The flag variety: Why is this cotangent bundle different ...

The flag variety:Why is this cotangent bundle different

from all other cotangent bundles?

Ben Webster

Northeastern University

May 7, 2012

Ben Webster (Northeastern) The flag variety May 7, 2012 1 / 19

This talk is online at

http://www.math.neu.edu/˜bwebster/TorontoI.pdf


Roadmap

For the next 3 days, I have a 3 step plan:

First, I will tell you my favorite story (the Flag Variety’s Tale). This ismostly “old” stuff, but of course, that doesn’t mean the news has gottenout.

Second, after telling you why the flag variety is so special, I want tellwhy it’s not as special as you might think. The representation theory ofLie algebras has a lot less to do with Lie algebras than you might think.Some surprisingly similar features show up in some very differentcontexts.

Thirdly, I to show you what motivated me to think about this stuff in thefirst place, which is a surprising (and not yet fully defined) duality ingeometry.


Step 0: notation

Let G be your favorite semi-simple Lie group over C. Most of the time,I’ll be imagining that G = SLn, so there’s no reason you shouldn’t.

Let B be a Borel. Of course, these aren’t unique, but they’re allconjugate, so you shouldn’t worry so much about choosing one. I mean,eventually you should, but not right now.

Let X = G/B. This is usually called the flag variety, since for G = SLn,this is the moduli space of flags

{0} ⊂ V1 ⊂ · · · ⊂ Vn = Cn.

It’s maybe more canonical to think about this as the variety of Borels;B′ ↔ XB′ is a bijection between the class of Borels and points of X.


Lie algebras and differential operators

Whenever G acts on a manifold Y , we have an “infinitesimal action,” a Liealgebra map

α : g→ X(Y)

by pushing forward right-invariant vector fields.

Taking “enveloping algebras,” we obtain a map of algebras

α : U(g)→ DY

from the universal enveloping algebra of g to differential operators on Y .

This map doesn’t have to be injective or surjective. But it could be....


Lie algebras and differential operators

When you have a map of U(g) to anything, the first thing you should askyourself is “what happens if I take associated graded (and then Spec)?”

Z(g) U(g) DY

h∗/W ∼= g∗/G g∗ T∗Y

α

µq

The map µ has a special name: “the moment map.” Moment maps aren’tunique, but in this case, there’s only one that intertwines the naturalC∗-actions on both sides (induced by the associated grading!).


Springer resolution

So, what does this map look like for X? Well, what is T∗X?

Tb′X = g/b′ ⇒ T∗b′X = (g/b′) = (b′)⊥ = [b′, b′] = n′

Said in a more universal way:

T∗X = G×B n = {(b′, 〈n,−〉) ∈ X × g∗|n ∈ b′, n nilpotent}

Since the action map α is the obvious map g→ Tb′X = g/b′, we have thatµ(x, ξ) = ξ.

Theorem (Springer)

The image of the map µ : T∗X → g∗ has image given by the nilconeN = q−1(0) and µ is a resolution of singularities of this locus.


Global differential operators

LetU(g)0 = U(g)⊗Z(g) C.

Since the center of differential operators is C, the map U(g)→ DX factorsthrough U(g)0.

The associated graded of the map U(g)0 → DX is the map C[N ]→ C[T∗X].Since the latter is an isomorphism, the former is as well.

Proposition

U(g)0 ∼= Γ(X; DX).

I should emphasize that this is very special for (partial) flag varieties:

Conjecture

If X is a projective variety, then there is a resolution of singularities T∗X → Ywith Y affine if and only if X = G/P for G a semi-simple complex Lie groupand P a parabolic.


Twisted differential operators

What about other maximal ideals of Z(g)?

The answer is twisted differential operators; if X is a variety withH2(X) = H1,1(X), then there is a “line bundle” with Chern class λ and thereis a sheaf of differential operators Dλ

X valued in this line bundle.

Generated by∇v,mf for v ∈ X(U), f ∈ OX(U) with relations

[∇v,mf ] = mLvf [mf ,mg] = 0 [∇v,∇w] = ∇[v,w] + mλ(v,w)

Let U(g)λ be U(g) modulo the annhilator of the Verma module Mλ (which isgenerated by a maximal ideal of the center).

Proposition

U(g)λ ∼= Γ(X; DλX).


Localization

Thus we have a functor

DλX -mod U(g)λ -mod

Γ(X;−) = HomDλX(Dλ

X,−)

Loc = DλX ⊗U(g) −

By general nonsense, we have a left adjoint.

Theorem (Beilinson-Bernstein)

The functor Loc is an equivalence if and only for every positive coroot α∨ wehave α∨(λ) /∈ Z<0. The derived functor LLoc is an equivalence if and onlyα∨(λ) 6= −1.

These conditions are easy to remember if you know Borel-Weil-Bott.

For λ integral, localization holds iff the corresponding line bundle hasnon-zero sections; derived localization holds iff it has non-zero cohomology.


Categories of representations

The motivation behind this work was to make it easier to understandcategories of representation. Given a representation M, you can consider thesheaf Loc M.

Of course, it would be interesting to look at, say the support of this on G/B,but that not’s the most interesting thing.

Any D-moduleM has a “good filtration” compatible with the degree filtrationon differential operators, whose associated graded is a coherent sheaf onT∗G/B.

DefinitionThe support of this associated graded (interpreted correctly to deal with thelack of uniqueness) is the microsupport ofM.


Microlocalization

You can actually “spread out” a D-module over the cotangent bundle.Consider the Rees algebra of differential operators

DλX (U) = C[h] · 〈h−1∇v,mf |v ∈ X(U), f ∈ O(U)〉 ⊂ C[h, h−1] · Dλ

X(U)

Adding the h−1 makes a big difference, since [DλX ,D

λX ] ⊂ hDλ

X . Thus,

DλX/hDλ

X∼= Sym•OX

(TX) (DλX )C

∗ ∼= DλX.This is local on X, but what about T∗X? We define a sheaf Qλ on T∗X byinverting sections invertible mod h; we need to complete in h to make sense.

Theorem

Qλ is a C∗-equivariant sheaf of flat C[[h]]-algebras with Qλ/hQλ.

Qλ[h−1] -modC∗

DλX -mod

f.g. C[[h]]-lattice good filtration


Microsupport

Suddenly, for every C∗-invariant subvariety C ⊂ T∗X, we can define asubcategory of (regular) modules CC with microsupport in C. Popularoptions:

C = X = µ−1(0): this corresponds to finite dimensional modules.

C = O := µ−1(b⊥): this is the union of the conormal bundles toSchubert cells. This corresponds to “category O .”

It’s also pretty interesting to think about bimodules with

C = Z := (µ× µ)−1(N∆) = T∗X ×N T∗X.

These are “Harish-Chandra bimodules.”

Taking support with multiplicities gives a map K(CC)→ HBMtop (C).


The convolution action

The derived category of bimodules D(CZ) are a monoidal category underderived tensor product

−L⊗− : D(CZ)× D(CZ)→ D(CZ).

If C = µ−1(µ(C)), then we get a module category

−L⊗− : D(CZ)× D(CC)→ D(CC).

TheoremThere are commuting maps of algebras and modules

K(CZ) HBMtop (Z) C[W]

K(CC) HBMtop (C)


K(CX) HBMtop (X) Csgn


K(CO) HBMtop (O) C[W]


Category O

The category CO is a well-known one. This is a category of modules over aLie algebra g which are “nice” with respect to a Borel b:

a module M lies in O if b acts locally finitely and the Cartan h actssemi-simply.a module M lies in O if it is the sections of a strongly B-equvariantuniversal D-module on G/B.

The principal block O ′0 is equivalent to a principal block O0.

The category O ′0 has a mixed lift O ′0.

That is, there is a positively graded algebra A such that

O ′0 is all A-representations

O ′0 is graded A-representations

Each simple graded A-representation occurs in a single degree, which we callits weight.


Category O

The category CO is a well-known one. This is a category of modules over aLie algebra g which are “nice” with respect to a Borel b:

a module M lies in O ′ = CO if b acts locally finitely and the center ofU(g) acts semi-simply.a module M lies in O ′ = CO if it is a sum of sections of regular twistedD-modules on G/B smooth along the Schubert cells.

The principal block O ′0 is equivalent to a principal block O0.

The category O ′0 has a mixed lift O ′0.

That is, there is a positively graded algebra A such that

O ′0 is all A-representations

O ′0 is graded A-representations

Each simple graded A-representation occurs in a single degree, which we callits weight.


Highest weight and Koszul

DefinitionA category C highest weight if it has an exceptional collection of standardobjects which generate C .

A mixed category C is Koszul if the two natural gradings on Ext∗C (L,L′) fortwo weight 0 simples L,L′ agree.

The (abelian!) category C ! of complexes of projectives where the ith grouphas head of weight i is called the Koszul dual of C .

There’s an equivalence of derived categories D(C ) ∼= D(C !).

TheoremThe category O ′ is highest weight.

The category O ′ is Koszul and self-Koszul dual.

The center of O ′ is H∗(X).


Twisting and shuffling

Remember, we have a monoidal action of D(CZ) (which is the homotopycategory of coinvariant Soergel bimodules) on D(O ′0); this commutes with thesame action twisted by Koszul duality.

This categorifies the left and right actions of C[W] on itself.

K(CZ) K(O ′0) K(O ′0) K(CZ)

C[W] C[W] C[W] C[W]

Koszul

w↔ w−1

We know that there are braid groups hiding in these categories. Many peoplehere know this because they’ve read papers on Soergel bimodules. This is thereverse of history; the actions were known in representation theory first.


Twisting and shuffling

DefinitionFor w ∈ W, let the shuffling functor for w be

Ψw = w∗ ◦ ι!wO ◦ ιO : DλX -mod→ Dλ

X -mod

where ιO : D(CO)→ D(DλX -mod) is the usual inclusion and ι!O its left adjoint.

For w ∈ W, let twisting functor for w be

Φw = Lλ−wλ ⊗OX LLocwλ(Γλ(X;−)) : DλX -mod→ Dλ

X -mod

where LLocwλ is localization for wλ-twisted D-modules.

TheoremThe shuffling and twisting functors give commuting actions of the Artin braidgroup π1(hreg/W).


The “point”

So, you can safely forget most of what I’ve said, but I want you to come awaywith a few essential points.

Category O is a category with three phases:

Algebraic: as modules over U(g).

Geometric: as sheaves of modules over (microlocal) differentialoperators.

Combinatorial: Soergel (bi)modules, the Elias-Khovanov realization.

It has a number of surprising and unusual features:

the highest weight structure/Koszulity

the categorification of convolution actions

a pair of commuting braid group symmetries

Tomorrow, we’ll discuss how surprising these really are.


The flag variety: Why is this cotangent bundle different ...

Documents

Transcript of The flag variety: Why is this cotangent bundle different ...