Category O for quiver varieties

Category O for quiver varieties

Ben Webster

Northeastern University

July 10, 2012

Ben Webster (Northeastern) Category O for quiver varieties July 10, 2012 1 / 28

This talk is online at

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


Where we start...

Tom’s talk set up some machinery.

One starts with a conic symplectic singularity M0, and

one pulls out an algebra Aλ and

a sheaf of algebras Dλ on M, any resolution of M0.

When you add in a Hamiltonian C∗-action, you get a special category ofmodules over either the sheaf or the algebra, which we call a “category O .”

But why should you care?


Duality

We stumbled across these ideas from a slightly funny direction.

Conjecture

There is a notion of “S-dual pairs” (M,M!) of symplectic singularities suchthat:

the quantization parameters of M are isomorphic to the space ofHamiltonian C∗-actions commuting with S on M!.

integral blocks of category O for M are Koszul dual to those for M!

(with parameters matching as above).

the twisting functors for M (defined by switching parameters) areintertwined with shuffling functors M! (defined by switching C∗-actions).

For now, we can take this as a definition of a S-dual pair.


Examples of duality

So here’s the list of symplectic cones thus far that we believe we have foundthe dual to:

hypertoric variety: MA X Gale dual: MA∨

nilcone: NgX Langlands dual: NLg

symmetric power: Symn(C2) X symmetric power: Symn(C2)

GI-instantons on C̃2/ΓJ? GJ-instantons on C̃2/ΓI

ΓI GIooMcKay//

quiver variety: Qλµ

? affine Grass. slice: Wλµ

Simplest interesting example: T∗Pn−1 ⇔ C̃2/Zn or, in terms of cones,M rk 1

n×n ⇔ C2/Zn.


Examples of duality






ΓI GIooMcKay//




n×n ⇔ C2/Zn.


Examples of duality






ΓI GIooMcKay//




n×n ⇔ C2/Zn.


How do we get evidence?

We haven’t get even begun to seriously dream about proving such aconjecture. We would be happy to just to check that it holds in many specialcases.

So how does one even collect evidence for such a conjecture? The sad fact isthat one has to get to know examples really well.


What would you like to know?

Of course, there’s some kind of deep philosophical question about what“knowing” a category means. Do we just want to look for interestingcategories that are equivalent to it (or maybe derived equivalent?).

Category O isn’t just a category; it’s a natural module over a monoidalcategory λHCa

λ of Harish-Chandra bimodules—the bimodules over Aλwhose associated graded (for some good filtration) has left and right actionsthat coincide.

There’s also a geometric counterpart of this category λHCgλ; all sheaves in

this category are supported on Z = M×M0 M.

If you believe the duality conjecture, there are actually two actions; one byHC bimodules and one by the HC bimodules on the Koszul dual category.

ConjectureThese actions commute.


What would you like to know?

Of course, there’s some kind of deep philosophical question about what“knowing” a category means. Do we just want to look for interestingcategories that are equivalent to it (or maybe derived equivalent?).

Category O isn’t just a category; it’s a natural module over a monoidalcategory λHCa

λ of Harish-Chandra bimodules—the bimodules over Aλwhose associated graded (for some good filtration) has left and right actionsthat coincide.

There’s also a geometric counterpart of this category λHCgλ; all sheaves in

this category are supported on Z = M×M0 M.

If you believe the duality conjecture, there are actually two actions; one byHC bimodules and one by the HC bimodules on the Koszul dual category.

ConjectureThese actions commute.


Twisting and shuffling

Inside Harish-Chandra bimodules, there are special bimodules, given bysections of quantizations of line bundles νTν′ for ν − ν ′ ∈ H2(M;Z). We alsohave isomorphisms Aν ∼= Aw(ν) for the action of a finite group W on H2(M).

Theorem (Braden-Proudfoot-W.)

Assume M0 has resolutions whose ample cones cover H2(M;R). The

functors νTν′L⊗− induce an equivalence between D(Aν -mod) and

D(Aν′ -mod) if ν, ν ′ avoid some small bad region.

Compositions of these functors with W-isomorphisms are called twistingfunctors and induce an action of π1(H2(M;C)reg/W).

There are some bad hyperplanes in H2(M;R); outside some region aroundthe hyperplanes, all the categories (in the same coset) in a chamber are thesame, related by these functors. On the other hand, crossing a wall andcoming back gives an interesting automorphism.



Let G be a maximal reduction subgroup of Hamiltonian transformationscommuting with S (which is a finite dimensional Lie group), T a maximaltorus and W its Weyl group.

For a T-action on M, consider the inclusion ιT : D(Oa)→ D(A -mod) and itsleft (right) adjoint ι!T (ι∗T). The shuffling functors are the functors generatedby ι!T′ ◦ ιT for T,T′ ⊂ T , along with the action of pull-back by W bysymmetries of Aλ.

Conjecture

The shuffling functors define an action of π1(treg/W), the set of actions withgeneric fixed points. Koszul duality interchanges shuffling and twistingfunctors.

As with twisting functors, there is a hyperplane arrangement (where we haveextra fixed points), but now things flip on exactly on the hyperplanes.


Index theory

Kashiwara and Schapira have defined an “Euler class” (or perhaps“characteristic cycle” would be more evocative) associating to a holonomicDλ-module N a class in HBM

top (suppN ).

Proposition

Applied to category O , we get an isomorphism

which has a counterpart forHC bimodules, which may not be an isomorphism.

K(Ogλ) HBM

top (M+)

K(λHCgλ) HBM

top (Z)

∼

These maps intertwine the (derived) tensor action of categories with theconvolution action on Borel-Moore homology.


Index theory


top (suppN ).

Proposition

Applied to category O , we get an isomorphism which has a counterpart forHC bimodules, which may not be an isomorphism.

K(Ogλ) HBM

top (M+)

K(λHCgλ) HBM

top (Z)

∼



Index theory


top (suppN ).

Proposition

Applied to category O , we get an isomorphism which has a counterpart forHC bimodules, which may not be an isomorphism.

K(Ogλ) HBM

top (M+)

K(λHCgλ) HBM

top (Z)

∼



Cotangent bundles

If M = T∗G/P is a cotangent bundle, then we at least understand in principlehow these should go (in the geometric case).

A completely combinatorial description of the category was given byElias-Khovanov in type A and Elias-Williamson in arbitrary type,building on work of Soergel. It involves some slightly complicated butpretty pictures.

Better yet, this description is completely compatible with the HCbimodule action; in fact, you use the bimodules to build the objects in thecategory.

It’s been known for a while that the action of projective functors oncategory O categorifies the regular action of C[W] on itself, so this fitsinto a known circle of ideas.


Cotangent bundles

In all of these, one uses geometric techniques to understand what’s going on:

Ext groups of semi-simple sheaves pushed forward by proper maps canbe calculated using the Borel-Moore homology of the fiber product. (TheElias-Khovanov presentation is essentially applying this toBott-Samelson resolutions.)

Hom spaces of projectives can be calculated using vanishing cycles andmicrolocal geometry or using functors on the category of projectives.

the grading we would like to be Koszul arises from Hodge theory.

For Koszul duality, we can hope to calcuate these for two different spaces, andthen notice that they are the same.

Unfortunately, these techniques are in the long list of things we don’t knowhow to properly generalize to an arbitrary resolution.


Symplectic reduction

Philosophically, I like to think that conic symplectic resolutions are “secretcotangent bundles” and if you find some way of teasing out theircontangentness, you’ll be cooking with gas.

Luckily, most of the singularities we care about (with an important exception)are reductions of cotangent bundles.

Pick your favorite C-vector space V with a reductive algebraic group G actingon it. It’s quite interesting to consider the GIT quotient V//αG for a genericcharacter α : G→ C∗ (so V//αG is orbifold).

However, things involving cotangent bundles of these aren’t quite “right”(toric varieties aren’t D-affine). So, one can set out to fix the cotangent bundle.


Hyperkähler analogues

The cotangent bundle T∗V is holomorphic symplectic (actually hyperkähler),with the action of G being Hamiltonian with moment map

〈X, µ(m, ξ)〉 = 〈ξ,X · m〉

This quantizes the natural map X ∈ g 7→ XM ∈ Γ(V; TV).

Definition

The hyperkähler analogue of V//αG is the reduction M = µ−1(0)//αG. Thiscontains to cotangent bundle T∗(V//αG) as an open subset, but also has somepoints coming from cotangent vectors at unstable basepoints.

As long as µ is regular at α, this variety is

an orbifold (often smooth)

a orbifold resolution of the affine GIT quotient M0 = µ−1(0)//0G

has topology independent of αBen Webster (Northeastern) Category O for quiver varieties July 10, 2012 14 / 28

Hyperkähler analogues

The cotangent bundle T∗V is holomorphic symplectic (actually hyperkähler),with the action of G being Hamiltonian with moment map

〈X, µ(m, ξ)〉 = 〈ξ,X · m〉

This quantizes the natural map X ∈ g 7→ XM ∈ Γ(V; TV).

Definition

The hyperkähler analogue of V//αG is the reduction M = µ−1(0)//αG. Thiscontains to cotangent bundle T∗(V//αG) as an open subset, but also has somepoints coming from cotangent vectors at unstable basepoints.

As long as µ is regular at α, this variety is

an orbifold (often smooth)

a orbifold resolution of the affine GIT quotient M0 = µ−1(0)//0G

has topology independent of αBen Webster (Northeastern) Category O for quiver varieties July 10, 2012 14 / 28

Hamiltonian reduction

Choose λ ∈ H2(BG) = (g∗)G. Let K(λ) be its image under the Kirwan mapH2(BG)→ H∗(M).

TheoremLet ν = λ+ ρ. The algebra AK(ν) = Aν is a quantum hamiltonian reductionof the differential operators D(V)

Aλ = D(V)G/(∑

X∈gD(V) · (XM − λ(X))

)G

The sheaf DK(ν) is a sheafy version of this reduction.

In both the algebraic and geometric contexts, there is a reduction functor

reda : D(V) -mod→ Aν -mod redg : D(V) -mod→ Dν -mod

It’s a bit nicer to restrict these functors to strongly equivariant D-modules (i.e.where XM − λ(X) integrate to a G-action).


Hamiltonian reduction

Choose λ ∈ H2(BG) = (g∗)G. Let K(λ) be its image under the Kirwan mapH2(BG)→ H∗(M).

TheoremLet ν = λ+ ρ. The algebra AK(ν) = Aν is a quantum hamiltonian reductionof the differential operators D(V)

Aλ = D(V)G/(∑

X∈gD(V) · (XM − λ(X))

)G

The sheaf DK(ν) is a sheafy version of this reduction.

In both the algebraic and geometric contexts, there is a reduction functor

reda : D(V) -mod→ Aν -mod redg : D(V) -mod→ Dν -mod

It’s a bit nicer to restrict these functors to strongly equivariant D-modules (i.e.where XM − λ(X) integrate to a G-action).


Reduction functors

On the category of strongly equivariant modules

TheoremThese functors are exact, essentially surjective, and sends projectives toprojectives.

They possess left and right adjoints reda! , reda

∗ and redg! , redg

∗ which splitthe appropriate reduction functor.

Thus, Aν -mod or Dν -mod can be realized as a quotient category of stronglyequivariant D(V)-modules. So one can do calculations in the latter category,where one has the familiar geometric tools.

You might rightly protest that this doesn’t sound very helpful. How the helldo I identify the D-modules that correspond to reductions? Doesn’t sound soeasy.


Reduction functors

On the category of strongly equivariant modules

TheoremThese functors are exact, essentially surjective, and sends projectives toprojectives.

They possess left and right adjoints reda! , reda

∗ and redg! , redg

∗ which splitthe appropriate reduction functor.

Thus, Aν -mod or Dν -mod can be realized as a quotient category of stronglyequivariant D(V)-modules. So one can do calculations in the latter category,where one has the familiar geometric tools.

You might rightly protest that this doesn’t sound very helpful. How the helldo I identify the D-modules that correspond to reductions? Doesn’t sound soeasy.


Category pre-O

It’s better to think about whole categories together. Rather than trying to findD-modules that land on individual modules, look for a natural categorylanding on category O .

We’ll only be interested in C∗-actions that are induced by G-invariantC∗-actions on V; however, it’s important to remember that many of these caninduce the same action on M.

DefinitionCategory pre-O is the subcategory pOν of f.g. strongly (G, λ) equivariantD(V)-modules M such that

M is generated by [G,G]-invariant elements.

for any x ∈ MGss , the generalized ξ-eigenvalues on D(V)G · x arebounded above.


Category pre-O

TheoremThe reduction functors send pre-O to O and their adjoints send O to pre-O .In particular, Oa

ν or Ogν is a quotient of pOν by the subcategory killed by reda

or redg.

Assume you can find a projective generator P, and semi-simple generator S inpOν ; let H = Hom(P,P) and E = Ext•(S, S).

PropositionThere are idempotents e ∈ H and ε ∈ E, and projective/s.s. generatorsPred = reda(Pe) and Sred = reda(S) in Og

ν (and similarly for Oaν , with different

idempotents) such that

eHe = Hom(Pred,Pred) and E/EεE = Ext•(Sred, Sred).

In this formalism, we can also easily calculate twisting functors (and if youcan calculate E or H, you can probably calculate shuffling functors as well).


Category pre-O

TheoremThe reduction functors send pre-O to O and their adjoints send O to pre-O .In particular, Oa

ν or Ogν is a quotient of pOν by the subcategory killed by reda

or redg.

Assume you can find a projective generator P, and semi-simple generator S inpOν ; let H = Hom(P,P) and E = Ext•(S, S).

PropositionThere are idempotents e ∈ H and ε ∈ E, and projective/s.s. generatorsPred = reda(Pe) and Sred = reda(S) in Og

ν (and similarly for Oaν , with different

idempotents) such that

eHe = Hom(Pred,Pred) and E/EεE = Ext•(Sred, Sred).

In this formalism, we can also easily calculate twisting functors (and if youcan calculate E or H, you can probably calculate shuffling functors as well).


Hypertoric varieties

For G a torus (i.e. hypertoric varieties) and V = Cn, this results in aparticularly simple description. We only consider λ integral.

All modules in pOν are smooth along the coordinate stratification. Thesimples in this category are in bijection with sign sequences, andD-equivariant Ext’s are the cube algebra described by Tom (D=diagonals).

In order to pass to pOν for a subtorus G we must:impose linear relations on equivariant parameters for change D→ G.remove some simples with unbounded weights (E = aE′a for a anidempotent projecting to the allowed simples).

Reducing further to Oaν or Og

ν involves quotienting out by simples with trivialreduction (Ered = E/EεE).

The Koszul duality theorem for hypertoric varities is just noting that thereductions one does to a projective generator are Gale dual to those done here.


Hypertoric varieties

For G a torus (i.e. hypertoric varieties) and V = Cn, this results in aparticularly simple description. We only consider λ integral.

All modules in pOν are smooth along the coordinate stratification. Thesimples in this category are in bijection with sign sequences, andD-equivariant Ext’s are the cube algebra described by Tom (D=diagonals).

In order to pass to pOν for a subtorus G we must:impose linear relations on equivariant parameters for change D→ G.remove some simples with unbounded weights (E = aE′a for a anidempotent projecting to the allowed simples).

Reducing further to Oaν or Og

ν involves quotienting out by simples with trivialreduction (Ered = E/EεE).

The Koszul duality theorem for hypertoric varities is just noting that thereductions one does to a projective generator are Gale dual to those done here.


Quiver varieties

Nakajima quiver varieties are particularly interesting from this perspectivesince they are a geometric incarnation of the representation theory of Liealgebras.

Recall that Nakajima quiver varieties are “the moduli space of framedrepresentations of the preprojective algebra.”

More precisely, fix a quiver Γ, and dimension vectors v and w. Let

E =⊕i∈Γ

Hom(Cvi ,Cwi)⊕⊕i→j

Hom(Cvi ,Cvj).

DefinitionLet Mw

v = T∗E//detGv and Nwv = T∗E//0Gv; this is the quotient of the set of

points that satisfy the preprojective condition and are stable: they have nosubmodule of the Cvi’s killed by the map to the Cwi’s. We’ll be interested inMw = tvM

wv .


Quiver varieties

So, I’d like to understand the categories O for quiver varieties M. First, I needto know what my options for C∗ actions are; the obvious source is AutGv(E).

PropositionIf Γ has no bigons, then we have exact sequences

0→ Gw → AutGv(E)→ (C∗)E(Γ)

0→ Gw/C∗ → AutGv(E)/Z(Gv)→ H1(Γ;C∗)

Two cases are of particular interest:

if T ⊂ Gw, then this is called a tensor product action.

when Γ is a cycle on n ≥ 1 vertices.


The categorical G-action

The categories Dν(v) -mod for the different dimension vectors don’t just havea separate monoidal category on each. Rather, one can define a larger categoryνHCg

ν of Dν(v)−Dν(v′) HC bimodules using the “stable Steinberg” Z.

There is a universal affine variety Nw∞ such that Nw

v is a stratum there. We candefine

Z =⊔v,v′

Mwv ×Nw

∞ Mwv′ .

Proposition

The action of D(νHCgν) preserves D(Og

ν) = ⊕vD(Ogν(v)) for any C∗ action.

The variety Z is particularly interesting because Nakajima defined a naturalalgebra map U̇(g)→ HBM

top (Z).



The categories Dν(v) -mod for the different dimension vectors don’t just havea separate monoidal category on each. Rather, one can define a larger categoryνHCg

ν of Dν(v)−Dν(v′) HC bimodules using the “stable Steinberg” Z.

There is a universal affine variety Nw∞ such that Nw

v is a stratum there. We candefine

Z =⊔v,v′

Mwv ×Nw

∞ Mwv′ .

Proposition

The action of D(νHCgν) preserves D(Og

ν) = ⊕vD(Ogν(v)) for any C∗ action.

The variety Z is particularly interesting because Nakajima defined a naturalalgebra map U̇(g)→ HBM

top (Z).



Let G be the Kac-Moody algebra for Γ. Rouquier, Khovanov-Lauda, etc.have defined (several variants of) a 2-category U which is the “right”categorification of U̇(G) (I’m actually using version from Cautis-Lauda).

TheoremAssume λ is integral. There is a natural, geometrically-defined 2-functorψ : U → νHCg

ν , such that the diagram below commutes

U̇(g)

K(νHCgν) HBM

top (Z)

K(D(Ogν) HBM

top ((Mw)+)∼

ΨK(ψ)


Tensor product actions

If T is a tensor product action, then M+ is the tensor product varietydefined by Nakajima. Thus⊕

vHBM

top (M+v ) ∼= Vµ1 ⊗ · · · ⊗ Vµ` λj =

∑i∈Γ

(dimCwiτj

)ωi

where τ1, . . . , τ` are the distinct weights of T on ⊕iCwi .

TheoremAssume G is finite or affine. Then D(Og) is equivalent as a categoricalG-module to the dg-modules over the certain explicitly presented finitedimensional algebras Tλ (Koszul dual). We have an isomorphism

K(Og(v)) ∼= Vλ1 ⊗ · · · ⊗ Vλ` .

λ1

λ1

λ3

λ3

λ2

λ2



These category O’s carry actions by twisting and shuffling functors. Fromgeometry, we know that

shuffling functors should give an action of the braid group on `-strands,permuting the weights.

twisting functors should give an action of Artin braid group of G.

TheoremShuffling functors are the categorified R-matrix action defined algebraicallyon D(Tλ -mod). Twisting functors (combined with the Weyl groupisomorphisms between quiver varieties) are (probably) Chuang andRouquier’s categorification of the quantum Weyl group.


Weighted KLR algebras

The heart of the proofs of all of these is a construction of certain objects incategory O or HC bimodules as reductions of D-modules on E.

Key observation

If ϑ : T→ GL(E) is a lift of the T-action on the quotient, then there is anatural map

pϑ : Xϑ = (Eϑ × G)/Gϑ → E

where Eϑ is the negative weight spaces for ϑ, and Gϑ is the negative weightparabolic for ϑ.

Proposition

The pushforward (pϑ)∗OXϑis a semi-simple object in D(pO). For finite or

affine type quiver varieties or hypertoric varieties, all simples in pO aresummands.

Horizontal slices of pictures encode ϑ’s; crossings/dots are natural geometricmorphisms.


Weighted KLR algebras

For quivers, the Ext algebras of the sums of these pushforwards are calledweighted Khovanov-Lauda-Rouquier algebras. These include asexamples:

the usual KLR algebras, for w = 0, and T acting trivially on the quotient.

the tensor product algebras without the violating relation T̃λ, for a tensorproduct action. The quotient to Og

λ exactly imposes the violating relation.

the quiver Schur algebras (Stroppel-W.) for an affine type A quivervariety which are “close enough” to being tensor product. In this case,the image in Og

λ has Ext-algebra given by a cyclotomic q-Schur algebra.

These algebras have diagrammatic relations as in the original case; the bigdifference is that one must pay attention to the distance between strands, andthere can be “action at a distance” depending on the C∗-action.


The affine case

The only finite or affine case that has any other kind of C∗-action is affinetype A. In this case, we have the highest weights λi with associated weights πi

and weight k on cyclically oriented edges.

This is particularly interesting, because our category O is the usual categoryO for the spherical symplectic reflection algebra.

One can calculate pOλ for λ integral relatively easily here; the Ext-algebra ofthe simples is yet another weighted KLR algebra. It’s a bit hard to draw thediagrams, but the rules for understanding them are simple.

Unfortunately, the more interesting cases are when λ is not integral.Preliminary results suggest that pOν for λ not integral is actually equivalent topOν′ for a higher rank. Interestingly, a similar change occurs in the hypertoriccase for non-integral character.


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Category O for quiver varieties