On Andersen and Jantzen ltrations · 2012. 3. 26. · Jantzen ltration On Andersen and Jantzen...

The category OASheaves on moment graphs

Andersen filtrationJantzen filtration

On Andersen and Jantzen filtrations

Johannes Kubel

University of Erlangen-Nurnberg

DFG Schwerpunkttagung, Schloss Thurnau

March 20, 2012

1 The category OA

2 Sheaves on moment graphs

3 Andersen filtration

4 Jantzen filtration

Basic setting

g ⊃ b ⊃ h semisimple complex Lie algebra with Borel andCartan subalgebra

S = S(h) = U(h)

S(0) the localization of S at the maximal ideal generated byh ⊂ S .

R+ ⊂ R ⊂ h∗ the root system of g with positive roots corresp.to b and ρ the half sum of positive roots

For α ∈ R+ we denote its coroot by α∨ ∈ h and the corresp.reflection by sα

W Weyl group with ρ-shifted dot-action on h∗

A a local, commutative S-algebra with structure mapτ : S → A(in this talk A = S(0) or A = S(0)/S(0)h = C)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Basic setting

S = S(h) = U(h)

Definition

The deformed category OA is the full subcategory ofU(g)-A-bimodules whose objects M satisfy

M is finitely generated over g⊗C A

M ∼=⊕

λ∈h∗ Mλ whereMλ = {m ∈ M | hm = (λ+ τ)(h)m ∀h ∈ h}M is locally b-A-finite

OA is abelian

simple modules: highest weight modules LA(λ) (λ ∈ h∗)

projective covers PA(λ) of LA(λ) exist in OA

OA∼=

⊕λ OA,λ where λ runs over anti-dominant weights

(block decomposition).

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

λ∈h∗ Mλ whereMλ = {m ∈ M | hm = (λ+ τ)(h)m ∀h ∈ h}

M is locally b-A-finite

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

Definition

M ∼=⊕

OA is abelian

OA∼=

d : OA −→ OopA duality on modules which are free over A

OA contains deformed Verma modules:∆A(λ) := U(g)⊗U(b) Aλ where Aλ is the b-A-bimodule A onwhich h acts by λ+ τ and [b, b] by 0.

∇A(λ) := d(∆A(λ))

Definition

A deformed tilting module is an object K ∈ OA which has a ∆A-and a ∇A-flag, i.e. a finite filtration with subquotients isomorphicto Verma (resp. Nabla) modules.

indecomposable deformed tilting modules are parametrized bytheir highest weight

KA(λ) := indecomposable deformed tilting module withhighest weight λ ∈ h∗

∇A(λ) := d(∆A(λ))

Definition

∇A(λ) := d(∆A(λ))

Definition

∇A(λ) := d(∆A(λ))

Definition

∇A(λ) := d(∆A(λ))

Definition

∇A(λ) := d(∆A(λ))

Definition

∇A(λ) := d(∆A(λ))

Definition

For A = S(0) and δ : A→ A/Ah ∼= C we get a base change functor· ⊗A C : OA → OC and the category OC coincides with the usualBGG-category O of g.

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

HomOA(P,∆A(λ))⊗A C ∼→ HomO(P ⊗A C,∆(λ))

HomOA(∆A(λ),K )⊗A C ∼→ HomO(∆(λ),K ⊗A C)

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

∆A(λ)⊗A C ∼= ∆C(λ) = ∆(λ),

PA(λ)⊗A C ∼= PC(λ) = P(λ),

KA(λ)⊗A C ∼= KC(λ) = K (λ),

The associated moment graph

For simplicity: λ ∈ h∗ integral, regular and anti-dominant.⇒ categorical structure of OA,λ governed by W.Associate to OA,λ an ordered, labeled graph G = (V,E, α,≤)

V := W vertices

E := {{x , y} ∈ P(V) | ∃β ∈ R+ : x = sβy } edges

α({x , sβx}) = β∨ labeling

Define the order on V by

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

for all w ,w ′ ∈W.

For simplicity: λ ∈ h∗ integral, regular and anti-dominant.

⇒ categorical structure of OA,λ governed by W.Associate to OA,λ an ordered, labeled graph G = (V,E, α,≤)

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

For simplicity: λ ∈ h∗ integral, regular and anti-dominant.⇒ categorical structure of OA,λ governed by W.

Associate to OA,λ an ordered, labeled graph G = (V,E, α,≤)

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

V := W vertices

w ≤ w ′ ⇔ w · λ ≤ w ′ · λ

Example:

For g = sl3(C) and λ = −2ρ

sα sβ

sαsβ sβsα

sαsβsα = sβsαsβ

HHHHHY

��*

��

��*

��

��*

α∨β∨

α∨(α + β)∨

β∨(α + β)∨

α∨ β∨(α + β)∨

Example:

sα sβ

sαsβ sβsα

HHHHHY

��*

��

��*

��

��*

α∨β∨

α∨(α + β)∨

β∨(α + β)∨

α∨ β∨(α + β)∨

Example:

sα sβ

sαsβ sβsα

HHHHHY

��*

��

��*

��

��*

α∨β∨

α∨(α + β)∨

β∨(α + β)∨

α∨ β∨(α + β)∨

Sheaves on G

Definition

An A-sheaf on the moment graph G is a tupelM := ({Mx}, {ME}, {ρx ,E}) with the properties

Mx is an A-module for any x ∈ V

ME is an A-module for all E ∈ E with α(E )ME = 0

ρx ,E : Mx →ME is a homomorphism of A-modules for x ∈ V,E ∈ E with x ∈ E .

For A = S consider all S-modules and S-linear maps as gradedmodules and graded morphisms in degree zero.

Sheaves on G

Definition

Sheaves on G

Definition

Sheaves on G

Definition

Sheaves on G

Definition

Sheaves on G

Definition

CA(G) := category of A-sheaves on G that admit a ”Verma flag”.There is an exact structure on CA(G) depending on the order on G.⇒ Notion of projective objects makes sense. Different order leadsto different projectives.

Theorem ((Braden-MacPherson), (Fiebig))

For every w ∈ V there exists an up to isomorphism unique gradedS-sheaf B↑(w) ∈ CS(G) with the properties

B↑(w)w ∼= S and B↑(w)x graded free with B↑(w)x = 0 unlessw ≤ x

B↑(w) is indecomposable and projective in CS(G)

Definition

Denote by B↓(w) the Braden-MacPherson sheaf on the samemoment graph G with reversed order.

CA(G) := category of A-sheaves on G that admit a ”Verma flag”.

There is an exact structure on CA(G) depending on the order on G.⇒ Notion of projective objects makes sense. Different order leadsto different projectives.

Definition

CA(G) := category of A-sheaves on G that admit a ”Verma flag”.There is an exact structure on CA(G) depending on the order on G.

⇒ Notion of projective objects makes sense. Different order leadsto different projectives.

Definition

An equivalence of categories

OVFA,λ ⊂ OA,λ full subcategory of modules with a deformed Verma

flag, A = S(0).

Theorem (Fiebig-localisation)

There is an equivalence of exact categories

V : OVFA,λ → CA(G)

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

V(∆A(x · λ)) = VA(x) (sky scraper sheaf)

flag, A = S(0).

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

flag, A = S(0).

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

flag, A = S(0).

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

flag, A = S(0).

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

flag, A = S(0).

V(PA(x · λ)) = B↑(x)⊗S A =: B↑A(x)

V(KA(x · λ)) = B↓(x)⊗S A =: B↓A(x)

Let w◦ ∈W be the longest element.

Proposition ((Fiebig), (Lanini))

There is an equivalence of categories

F : CS(G)→ CopS (G)

F (B↑(x)) = B↓(w◦x)

F (VS(x)) = VS(w◦x)

Lift F via Fiebig-localisation to an equivalence

T : OVFA,λ → (OVF

A,λ)op

with T (PA(µ)) = KA(w◦ · µ) and T (∆A(µ)) = ∆A(w◦ · µ)(µ ∈W ·λ)

F (B↑(x)) = B↓(w◦x)

A,λ)op

F (B↑(x)) = B↓(w◦x)

A,λ)op

F (B↑(x)) = B↓(w◦x)

A,λ)op

F (B↑(x)) = B↓(w◦x)

A,λ)op

F (B↑(x)) = B↓(w◦x)

A,λ)op

F (B↑(x)) = B↓(w◦x)

A,λ)op

Andersen filtration

A = C[[t]] with structure morphism τ : S = C[h∗]→ C[[t]]induced by Cρ ⊂ h∗.K ∈ OA a tilting object.Composition gives a non-degenerate bilinear form of freeA-modules:

HomOA(∆A(λ),K )× HomOA

(K ,∇A(λ))→

→ HomOA(∆A(λ),∇A(λ)) ∼= A

Andersen filtration

A = C[[t]] with structure morphism τ : S = C[h∗]→ C[[t]]induced by Cρ ⊂ h∗.

K ∈ OA a tilting object.Composition gives a non-degenerate bilinear form of freeA-modules:

(K ,∇A(λ))→

Andersen filtration

A = C[[t]] with structure morphism τ : S = C[h∗]→ C[[t]]induced by Cρ ⊂ h∗.K ∈ OA a tilting object.

Composition gives a non-degenerate bilinear form of freeA-modules:

(K ,∇A(λ))→

Andersen filtration

A = C[[t]] with structure morphism τ : S = C[h∗]→ C[[t]]induced by Cρ ⊂ h∗.K ∈ OA a tilting object.Composition gives a non-degenerate bilinear form of freeA-modules:

(K ,∇A(λ))→

Define filtration on HomOA(∆A(λ),K ) by

F iA(K , λ) := {ϕ ∈ Hom(∆A(λ),K ) |ψ◦ϕ ∈ t iA ∀ψ ∈ Hom(K ,∇A(λ))}

Definition

The image F i (K ⊗A C, λ) of the filtration F iA(K , λ) under the

surjection HomOA(∆A(λ),K ) � HomO(∆(λ),K ⊗A C) is called

Andersen filtration on HomO(∆(λ),K ⊗A C)

Definition

Jantzen filtration

Theorem (Jantzen)

Let µ ∈ h∗. Then ∆(µ) has a filtration:

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

∆(µ)1 is the maximal submodule of ∆(µ)

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Let P ∈ O be projective. Then the Jantzen filtration on ∆(µ)induces a filtration on Homg(P,∆(µ)).

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Jantzen filtration

Theorem (Jantzen)

∆(µ) = ∆(µ)0 ⊃ ∆(µ)1 ⊃ ∆(µ)2 ⊃ ... ⊃ 0

with the properties

Sum formula:∑i>0

ch∆(µ)i =∑

α∈R+,sα·µ<µ

ch∆(sα · µ)

Example

g = sl3(C), λ ∈ h∗ regular, integral and anti-dominant. sα, sβ ∈W

simple reflections, w = sαsβ.Q: Simple subquotients of ∆(w · λ) and its multiplicities?

L(λ) and L(w · λ) occur exactly once.

sum formula:∑i>0

ch∆(w · λ)i = ch∆(sα · λ) + ch∆(sβ · λ)

= chL(sα · λ) + chL(sβ · λ) + 2chL(λ)

⇒ ∆(w · λ)2 = L(λ)⇒ ∆(w · λ) has composition factors L(w · λ), L(sβ · λ), L(sα · λ)and L(λ) each occuring with multiplicity one.

Example

simple reflections, w = sαsβ.

Q: Simple subquotients of ∆(w · λ) and its multiplicities?

sum formula:∑i>0

Example

sum formula:∑i>0

Example

sum formula:∑i>0

Example

sum formula:∑i>0

Example

sum formula:∑i>0

⇒ ∆(w · λ)2 = L(λ)

⇒ ∆(w · λ) has composition factors L(w · λ), L(sβ · λ), L(sα · λ)and L(λ) each occuring with multiplicity one.

Example

sum formula:∑i>0

A,λ)op induces an isomorphism:

T : HomOA(PA(x ·λ),∆A(y ·λ))

∼→ HomOA(∆A(w◦y ·λ),KA(w◦x ·λ))

Theorem

The isomorphism ϕ := T ⊗A idC we get after base change:

ϕ : Homg(P(x · λ),∆(y · λ))∼→ Homg(∆(w◦y · λ),K (w◦x · λ))

identifies the Jantzen filtration on Homg(P(x · λ),∆(y · λ)) withthe Andersen filtration on Homg(∆(w◦y · λ),K (w◦x · λ)).

Theorem

References

A. Beilinson and J. Bernstein, A proof of Jantzen conjectures,Adv. Soviet Math., 1993.

T. Braden and R. MacPherson, From moment graphs tointersection cohomology, Math. Ann., 2001

P. Fiebig, Sheaves on moment graphs and a localization ofVerma flags, Adv. Math., 2008.

J.C. Jantzen, Moduln mit einem hochsten Gewicht, LectureNotes in Mathematics, Springer, 1979.

J. Kubel, From Jantzen to Andersen filtration via tiltingequivalence, preprint 2010, to appear in Math. Scand.

J. Kubel, Tilting modules in category O and sheaves onmoment graphs, preprint 2012

W. Soergel, Andersen Filtration and hard Lefschetz, Geom.Funct. Anal., 2008

On Andersen and Jantzen ltrations · 2012. 3. 26. · Jantzen ltration On Andersen and Jantzen...

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