.
R
Proof. If λ is not contained in the equivalence class of µ under ∼R=∼C all Hom-
spaces occurring in the diagram are 0 by block decomposition and the assertion of
the proposition is true.
So let us assume λ to be in the equivalence class of µ. It is easy to see that J
and E commute with base change and we have also verified this property for t in
Lemma 3.5already. Let p⊂R be a prime idealof height1. We abbreviateHom=
Hom ,P = P (λ),K = K (λ′),∆ = ∆ (µ),∇ = ∇ (µ),∆′ = ∆ (µ′) and
ORp R R Rp Rp Rp
∇′ =∇ (µ′). After applying·⊗ R to ourdiagramandthe basechangeisomor-
Rp R p
phisms of Theorem 2.4 and Corollary 2.2 we get a diagram of R -modules
p
J
Hom(P ⊗ R ,∆) −−−−→ Hom(P ⊗ R ,∇)
R p R p
(4.2)
t

Hom(∆′,yK⊗RRp) −−−E−→ (Hom(K⊗RRp,∇′))∗Rp
where we omit the index R for t,J and E.
p
We want to show that we get an isomorphism L of R -modules as the missing
Rp p
right vertical of the upper diagram to make it commutative for every prime ideal
p⊂R of height 1. By block decomposition we get
n
PR(λ)⊗RRp ∼= PRp(λi)
Mi=1
for certain indecomposable projective objects P (λ ) ∈ O and λ ∈ λ where
Rp i Rp i
λ denotes the equivalence class of λ under ∼R=∼C. Since t is fully faithful and
respects base change we also get a decomposition
n
KR(λ′)⊗RRp ∼=tRp(PR(λ)⊗RRp)∼= tRp(PRp(λi))
Mi=1
It is easy to see that J, E and t respect these decompositions. Hence, we get in
formulas
J(Hom (P (λ ),∆ (µ))) ⊂ Hom (P (λ ),∇ (µ))
ORp Rp i Rp ORp Rp i Rp
t(Hom (P (λ ),∆ (µ))) = Hom (t(∆ (µ)),t(P (λ )))
ORp Rp i Rp ORp Rp Rp i
E Hom (t(∆ (µ)),t(P (λ ))) ⊂ Hom (t(P (λ )),∇ (µ′)) ∗Rp
ORp Rp Rp i ORp Rp i Rp
(cid:16) (cid:17) (cid:16) (cid:17)