category o for quiver varietiesmathserver.neu.edu/~bwebster/auslander.pdf · category o the...

Category O for quiver varieties

Ben Webster

Northeastern University

April 28, 2012

Ben Webster (Northeastern) Category O for quiver varieties April 28, 2012 1 / 21

This talk is online at

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


Category O

Everyone’s favorite category of infinite dimensional representations iscategory O . This is a category of modules over a Lie 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. Trust me, it’sbetter this way.


Category O

Everyone’s favorite category of infinite dimensional representations iscategory O . This is a category of modules over a Lie algebra g which are“nice” with respect to a Borel b:

a module M lies in O ′ if b acts locally finitely and the center of U(g)acts semi-simply.

a module M lies in O ′ 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. Trust me, it’sbetter this way.


Category O

The principal block O ′0 has an exceptional list of properties:

O ′0 is highest weight (quasihereditary).

O ′0 has a graded lift which is Koszul.

The center of O ′0 and “pure” Hochschild cohomology are H∗(G/B;C).

The simples of O ′0 carry natural partitions into left, right, and 2-sidedcells.

There are two collections of derived auto-equivalences of O ′0 (shufflingand twisting). They define commuting actions of the Artin braid groupπ1(hreg/W) of g.O ′0 is Koszul self-dual. The corresponding derived auto-equivalence

exchanges (graded versions of) the shuffling and twisting functors.switch left cells to right cells and sends two-sided cells to two-sided cells.It is order-reversing for the highest weight ordering of the simple objects,and also order-reversing on the set of two-sided cells.


But why?

How are we supposed to explain these properties? Are they just a remarkablecoincidence? Or is there some general class of categories such that O ′0 is justthe first one we happened across?

For the last ≈ 5 years, I’ve been trying to show that there’s something behindDoor #2.

So, how might we generalize the construction of O ′0? Where did I insert aparameter I could vary? How about G/B?

Joseph, ’83 (paraphrased)“One particular protagonist has suggested that enveloping algebras shouldnow be relegated to a subdivision of the theory of rings of differentialoperators.”


But why?

But G/B is very wierd as a projective variety; it has many global differentialoperators, most varieties have very few. The most special thing is theexistence of the Springer resolution T∗G/B ∼= G×B b⊥ → g∗.

Conjecture

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

Lots of functions on T∗X means lots of differential operators on X. So, thatdoesn’t happen so often.

So, maybe the differential operators part isn’t so important. Maybe we shouldbe looking for something to replace the Springer resolution with.


Symplectic resolutions

DefinitionIf p : X → Y is a resolution of singularities (Y normal) and X carries analgebraic symplectic form ω, we say that p is a symplectic resolution ofsingularities.

We call such a resolution conical if X and Y carry a C∗ ∼= S-action withω having weight n > 0.

If X carries a closed algebraic 2-form which only degenerates on theexceptional locus Y is a symplectic variety.

So, the Springer resolution is a symplectic resolution of N , which hassymplectic singularities.

Conical symplectic singularities are very nice from the algebro-geometricperspective. They are generally very hard to find.


Symplectic resolutions

Examples:

Y ∼= C2/Γ has a unique symplectic resolution.

X = T∗G/P always resolves something affine, but when G 6= SLn, itmight not be a nilpotent orbit.

You can intersect T∗G/P with a Slodowy slice; we call theseS3-varieties.

Y ∼= C2n/(Sn o Γ) has a resolution by Hilbn(C̃2/Γ).

slices between Grλ and Grµ in the affine Grassmannian are symplectic,but don’t necessarily have a resolution.

hypertoric varieties; these “fix” the cotangent bundles of toric varieties.

Nakajima quiver varieties (this is the case I want to concentrate on).


Quantizations

So, how do we get an analogue of the universal enveloping algebra anddifferential operators? Magic (by which I mean, formal geometry).

Theorem (Bezrukavnikov-Kaledin, Braden-Proudfoot-W.)A conical sympletic resolution X carries a unique family of C∗-equivariantdeformation quantizations Qλ of the structure sheaf SX parameterized byλ ∈ H2(X;C).

If X = T∗G/P, then Qλ is the sheaf of twisted microlocal differentialoperators for the twist λ− 1/2 · c1(G/P).

If X is a hyperkähler quotient, then Qλ is a non-commutativeHamiltonian reduction (of sheaves).

We let Aλ = Γ(X;Qλ)C∗; this is a noncommutative filtered algebra with

associated graded C[Y]. It also only depends on Y .


Quantization

Examples:

If Y is a nilcone, then Aλ is a quotient of the associated universalenveloping algebra by a maximal ideal of the center.

If Y is a Slodowy slice, then Aλ is a central quotient of a W-algebra(Gan-Ginzburg).

If Y ∼= C2n/(Sn o Γ), then Aλ is the symplectic reflection algebra forSn o Γ (Etingof-Gan-Ginzburg-Oblomkov, Gordon).

If Y is a slices between Grν and Grµ in the affine Grassmannian, then Aλis a (probably) a quotient of a shifted Yangian (Kamnitzer-W.-Weekes-Yacobi).

If Y is a hypertoric variety, Aλ is what’s called the “hypertoricenveloping algebra” (Musson-v. d. Bergh, Bellamy-Kuwabara, BLPW).


Categories O

We have algebras now, but what are the right categories of representationsover them?

In order to define a category O , we have to make a choice, analogous to thechoice of Borel. We fix a Hamiltonian action of T ∼= C∗ on X commutingwith S.

Such an action always lifts to an inner action on the algebra Aλ for a gradingelement ξ ∈ Aλ.

DefinitionAlgebraic category O is the full subcategory of finitely generatedAλ-modules where ξ acts with finite dimensional generalizedeigenspaces which are bounded above.

Geometric category O is the full subcategory of S-equivariant regulargood Qλ[h−1/n]-modules where every point in the support has a limit asT× S 3 (t, t)→ 0.


Localization?

As in the Lie theoretic case, there are adjoint localization and sectionsfunctors.

Q[h−1/n] -modgdS Aλ -mod

Q[h−1/n]⊗Aλ −

Γ(X;−)S

Theorem (Braden-Proudfoot-W.)These functors are “usually” equivalences, in which case we say localizationholds at λ. In particular, localization holds when λ is “deep enough” in thenef cone of X.

Theorem (Braden-Proudfoot-Licata-W.)

These functors preserve category O’s. Thus, if localization holds Oaλ∼= Og

λ.


Hypertoric varieties

Let X be hypertoric, and let O be any generic block of category Og or Oa.

Theorem (Braden-Licata-Proudfoot-W.)O is highest weight (quasihereditary).

O has a graded lift which is Koszul.

The pure Hochschild cohomology is H∗(X;C); the center is H∗(X!;C).

The simples of O carry partitions into left, right, and 2-sided cells.

There are two collections of derived auto-equivalences of O (shufflingand twisting). They define commuting actions of π1(H2(X;C)reg/W) andπ1(H2(X!;C)reg/W !).

O is Koszul dual to O !, hypertoric category O for X!. The correspondingderived auto-equivalence

exchanges (graded versions of) the shuffling and twisting functors.switch left cells to right cells and sends two-sided cells to two-sided cells.It is order-reversing for the highest weight ordering of the simple objects,and also order-reversing on the set of two-sided cells.


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; this is the quotient of the set of points that satisfy thepreprojective condition and are stable: they have no submodule of the Cvi’skilled by the map to the Cwi’s.


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

Let G be the Kac-Moody algebra for Γ. Rouquier, Khovanov-Lauda, etc.have defined a notion of a “categorical G-action.”

In very crude terms, this consists of

a category with “weight decomposition” C ∼= ⊕µ∈H∗Cµ

functors Fi,Ei : Cµ → Cµ∓αi and

certain natural transformations between compositions of these functorswhich force the Fi,Ei to satisfy versions of the relations of G.

Let Og(v) be a block of the category O of Mwv for a fixed T-action (and a

central character which is “integral” in a sense I won’t explain).

Theorem (W.)

The sum of derived categories ⊕vD(Og(v)) carries a natural,geometrically-defined, categorical G-action.


Decategorification

LetM+ = {m ∈M| lim

t→0(t, t) · m exists.}

This is an isotropic subvariety, where the support of any object in Og lives.We have a natural characteristic cycle map

CC : K(Og)→ HBMtop (M+).

Theorem (Braden-Proudfoot-W.)This map intertwines the G-action induced by the categorical G-action withthe G action on HBM

top (M+) defined by Nakajima.

Thus, we have a categorification of Nakajima’s construction.


Tensor product actions

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

vHBM

top (M+v ) ∼= Vλ1 ⊗ · · · ⊗ Vλ` λj =

∑i∈Γ

(dimCwiτj

)ωi

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

TheoremThe sum of derived categories ⊕vD(Og(v)) 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


Analogies to category O

Theorem (W.)Og is standardly stratified (highest weight if λi’s are miniscule).

If all λ’s are miniscule, Og has a graded lift which is Koszul.

The “pure” Hochschild cohomology is (probably) H∗(M;C) ∼= Z(Tλ).

The simples of Og carries natural partitions into left, right, and 2-sidedcells (corresponding to isotypic decomposition of tensor product).

There are two collections of derived auto-equivalences of O ′0 (shufflingand twisting). They define commuting actions of the Artin braid groupπ1(Hreg/W) of G and the (usual type A) pure braid group. Thesecategorify the quantum Weyl group and R-matrix braiding actions.

What happens with Koszul duality is a tale of its own...should correspondto same story for affine Grassmannian.


The affine case

For the cyclic quiver, we have an extra degree of freedom; we can choose T sothat we have the highest weights λi with associated weights πi and weight kon one of the edges. As long as k > 0, you get a Grothendieck group biggerthan Vλ1 ⊗ · · · ⊗ Vλ` .

Proposition

⊕vK(Og(v)) is a higher level Fock space. Accounting for graded lift, it’s(probably) Uglov’s q-deformed higher Fock space.

If |k| � |πi| for all i, then it’s a tensor product of level 1 q-Fock spaces.

If |k| � |πi|, it’s JMMO q-Fock space.

Understanding these categories well should lead to interesting connectionswith Cherednik algebras, level-rank duality, etc.

All the analogies to category O that hold for miniscule tensor product cases(should) also hold here.


To do list

There are still quite a few issues to be worked out about the basic structures ofthese categories.

number of simples, dependence on parameters, character formulas, etc.

better understanding of particular examples, especially in the case withno resolutions (affine Grassmannian slices, finite group quotients)

Koszulity conjecture: necessitates formalizing graded lifts/Hodgestructures

duality conjecture: each category O for one symplectic singularityshould be Koszul duality to that for another singularity. But at themoment, we don’t even have a good description of the other singularity.


category o for quiver varietiesmathserver.neu.edu/~bwebster/auslander.pdf · category o the...

Documents