symplectic duality and klr algebras...the higgs presentation theorem (w.) 1 when the grading which...

Physics Representation theory Diagrams

Symplectic duality and KLR algebras

Ben Webster

University of WaterlooPerimeter Institute for Mathematical Physics

December 4 2017

Ben Webster UWPI



Philosophy

What is the correct context for geometric representation theory

I think I may be contractually obligated to now say that itrsquosmathematical physics

Thatrsquos fine because I mostly think thatrsquos true

Ben Webster UWPI



Philosophy

Powerful aspects of physics (from the perspective of amathematician)

Natural geometric input (and by that I mean PDEs) suggestssurprising algebraic structures especially when dimensionallyreduced or sent to a limit When algebras and categories arisethis way the underlying quantum field theory provides insight ontheir structure

In particular the same quantum field theory might have manydifferent realizations and a given question might be easy in oneand hard in another A particularly common version of this isvarious dualities

Mirror symmetry relating Fukaya categories and coherent sheaves isan especially famous example of this but various versions ofgeometric Langlands can also be thought of this way

Ben Webster UWPI



Gauge theories

Lots of interesting representation theory comes out this way Thereare different configurations of the physics you can put in but for mypurposes Irsquom interested in a very specific example

Fix a connected reductive C-algebraic group G and let V be arepresentation Out of this data we can create a gadget called anldquoN = 4 supersymmetric 3-d gauge theoryrdquo

You should think somewhere in the background there is an actualstate space whose objects are sections of certain bundles on a3-manifold and one calculates ldquoexpectationsrdquo of ldquoobservablesrdquo byintegrating a ldquoprobability measurerdquo on this space of section Youshould then forget you thought that

Ben Webster UWPI



Gauge theories

The important thing for us is that by looking at this picture and doingsome physicsy transformations we can get two singular affinevarieties and quantizations of those varieties

These are called the Higgs branch and the Coulomb branchDespite all the fancy physics terminology these have relativelystraightforward definitions in mathematics as algebraic varietiesMHMC with a symplectic form and families of algebras AHAC

quantizing these

I keep thinking it should be possible to actually explain thesedefinitions in a talk Hard experience suggests this is not actually thecase

Ben Webster UWPI



Higgs and Coulomb

Luckily Hiraku defined these for me so I donrsquot have to go too deepinto the details

The Higgs branch is the categorical quotient by G of the zerolevel of the moment map micro TlowastV rarr glowast Its quantization is thenon-commutative Hamiltonian reduction of the differentialoperators on V The Coulomb branch is birational to TlowastTorW The birationalmodifications require a bit of explanation but they follow somesimple combinatorial rules

Many interesting symplectic varieties appear this waythe nilcones Ng of simple Lie algebras (U(g))Slodowy slices (W-algebras)Symn(C2Γ) (symplectic reflection algebras)Nakajima quiver varietieshypertoric varieties

Ben Webster UWPI



Higgs and Coulomb

Definition

We call the Higgs and Coulomb branch of a single theory symplecticdual varieties Itrsquos not obvious why this would be a symmetricproperty but physics suggests it actually should be (though maybethat duals are not unique)

The nilcones Ng and NLg are dual

Special Slodowy slices and special nilpotent orbitscorresponding under the Spaltenstein involution are dual

Hypertoric varieties come in dual pairs indexed by a bijection ofunderlying combinatorial data Gale duality

Quiver varieties in finite and affine types are dual to slices in theaffine Grassmannian

Ben Webster UWPI



The toolbox

But everything wersquove said up to this point has no real mathematicalcontent What do we learn about an algebra by realizing it this way

Wersquore at a representation theory conference so obviously the thing wewould like to understand is the category of representations of thequantizations of these varieties

This means that wersquore categorifying Hirakursquos talk he was dealingwith Lagrangian cycles wersquoll want representations ofnon-commutative algebras whose characteristic varieties are of thisform

In particular his condition about possessing limits under Clowast actionsbecomes a ldquocategory Ordquo condition when you think about it as asupport condition

Ben Webster UWPI



The toolbox

The conjecture in Hirakursquos talk is essentially that two Lagrangiansubvarieties have the same number of components

The category of modules with these as supports are related with thebijection of components being a consequence

The relation between them is a very special one which appears manyplaces in representation theory Koszul duality

Definition

The Koszul dual C of a mixed graded category C is the abeliancategory of linear projective complexes in C

Proposition

We have Cprime sim= C if there is a semi-simple generator L such thatC sim= Extlowast(L L) -mod

Ben Webster UWPI



The toolbox

To prove this we need tools to analyze the representation theory ofboth of these algebras AH and AC For the two sides theyrsquore quitedifferent in flavor

They roughly correspond to the main sections of Soergelrsquos ldquoKategorieO perverse Garben und Moduln uumlber den Koinvarianten zurWeylgrupperdquo

ldquoArgumente aus der Topologierdquo this shows that the Ext ofsimple modules in category O are controlled by Soergelbimodules

ldquoArgumente aus der Darstellungstheorierdquo this shows that theendomorphisms of projectives in category O are controlled bySoergel bimodules

Thus together they establish a Koszul duality

Ben Webster UWPI



The toolbox

Now letrsquos see how this works in our context

Given a cocharacter χ Clowast rarr H = NGL(V)(G) we let Vχ be the sumof the non-negative weight spaces for this character this is invariantunder the parabolic Pχ sub G corresponding to non-negative weightspaces in g

Functions on Vχ give a (semi-simple regular holonomic) D-moduleon V this can be averaged from Pχ-equivariant to an G-equivariantD-module Yχ with the same properties

Theorem

ExtlowastG(Yχ Yχprime) = HBMGlowast (χXχprime)

χXχprime = (gPχ gprimePχprime v) | v isin gVχ cap gprimeVχprime sub GPχ times GPχprime times V

Ben Webster UWPI



The toolbox

On the other hand the Coulomb branch can be attacked withanalogues of Soergelrsquos V-functor

The quantum Coulomb branch AC contains a polynomial subalgebraS = Sym(hlowast)H arising from the integral system TlowastTorW rarr tW

We can analyze the representation theory of AC using the action ofthis subalgebra

Definition

Given a maximal ideal m sub S we let

Wm(M) = m isin M | mNm = 0 for N ≫ 0

Ben Webster UWPI



The toolbox

Given a cocharacter χ Clowast rarr H let m be the corresponding maximalideal of S

Theorem

983153ExtlowastG(Yχ Yχprime) sim= Hom(Wmχ Wmχprime )

The proof of this result is completely combinatorial in nature youjust compute both sides The computation is fiddly but notparticularly deep or difficult

Ben Webster UWPI



The toolbox

At least one interpretation of it is as a Koszul duality the Ext-algebraof a semi-simple set of objects matches the endomorphism algebra ofa projective object in another category

Corollary

The category of weight modules over the quantum Coulomb branchhas a graded lift with the classes of indecomposable projectives givenby a ldquocanonical basisrdquo

Ben Webster UWPI



Category O

We can give a more symmetric definition of this duality if we pass tocategory O

The choice of a particular quantization of the HiggsCoulomb branchcorresponds to the choice of an internal grading on the dualCoulombHiggs branch that is a grading with [ξ a] = deg(a)a forsome ξ

Given such an internal grading we can define a category O for it

Definition

Let category O be the full subcategory of modules such that ξ actslocally finitely with generalized eigenspaces finite dimensional andbounded above

Ben Webster UWPI



Category O

Theorem

We have an isomorphism of OHiggs with the Koszul dual OCoulomb in

all the cases wersquove discussed with quantization and gradingparameters chosen to match

In the case of flag varieties and Slodowy slices this isparabolic-singular duality for the original category O

In the case of hypertoric varieties this was proven byBraden-Licata-Proudfoot-W by a more hands-on argument

The most interesting case is that of quiver gauge theories in thiscase the Higgs branch is a Nakajima quiver variety and at leastin ADE type the Coulomb branch is an affine Grassmannianslice The details in this case are work in progress ofKamnitzer-Tingley-W-Weekes-Yacobi

Ben Webster UWPI



The Higgs presentation

So letrsquos try to say a tiny amount about those details in the quivergauge theory case

On the Higgs side the presentation we need for the proof is of ldquoKLRtyperdquo it requires just a small modification of work of RouquierVasserot and Varagnolo (and in fact is basically equivalent to apresentation given by Sauter)

In this case

G =983143

GL(vi) V =983131

irarrj

Hom(Cvi Cvj)oplus983131

i

Hom(Cvi Cwi)

We have an inclusion

(G times Gw)Clowast rarr NGL(V)(G)

We interpret a cocharacter into this subgroup as a vi-tuple of ldquoblackrdquoweights from GL(vi) and a wi-tuple of ldquoredrdquo weights from GL(wi)

Ben Webster UWPI




We represent these by putting dots on a horizontal line using theweights as x-positions

We can then represent morphisms between Yχ and Yχprime by immerseddiagrams interpolating between top and bottom decorated with dots(which represent Chern classes)

i j iλ1 λ2

Ben Webster UWPI




The relations between these are readily calculated geometricallyThey are given by

i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j

Qij(y3 y2)minus Qij(y1 y2)

y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)

1 When the grading which defines category O is ldquotensor productrdquo(in the sense of Nakajima) the resulting Higgs category O is thequadratic dual of the corresponding (unique) categorified tensorproduct category

2 In the case of an affine type A quiver variety the result is thequadratic dual of the corresponding (unique) categorified Fockspace this can be identified with a block of category O for acyclotomic Cherednik algebra

In both cases the action of the Lie algebra is via bimodulesquantizing Nakajimarsquos Hecke correspondences

This shows that simple modules in these categories match thecanonical basis

Ben Webster UWPI



The Coulomb presentation

On the Coulomb side this presentation has a less familiar flavor butinvolves diagrams of a similar style on a cylinder instead of a planewith full rotation around the cylinder giving a shift in the dotsi

ik

k

i

i

You should glue this intoa cylinder by attaching thefringed edges

Ben Webster UWPI




These also come with a slightly inscrutable slide full of relations

i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j

qij(y3 y2)minus qij(y1 y2)

y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii

pi+(yn)minus piminus(y1)

yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)

1 When the quantization parameter is ldquotensor productrdquo (in thesense of Nakajima) the resulting Coulomb category O is thecorresponding (unique) categorified tensor product category

2 In the case of an affine type A quiver variety the result is thecorresponding (unique) categorified Fock space this can beidentified with a block of category O for a cyclotomic Cherednikalgebra

In both cases the action of the Lie algebra is via Whittaker inductionand restriction functors (generalizing those of Bezrukavnikov andEtingof for Cherednik algebras)

Thus the canonical basis in these cases match the projectives (ortiltings depending on conventions)

Ben Webster UWPI




Another cool consequence of this presentation is the ability toexplicitly carry out Kaledinrsquos construction of a tilting generator

Theorem

For a quiver gauge theory of finite or affine type A there is a tiltinggenerator whose endomorphisms are a cylindrical weighted KLRalgebra this means you take the diagrams and relations from theHiggs presentation but you draw them on a cylinder rather than aplane

Since Slodowy slices in type A are finite type A quiver varieties thisproduces tilting bundles on all of these varieties generalizing forexample work of Anno and Nandakumar in the two-row Slodowyslice case (these are the Coulomb branches of sl2 quiver gaugetheories)

Ben Webster UWPI




Thanks

Ben Webster UWPI



Philosophy

What is the correct context for geometric representation theory

I think I may be contractually obligated to now say that itrsquosmathematical physics

Thatrsquos fine because I mostly think thatrsquos true

Ben Webster UWPI



Philosophy





Ben Webster UWPI



Gauge theories




Ben Webster UWPI



Gauge theories



quantizing these


Ben Webster UWPI



Higgs and Coulomb




Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Philosophy





Ben Webster UWPI



Gauge theories




Ben Webster UWPI



Gauge theories



quantizing these


Ben Webster UWPI



Higgs and Coulomb




Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Gauge theories




Ben Webster UWPI



Gauge theories



quantizing these


Ben Webster UWPI



Higgs and Coulomb




Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Gauge theories



quantizing these


Ben Webster UWPI



Higgs and Coulomb




Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Higgs and Coulomb




Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Higgs and Coulomb

Definition






Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox





Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox




Definition


Proposition


Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox






Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox




Theorem



Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox




Definition



Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox


Theorem



Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



The toolbox


Corollary


Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Category O




Definition


Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI



Category O

Theorem






Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI






In this case

G =983143

GL(vi) V =983131

irarrj


i

Hom(Cvi Cwi)




Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI






i j iλ1 λ2

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

ij λ

=

ij k

+

ij k

δijk

=

=

i λ

=

ki

δik

λ i

=

ik

δik

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI




Theorem (W)





Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI





ik

k

i

i


Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI





i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

qij(y1 y2)

ki j

=

ki j

unless i = k ∕= j

ii j

=

ii j

+

ii j


y3 minus y1

= minush = +h

i

= pi+

983086

i

983087

i

= piminus

983086

i

983087

ii

=

ii

+

ii


yn minus y1 minus h

ji

=

ji

if i ∕= j

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI




Theorem (KTWWY)





Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI





Theorem



Ben Webster UWPI




Thanks

Ben Webster UWPI




Thanks

Ben Webster UWPI


symplectic duality and klr algebras...the higgs presentation theorem (w.) 1 when the grading which...

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