Langlands' Philosophy and Koszul Duality

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

WOLFGANG SOERGEL

Contents

Introduction

The basic conjecture

The easiest example

The geometry of AdamsBarbaschVogan

Thanks

Notations and conventions

Groups and varieties

Grothendieck groups

Equivariant derived categories

Geometric extension algebras

Examples

Tori

Generic central character

The case G SL

Complex groups

Compact groups

Tempered representations

The center and geometry

Further conjectures

Generalities on Koszul duality

Application to our situation

Motivation

References

Introduction

In this article I will formulate conjectures relating representation

theory to geometry and prove them in some cases I am trying to

extend work of AdamsBarbaschVogan ABV and some joint work

of myself with Beilinson and Ginzburg BGS All this should b e

regarded as a contribution to the lo cal case of Langlands philosophy

SOERGEL

The lo calization of BeilinsonBernstein also establishes a relation

b etween representation theory and geometry This is however very

dierent from the relation to b e investigated in this article Whereas

lo calization leads to geometry on the group itself or its ag manifold

the results in ABV to b e extended in this article lead to geometry

on the Langlands dual group

The basic conjecture Let G b e only for a short moment a

real reductive Zariskiconnected algebraic group We want to study

admissible representations of the asso ciated Lie group GR of real

p oints of G More precisely we want to study the category MGR of

smo oth admissible representations of GR Remark that by a theorem

of CasselmanWallach this is nothing else but the category of Harish

Chandra mo dules for GR mo dulo the choice of a maximal compact

subgroup K GR Let g b e the complexication of the Lie algebra

of GR U mo d U U mo d U g the enveloping algebra of

g and Z U mo d U the center of U mo d U Then MGR

decomp oses according to innitesimal character into

MGR MGR

where runs over the set Max Z of maximal ideals of Z

There are several classications of the simple ob jects in MGR

The most p opular ones are by Langlands Lan VoganZuckerman

Vog and BeilinsonBernstein BB In this article I will exploit a

version of the rst classication scheme due to Langlands and Shelstad

and more precisely the form given to it by AdamsBarbaschVogan

ABV

Let me roughly explain its structure First of all we have to forget

ab out the real algebraic group G and just remember the inner class of

real forms for G C given by G In other words and other notations

we start out with a complex connected reductive algebraic group G

endowed with an inner class of real forms To simplify the exp osition

let us furthermore choose a nite central subgroup F of G To these

data one asso ciates in a more or less unique way and to detail this

more or less takes most of the time

a nite family S of antiholomorphic involutions G G all

from the xed inner class

a nite connected algebraic covering G of the dual group G ie

of the complex connected reductive algebraic group G whose ro ot

system is dual to the ro ot system of G

a collection fX g of complex algebraic G varieties

MaxZ

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

I will give details in section For every antiholomorphic involution

G G let GR denote the xed p oint set of the induced map

GC GC and for any MaxZ let GR denote the

set of isomorphism classes of simple ob jects in MGR Then

AdamsBarbaschVogan ABV establish a bijection dep ending on

additional choices b etween

a the set GR and

b the set of isomorphism classes of pairs Y where Y is a G

orbit on X and an irreducible G equivariant lo cal system

on Y

Strictly sp eaking this classies the representations of a whole collec

tion of real forms simultaneously rather then representations of a single

group at a time To explain which geometric parameters Y b elong

to which real form S wouldnt make go o d reading for this intro

duction compare Vog Let me mention however that any

real form in the given inner class is equivalent to GR with S

if we choose F big enough hence the ab ove classication captures all

real forms

One p oint of setting up the ab ove classication scheme is the very

b eautiful and miraculous fact that the KLdata on the representation

theoretic side are mirrored by the ICdata on the geometric side This

is the essence of Vogans character duality Vog I conjecture that

this duality lifts to the level of categories To give more details I have

to introduce more notation

Let H b e a complex algebraic group and X a complex algebraic H

variety Supp ose H decomp oses X into nitely many orbits Then

we can form a graded C algebra Ext X as follows Take the equi

variant derived category D X as dened in BL This is a tri

angulated C category containing the category P X of equivariant

p erverse sheaves as a full ab elian sub category For F G D X let

us denote by Hom F G the space of homomorphisms in D X and

H H

F G Hom F G n Since H was supp osed to act with put Hom

nitely many orbits there are up to isomorphism only nitely many

simple ob jects in P X Let L P X denote the direct sum of

H H

all simple ob jects ie take one from each isomorphism class Then

put

Ext X Hom L L End L

H n

H H

There is our graded C algebra We will refer to it as the geometric

extension algebra

For any ring R let R M od b e the category of all left R mo dules For

a graded C algebra A A dene the full sub category A N il

SOERGEL

A M od by

A N il fM A M od j dim M A M for n g

So ob jects in A N il are not assumed to b e graded but only conti

nous along the graduation Recall now the classication scheme from

ABV The main ob jective of this article is to formulate motivate

and discuss the following

Basic conjecture There exists an equivalence of categories

Ext X N il MGR

Remarks

Take again our algebraic group H acting on our variety X with

nitely many orbits Then the simple ob jects of Ext X N il

are in an obvious bijection with the simple ob jects of P X and

these in turn are parametrized by pairs Y with Y an H orbit

on X and an irreducible H equivariant lo cal system on Y cf

BL Thus we see that the simple ob jects on the left and on

the right of our conjectured equivalence are parametrized by the

same set Certainly our equivalence should identify simple ob jects

which have he same parameter

In this article I will check the conjecture in the following cases

if G is a torus for arbitrary G and almost all central

characters under the additional assumption F for G

SL for complex groups more precisely when G G G for

c c

a semisimple simply connected complex algebraic group G and

F and the inner class of real forms is such that S consists just

of one and GR G C under the additional assumption

that the central character is integral regular

I will explain in the last section how I exp ect this conjecture to

b e related to Vogans character duality formulas Vog ABV

It seems reasonable to make analogous conjectures for nonar

chimedean elds as well In these cases X should b e replaced

by spaces of quasiadmissible morphisms of the WeilDeligne group

to the Langlands dual group I exp ect that one may identify in

sp ecial situations the geometric extension algebras with Lusztigs

graded ane Hecke algebras

The easiest example To give the reader a more concrete idea

of what is going on let me treat the easiest example Let us take

G C endowed with the inner class of real forms consisting of the

split form so that GR R and choose F

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

Then S will consist of the single element the covering G G

will b e the identity C C and X will consist just of two p oints

for every Max Z We nd that

Ext X Ext two p oints

G C

Ext p oint Ext p oint

C C

H B C C H B C C

H P C C H P C C

C t C t

where B C P C is the classifying space of C and deg t

On the other hand MR is just the category of continuous nite

dimensional representations of R I leave it to the reader to check for

every Max Z the equivalence

C t C t N il MR

This proves our conjecture in this sp ecial case

The geometry of AdamsBarbaschVogan I promised at the

b eginning that given a complex connected reductive algebraic group G

endowed with an inner class of real forms and a nite central subgroup

F I would construct a multiset S of real forms a nite con

nected covering G of the dual group and a collection fX g

MaxZ

of G varieties

To fullll this promise I have to make some choices In ABV the

authors investigate in detail how these choices aect the nal outcome

Let us choose an antiholomorphic involution G G in the

given inner class that stabilizes a Borel subgroup Then form

j j

the real Lie group G G GC if we wouldnt abuse

notation where Gal C R f g acts on G via this an

tiholomorphic involution For any G G with Z G

the conjugation int G G is an antiholomorphic involution

inner to Let S G G b e a system of representatives for

the Gconjugacy classes of G G with F As explained

we may consider S as a multiset of antiholomorphic involutions

G of G For example S H

I warn the reader that for a complex algebraic group I quite

often write just G for the complex p oints of G to b e denoted GC

in more careful notation I hop e this do es not lead to confusion

Let us choose in addition T B G a stable maximal torus

and Borel subgroup of G This gives rise to a based ro ot system

X R X R with an involution s X X stabilizing R

and In more detail X XT fhomomorphisms of complex

algebraic groups T C g and s X X is characterized

SOERGEL

by the formula st t t T The notation s is

explained in more detail in Then choose a triple G T

consisting of a connected reductive complex algebraic group G

a maximal torus T G and an isomorphism X XT

that identies the ro ot system of G with the dual of the ro ot

system of G

Now the nite central subgroups F of G are in onetoone

corresp ondence with nite quotients of XZR via F XF

But the nite connected coverings H G of G are also

in onetoone corresp ondence with nite quotients of XZR via

H XX T In this way any nite central subgroup F

of G gives rise to a nite connected covering G of G

This is the most subtle part We start introducing some notations

Let H b e any complex connected reductive algebraic group

h LieH its Lie algebra and h a semisimple element Put

h f h j ng

n h

n n

e exp i H

The canonical at through is the ane subspace F

n h One proves see ABV that the canonical ats

partition the set h of semisimple elements of h and even partition

each conjugacy class of semisimples Furthermore the function e

is constant on each canonical at Let F h denote the space of

all canonical ats through semisimple elements of h and for a

semisimple orbit O h put F O f F h j O g Then

e is a function e F h H

Now recall G and put g LieG We may and will identify

LieC C in such a way that the exp onential map of Lie groups

corresp onds to the usual exp onential map This determines an

embedding X HomT C HomLieT LieC LieT

hence an isomorphism X C LieT On the other hand de

termines an embedding X HomC T HomLieC LieT

LieT hence an isomorphism X C LieT

Let W b e the Weyl group We have bijections

Max Z LieT W X C W LieT W g G

induced resp ectively by the HarishChandra homomorphism our

ab ove isomorphisms and the embedding LieT g Denote by

O their comp osition

C g of isomorphisms x Now choose a collection fx

U of the additive group C onto the simple ro ot subgroups of

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

G The involution s X X has an adjoint s X X

We will abreviate s X X This gives rise to an

involutive automorphism G G of the complex algebraic

group G characterized by T T X XT

and x x Thus we may form the semidirect

pro duct G G the socalled Lgroup

Now for any MaxZ we dene

X fy j y G G F O such that y eg

This is in a natural way a complex algebraic variety on which G

acts via conjugation For more details see ABV This fulllls

my promise

It is useful to remark that by ABV e the comp onent

groups of isotropy groups of G acting on X are all commutative

Hence all lo cal systems app earing in our geometric parameters are of

rank one

Thanks I thank David Vogan Victor Ginsburg and Valery Lunts

for very helpful discussions I thank the DFG MPI and CNRS for

nancial supp ort and the latter two for providing a stimulating research

atmosphere

Notations and conventions

Groups and varieties Let G H b e complex algebraic groups

A holomorphic resp antiholomorphic morphism from G to H is a

morphism of group schemes G H such that the diagram

G H

S pecC S pecC

with the lower horizontal the identity resp the complex conjugation

commutes

Let G b e a complex algebraic group Two antiholomorphic involu

tions of G are said to b elong to the same inner class if and only

if is an inner automorphism of G ie intx with x G

Two antiholomorphic involutions of G are said to b e equivalent

if and only if they are conjugate by an inner automorphism of G ie

intx intx with x G Any two equivalent antiholomorphic

involutions b elong to the same inner class

Let A B b e groups and B AutA a group homomorphism Let

us write ba a a A b B We dene a new group A B

SOERGEL

j j

AB as follows As a set AB A B and the multiplication is given

by a ba b aa bb

Let G b e a complex algebraic group Then we denote by LieG its

Lie algebra This is a Lie algebra over C Let H b e a Lie group Then

we denote by Lie H the Lie algebra of H This is a Lie algebra over R

We denote its complexication by Lie H C Lie H Let G b e a

C R R

complex algebraic group Then LieG Lie GC canonically This

is an isomorphism of Lie algebras over R where on the left we quietly

restrict scalars from C to R Let b e an antiholomorphic involution on

G Then LieG Lie GR canonically

Let X b e a complex algebraic variety We often abuse notation and

write X instead of X C Sometimes we also abuse notation in the

other direction and write for example C C instead of G G

m C a C

Grothendieck groups Let A b e an ab elian resp triangulated

category Then we let A b e the Grothendieck group of A generated

by the ob jects with the usual relations for all short exact sequences

resp distinguished triangles Any ob ject A A gives an element

A A

Let A b e an additive category Then we let hAi b e the split Gro

thendieck group of A generated by the ob jects with the usual relations

for all split short exact sequences Any ob ject A A gives an element

hAi hAi

Equivariant derived categories My basic reference for what

follows is BL Let G b e a complex algebraic group and X a com

plex algebraic Gvariety To X we may asso ciate a triangulated C

category D X the b ounded constructible Gequivariant derived cat

egory called D X in BL We note Hom the morphisms

in D X put Hom Hom and use analogous notation End

G n G

G G

for endomorphisms If G we come back to our usual trian End

gulated categories D X D X D X and since Hom lo oks so

stupid we write Hom in this case

I cannot rep eat the details of the denition of D X To give the

reader an idea of what D X is like let me just remark that it embeds

as a full triangulated sub category in the b ounded derived category

D S hEG X of sheaves of C vectorspace s on EG X where

G G

EG is a contractible space with a free right Gaction eg the total

space of the universal bundle on the classifying space BG of G

The usual formalism of the six op erations of Grothendieck extends

to this equivariant situation For F G D X we may form F

G RH omF G D X and any equivariant morphism f X Y

of Gvarieties gives rise to adjoint pairs of triangulated functors f f

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

and f f relating D X and D Y The categories D X also

G G G

come equipp ed with a p erverse tstructure and we let P X b e its

core ie the Gequivariant p erverse sheaves for the middle p erversity

and H D X P X the p erverse cohomology functors Let

G G

pt P pt b e the constant ob ject C and put X p pt D X for

G G

any X where p X pt is the constant map

The irreducible ob jects of P X are parametrized by pairs Y

with Y a Gstable lo cally closed subset of X and an irreducible G

equivariant lo cal system on Y Namely to such a pair one asso ciates the

intersection cohomology complex with twisted co ecients IC Y

P X

Let H G b e a morphism of algebraic groups Then we may

dene the restriction functors R es D X D X They are trian

G H

gulated and compatible with all our internal op erations If is either

a nite covering or the inclusion of a reductive Levi factor then

R es induces isomorphisms Hom F G Hom R esF R esG for all

G H

F G D X

Geometric extension algebras Assume now again that X is

a H variety decomp osing into nitely many orbits Remember we de

L L where L ned the graded C algebra Ext X Hom

H n

IC Y is the sum over all simple ob jects of P X so that Y

runs over all H orbits in X and over all irreducible H equivariant

lo cal systems on Y

Supp ose H G is a closed subgroup Then the induction equiva

lence gives an isomorphism Ext X Ext G X

H G H

Supp ose H is connected Then Ext pt H BH C is just the co

homology ring of the classifying space of H If in particular T is a torus

then Ext pt S LieT is just the ring of regular functions on its

Lie algebra graded in such a way that deg LieT More generally

if H is any connected algebraic group then Ext pt H BH C

S LieT consists of the invariants of the Weyl group in the regular

functions on the Lie algebra of a maximal torus T G

Let N b e a complex algebraic group H N its identity comp onent

and W N H the quotient Then N acts on H by conjugation

hence it acts on H BH C and since H is connected this N action

factorizes to give an action of W on H BH C Let us denote this

action by W Aut H BH C

Now whenever we are given a C algebra A together with an action

W Aut A of a group W on A by algebra automorphisms we may

form the twisted group ring A W A C W with multiplication

a w b v a w b w v In the ab ove notations we prove

SOERGEL

Theorem Let N be a complex algebraic group H N its iden

tity component and W N H the quotient If W is commutative then

Ext pt H BH C W

If not these two rings are stil l Moritaequivalent

In fact we will prove even a more general theorem to prepare for the

calculations concerning SL Namely let H N W b e as ab ove let

X b e an N variety and L D X an ob ject in the equivariant derived

category We have the restriction functor R es D X D X Its

N H

right and left adjoints coincide in our situation so we denote them just

by I nd D X D X See BL for the denitions of I nd

H N

I nd Let me rst give the statement I want b efore I explain and prove

Theorem End I nd R esL End R esL W

N H

R esL Explanation I ought to explain the action of W on End

Let E EN denote the universal bundle over the classifying space of

N Let E X E X b e the pro jection We have the following

H N

two commutative squares

D X D S hE X

N N

Res I nd

D X D S hE X

H H

where stand for the functors usually denoted L R The

horizontal arrows are fully faithful embeddings In fact there is a right

W action on on E X given by w E X E X e x

H H H

en n x for any n N representing w W Our E X

E X is just the quotient by this right W action

Let the image of L by the upp er horizontal in the ab ove diagram b e

R esL End denoted L Thus End L But any w W gives

L L End L End w w End

where the rst map is induced by w the second by w There

is our W action on End L

Proof We pro cede in four steps

Dene a morphism End R esL End I nd R esL

H N

Dene a morphism C W End I nd R esL

Prove that they are compatible ie t together to dene a

morphism

End R esL W End I nd R esL

H N

Prove that this morphism is an isomorphism

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

So lets go to work

The rst step is clear by functoriality of I nd Translating into a

dierent picture the morphism is End L End L

Translating via our diagram we see that we need a morphism

C W End L Indeed any w W denes a morphism hw i

L L as the comp osition L w w L L

where the rst map comes from the adjointness transformation in

w w and the second from w One may check that hw ihv i

hw v i Thus we dened a morphism C W End L

Let f End L and w W We have to establish in End L

the equation hw i f w f hw i For this we consider the com

mutative diagram

L w w L w L L

f w w f w f w f

L w w L w L L

which gives us the required equation

We have to prove that the pro duct

L f w id hw i L C W End End

is an isomorphism or equivalently that the pro duct

L f w adj hw i L L C W Hom L Hom

is an isomorphism where adj is the map corresp onding to the identity

id via the adjointness But L L and under

w W

this isomorphism the W action on the left by the hw i corresp onds to

the regular action on the right So indeed we get an isomorphism

qed

Proof of the rst theorem Apply the second theorem to X pt L pt

qed

Examples

Tori To establish the conjecture for G a torus T is not to hard

We b egin with some denitions As is well known there is a contravari

ant equivalence of ab elian categories

diagonalizable nitely generated

complex algebraic commutative

groups groups

D iag

For morphisms of the left resp right category we will often write

just instead of X resp D iag The reason that this do es not

SOERGEL

lead us into a catastrophe is that we only have to work with pairwise

commuting involutions

Any diagonalizable complex algebraic group G admits a unique anti

holomorphic involution s s G G such that sg g g

G XG This antiholomorphic involution gives rise to what one

calls the split real form of G For every holomorphic or antiholomor

phic morphism of complex diagonalizable groups f G H we have

f s s f

G H

Both the holomorphic and antiholomorphic homomorphisms from

one complex diagonalizable group to another form an ab elian group

We write these groups additively For example if f G H is

holomorphic or antiholomorphic we have f g f g g G

Now let us get at our torus T It comes with an inner class of real

forms ie with an antiholomorphic involution This gives us an

involution s X s on X XT and thus an involution s

on the dual lattice X Let us describ e the varieties X They are

either empty or isomorphic to X which we henceforth abreviate to

X The isotropy group I is the same for all p oints of X and is in fact

the kernel of id T T Applying our exact contravariant

functor X we get XI X id X Thus LieI XI C

X C and Ext X is just a cartesian pro duct of say n copies

Z g

of the symmetric algebra S S X C over the dual of the Lie

algebra of the isotropy group

On the other hand T R admits a Cartan decomp osition T R

K A with K compact and A a vector group In fact A is just the one

comp onent of T R T R s and this intersection in turn consists

just of all xed p oints for some antiholomorphic involution on the kernel

of id s T T Thus Lie A Liekerid s X C In

C Z

other words the S from ab ove is also the enveloping algebra of Lie A

So for each Max Z either

M M

MT R S N il

n copies

for some n or the left hand side vanishes So to establish the con

jecture we just have to show that X MT R and

that if b oth sides do not vanish then n n But this is calculated

g r

in detail in ABV and essentially even in Lan for F

Generic central character For most Max Z the situation

is very simple Namely for outside a countable union of prop er closed

subvarieties of Max Z only the split real form admits representations

with innitesimal character and the irreducible ones are just the

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

principal series induced from a Borel I now test the conjecture on this

example

Let LieT f LieT j h i Z R g b e the set of all

not at all integral weights and Max Z MaxZ its image under

the HarishChandra homomorphism

Lemma If G is endowed with the inner class of real forms con

taining the split form and Max Z then X G T ft

T j t g as a G variety

Proof Since G is endowed with the split inner class G G

For MaxZ each element of O is its own canonical at hence

X fy j y G O such that y exp ig Let us

determine for one O all p ossible y Assume to simplify

LieT Since was supp osed to b e not at all integral the centralizer

of exp i in G is just T and we nd that necessarily y T

Thus a bijection ft T j t g fy G j y exp i g

t t exp i But every Max Z is in particular regular so

O G T and we nd indeed X G T ft T j t g

with G acting only on the rst factor qed

Let us now assume for simplicity that F

Lemma Suppose G is endowed with the inner class of real forms

containing the split form Then only one of the S gives a split form

of G

Proof If we identify S H G then S gives the split

form if and only if it maps to the identity element in H GZ G

However consider the exact sequence

H Z G H G H GZ G

of nonab elian cohomology Since the rst map factors over the coho

mology of a split torus T G and H T the preimage of

the identity element in H GZ G is just the identity element in

H G qed

Let s S represent the split form For Max Z we know that

MGR if S s So we have for MaxZ just

to establish an equivalene of categories

X N il MGR s Ext

r ank G

But b oth sides are just N il mo dules over a Cartesian pro duct of

copies of the symmetric algebra S S LieT of the Lie algebra of the

G T maximal torus For the right hand side this follows from Ext

pt H BT C S LieT where the grading is determined Ext T

SOERGEL

by deg LieT For the left hand side it follows from the fact that

for any LieT the functor

MT R s MGR s

of extending trivially to a Borel and inducing with the shift that en

sures that unitary representations go to unitary ones is an equivalence

of categories

The case G SL There is only one inner class of real forms

Certainly G P GL C Let us compute the X We distinguish

three cases

not integral Then X G T ftwo p ointsg as we showed

already more generally

singular Then O fg and X fy G j y g

But any element of nite order is semisimple hence contained in

a maximal torus and in a torus there are precisely two ro ots of one

The stabilizers in G of these two are G and N T which we

will henceforth abreviate N Thus X pt G N

regular integral Then exp i for all O hence

X is just the Cartesian pro duct of F O with fy G j

y g But if Max Z is regular integral we have quite

generally F O G B Hence in our situation fF j

O g G B and

X G B G B G N

where G acts diagonally on the second part

The subcase F Let us rst check the conjecture for F

Then S H SL C consists just of one element with

GR SL R For any MaxZ let us abreviate MSL R

M These categories break down according to K types into M

ev odd

M M Now we pro cede case by case

not integral This has already b een treated in general

singular It is known that M is isomorphic to the category of all

nite dimensional continuous mo dules over the completion Z of Z at

ev W

Via the HarishChandra homomorphism we nd M S N il

with S S LieT the symmetric algebra over LieT But on the other

W W

pt H BG C H BT C S so that indeed hand Ext

pt N il M Ext

odd

The structure of M is more complicated The irreducible ob jects of

this category are the two limits of principal series and these extend

among each other More precisely the picture is as follows Cho ose

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

in g a standard basis X H Y with H X X H Y Y

X Y H and such that C H g is the complexication of the Lie

algebra of the compact torus SO SL R For M M n Z

put M fv M j H v nv g Then

M M

odd

is an equivalence of M to nite dimensional representations of the

quiver

with relations nilp otent This is not dicult and follows for exam

ple from the more general results in Kho I leave it to the reader to

identify the quiver algebra with the twisted group ring S W where

W Aut S is the obvious action Putting things together we get

odd

M S W N il

But on the other hand

Ext G N H BN C by

H BT C W by theorem of section

S W

odd

So indeed M Ext G N N il and with our previous result

we obtain nally M Ext X N il for singular

regular integral By the translation principle we may and will

henceforth assume Ann C is just the annihilator of the trivial

odd

representation Then M can b e identied with the category of all

nite dimensional continuous mo dules over the completion Z of Z at

odd

and via the HarishChandra homomorphism we nd M S N il

odd

But Ext G B H BB pt S so M Ext G B

G G

N il

The category M is much more interesting Using the preceding

notations we may describ e it as follows The functor

Y Y

M M

- -

X X

establishes an equivalence of M to nite dimensional representations

of the quiver

SOERGEL

- -

with relations and nilp otent This is not dicult

and follows for example from the more general results in Kho Now

let us attack the geometric extension algebra Since

G B G N G G B

we have using the induction identity just to understand Ext G B

There are two N orbits on G B P C X namely Y f g

and U C Here Y supp orts just one irreducible N equivariant lo

cal system Y but U supp orts two of them with trivial and nontriv

ial N action say U and U The irreducibles in P X are corre

and X where i Y X is the inclu sp ondingly i Y X

sion and X resp X denotes just the constant sheaf C on X

with trivial resp nontrivial N action By denition Ext X

i Y X X But since we only need to know this End

ring up to Morita equivalence let us rather compute the equivariant

selfextensions of i Y i Y X X This ob ject has the ad

vantage to b e isomorphic to I ndR esi Y where we consider X

N as the semidirect pro duct N T W and use the notations of

so we may apply theorem of section to L i Y X The

R esL with its theorem tells us that we should rst compute End

W action This is not to hard so I just give the result

R esL is the quiver algebra of the quiver Prop osition End

l m mr

m r

- -

ml r m

with relations r mml l mmr where every arrow has de

gree one The reection of the quiver along the vertical symmetry axis

induces the lookedfor action of W on the quiver algebra

Proof Left to the reader Perhaps it would have b een more sugges

tive to call the extremal p oints and instead of r and l

The nite dimensional representations of Ext X can b e identied

by Morita equivalence with nite dimensional representations of

End I nd R esL End R esL W

N T

and these in turn are just nite dimensional representations of the

quiver with relations from the prop osition equipp ed with an action

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

of W on the representation compatible with the ab ove action of W on

the quiver Such an ob ject consists of a diagram of nite dimensional

spaces

l m mr

W W W

l m r

- -

ml r m

together with isomorphisms s W W s W W s W W

r l m m l r

such that the relations s id l mmr r mml smr

ml s sl m r ms are satised We got to identify the category of

such ob jects with the category of nite dimensional representations of

the related quiver

This can b e done as follows To a representation

M M

- -

of the related quiver we asso ciate the representation

M M

- -

of the related quiver with W action given by

M M M M s id M M s

This can b e seen to b e an equivalence of categories and induces an

equivalence of categories M Ext X N il So putting things

together we indeed established the desired equivalence of categories

M Ext X N il for regular integral

The subcase F Z G Now take for F the full center of SL C

ie F f g Then S can b e seen to consist of three elements

giving rise to the three real forms SU SU and SU Cer

tainly two of them are just the compact form SU and the middle

one is the split form SL R So on the representation theoretic side

of our conjecture only for regular integral there are some changes

and we have to consider instead of MSL R M the category

MSU i i C N il M C N il

SOERGEL

On the geometric side we get G SL C By homomor

phisms b etween ob jects in our equivariant derived categories do not

change when we pass from G to G However there might b e more

simple ob jects in P X than in P X and this happ ens pre

cisely when the isotropy groups of p oints of X with resp ect to P

have more comp onents than the isotropy groups with resp ect to P

In our situation this happ ens only for regular integral and for p oints

in the op en orbit on G B G N X To verify our conjecture

for F f g we just have to check that

Ext G B G N Ext G B G N C C

Via the induction identity this can b e reduced to showing

Ext X Ext X C C

N N

where N N T Let L P X b e the sum of all simple ob

F N

X In P X there are an additional jects L i Y X

two simple ob jects In more detail the isotropy group in N of a

p oint in the op en orbit U X is just ZZ thus there are four

simple ob jects F F F F in P X corresp onding to the four ir

reducible characters on ZZ Let j U X b e the inclusion Then

j F X j F X and our two new irreducible ob jects in

P X are j F j F and j F j F I leave it to the reader to

check that

L C C End L j F j F End

N N

This gives the conjecture for G SL R F f g

Complex groups Here comes one of the main motivations for

the conjecture Namely we check that it correctly predicts the structure

of part of the representation theory of G C for a semisimple connected

complex algebraic group where we consider G C as a real Lie group

In fact we will only give details for G simply connected and assume

this from now on To get in the setup of the conjecture choose a split

antiholomorphic involution s G G and put G G G with the

c c c c

quasisplit antiholomorphic involution G G x y sy sx

Then GR G C and H G S f g We prove under

the ab ove assumptions

Theorem Suppose is a regular integral central character Then

there is an equivalence of categories

X N il MGR Ext

ProofTheorem We b egin by simplifying the right hand side Abre

viate G G and let B G b e a Borel subgroup c

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

Lemma Suppose is regular integral Then X G

G B for a suitable embedding B G

Proof Left to the reader

Next we simplify the right hand side Let HC h denote the category of

HarishChandra bimo dules for g LieG investigated in So e

c c

Let Z U mo d U g b e the center and Z Ann C the central

c c Z

annihilator of the trivial representation of g Dene H fM HC h j

n n

Z M M Z for n g the category of HarishChandra

c c

bimo dules with trivial central character from b oth sides

Lemma For regular integral there is an equivalence of cate

gories

H MGR MG C

Proof This is contained in the literature BG

Mo dulo these two lemmas we see that we need to establish an equiv

alence of categories H Ext G B N il This was a conjecture in

BG It was essentially solved in So e however I did not write

down there the top ological part Merely I constructed in a combina

torial way a graded C algebra A and proved the equivalence of cat

egories H A N il So we just have to establish an isomorphism

A Ext G B of graded C algebras

Let T B b e a maximal torus and W N T T the Weyl group

Then the irreducible ob jects of P G B are just the intersection coho

mology complexes IC of the closures of the Schubert varieties B x B B

IC We got for x W Now by denition Ext G B End

xW x B

to understand this algebra b etter

The B equivariant cohomology ring of G B is well known to b e

G B End S S where S S LieT are the regular functions

on LieT graded in such a way that LieT is the degree two part of

S Now equivariant hypercohomology is a functor

H Hom G B D G B S S mod

B B

from the equivariant derived category to the category of graded mo d

ules over the equivariant cohomology ring

Prop osition For al l x y W equivariant hypercohomology

induces an isomorphism

H Hom IC IC Hom H IC H IC

B x y S S B x B y

Proof Postponed to

In particular we deduce an isomorphism

Ext G B End H IC

B S S xW B x

SOERGEL

But the algebra A was dened as

A End B

S S xW x

where the B are certain graded S S mo dules describ ed in a combi

natorial way in So e So to prove our theorem we need just

Lemma For al l x W there is an isomorphism of graded S S

B modules H IC

x B x

Proof The pro of of this lemma is very similar to the pro of of its

nonequivariant analogon in So e so I will rather give a sketch of

pro of

Let W S b e the Coxeter system asso ciated to G B and let H

H W S Zt t T b e the corresp onding Hecke algebra For

xW x

s S let P B b e the asso ciated parab olic Let G B G P

s s s

b e the pro jection One may make hD G B i into a H mo dule by the

prescription thF i hF i and T hF i h F i for all s S

s s

If fC g is the selfdual basis of the Hecke algebra dened in KL

then C hI C i hI C i for all x W

e x

Now let us sp ecify that we choose the identication H G B

S W S such that the pullback via G B G P corresp onds

s s

to the inclusion S W S S W S on equivariant cohomology

S S

rings Furthermore let us identify graded S S mo dules with graded

S bimo dules in the obvious way Let S mof S denote the category

of all graded S bimo dules which are nitely generated as left and as

right mo dules Then we may interpret H as a functor

H D G B S mof S

B B

and use essentially the same arguments as in So e to prove H F

B s

S H F for s S F D G B

S B B

Now recall from So e how we made hS mof S i into a H mo dule

The ab ove formula says then just that H induces a morphism of H

mo dules H hD G B i hS mof S i Certainly H hI C i

B B B e

hS i thus H hI C i H C hI C i C H hI C i C hS i hB i by

B x B e B e x

x x x

denition of B qedLemma and theorem

We still have to prove prop osition Let us start with some

general considerations Let X b e a top ological space ltered by closed

subspaces X X X X and let F D S hX b e an

ob ject in the b ounded derived category of sheaves on X To calculate

its hypercohomology H F we have to choose an injective resolution

I of F and take cohomology of the complex of global sections I

I I Certainly this is ltered by supp orts

X X

I So its cohomology is the limit of a sp ectral sequence with E

pq pq

term E H F the lo cal hypercohomology of F along X

X X

p+1

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

X Let j Y X b e the embedding of a lo cally closed subset

Then we can write the lo cal hypercohomology as H F H v F

ie as the hypercohomology of the derived lo cal cohomology sheaf

v F D S hY Let j X X X denote the inclusion We

p p p

may then write our E term as E H j F

Supp ose next that X is a complex algebraic B variety and that all X

are closed B stable subvarieties Applying the ab ove considerations to

E B X we nd that for F D X the equivariant hypercohomology

B B

pq pq

H F is the limit of a sp ectral sequence with E term E H j F

In particular if G D X is another ob ject the equivariant extensions

Hom F G H RH omF G are the limit of a sp ectral sequence

pq pq pq

with E term E H j RH omF G H RH omj F j G

p p p

B B

Supp ose that X is a B variety with a B stable stratication X

X into smo oth irreducible subvarieties Put jw j dimX and

w w

w W

jw j Let let C P X b e the constant p erverse sheaf C X

w B w w w

IC P X denote the intersection cohomology complex of the clo

w B

sure of X extended by zero to X Let j X X b e the inclusions

w w w

of the strata and supp ose there exist isomorphisms

j IC n C

w v

v v w

in D X for all v w W with suitable integers n dened by

B v

v w

these isomorphisms

Prop osition Suppose we have in addition to the above assump

tions parity vanishing ie n unless has the same parity

v w

X for al l w unless i is even Then as jv j jw j and H

IC IC Hom H IC H IC i H induces an injection Hom

x y C x y

B B B B

i j a

n n dim H X IC IC ii dim Hom

C z x y C

z v z w B B

aij z W

X Proof ii Consider the ltration on X given by X

x p

jxjpdimX

If we calculate Hom IC IC by the asso ciated sp ectral sequence we

x y

nd as E term

pq pq

H RH omj IC j IC E

x y

p p

H RH omj IC j IC

x y

jz jpdimX

z z

i j

H RH om n C i n C j

i z j z

jz jpdimX

z x z y

pq ij

i j

n n H X

jz jpdimX ij z

z x z y

By our parity assumptions this vanishes unless p q has the same

parity as jxj jy j In particular our sp ectral sequence degenerates at

this E term and ii follows

i Also H IC is the limit of a sp ectral sequence with E term

pq pq

E H j IC and again by parity vanishing it degenerates at this

term It is sucient to show that any nonzero morphism f IC x

SOERGEL

IC in D X induces for some p a nonzero morphism j f j IC

y B x

p p

j IC Let i X X b e the inclusion We may decomp ose X

y p p p

into an op en and a closed subset

i u

X X X X

p p p p

Then we have a distinguished triangle i i id u u i i i i u j

p p p p

which says that i f and j f imply i f So if all j f

p p p p

then i f f qed

ProofProposition It will b e sucient to prove injectivity

of the map and check that the Dimensions coincide on b oth sides

where for a graded vector space V V we understand its Dimension

i i

to b e the p olynomial DimV t dim V First we check that the

preceding prop osition applies in our situation ie for the stratication

of X G B by the Bruhat cells X B w B B We need

Lemma In D X the j IC decompose into the direct sum

B v w

of their perverse cohomology ie j IC n C for suitable

w v

v v w

integers n

v w

Proof In the nonequivariant situation this follows from p ointwise

purity Namely the j IC are pure hence the sum of their p erverse

cohomology groups

We may reduce our equivariant situation to a nonequivariant situa

tion as follows First choose a nite interval I Z such that IC

I I

D X Then we have to prove the lemma for the j IC D X

w v

B B

Now choose a smo oth nacyclic free right B space M where n is bigger

then the length of I Then by BL we have a commutative diagram

I I

D X D M X

j idj

I I

D X D M X

v B v

where the horizontal maps are fully faithful embeddings But the na

ture of the singularities is the same for M X and X Hence the

B w w

intersection cohomology complex of M X is also p ointwise pure

B w

and the lemma follows qed

Now we may in fact use prop osition to show that our map

H Hom IC IC Hom H IC H IC

B x y S S B x B y

is an injection So we just have to check that the Dimensions are equal

But

DimHom IC IC DimHom IC IC S

x y x y C

B D

by prop osition

LANGLANDS PHILOSOPHY AND KOSZUL DUALITY

and

DimHom H IC H IC DimHom B B by

S S B x B y S S x y

DimHom B B C S by

S S x y S C

DimHom B C B C S by

S x S y S C

DimHom H IC H IC S

S x y C

But we know already from So e that

Hom IC IC Hom H IC H IC

x y S x y

qed Proposition

Compact groups Supp ose under the ABVparametrization some

geometric parameter Y corresp onds to a representation of a com

pact form of G Our basic conjecture would say that

Homj IC Y IC Y n

unless Y Y and n The results of ABV on the duality

b etween ICdata and KLdata tell us already that the ab ove space of

homomorphisms vanishes unless Y Y We will verify in

addition that the isotropy group of p oints in Y is nite This implies

that there are now higher selfextensions of IC Y

So let Max Z b e integral regular Put I fy G G j

y g Clearly X I G B with the diagonal G action

In ABV it is shown that I decomp oses into a nite number of

G orbits that these orbits are all closed and that the corresp onding

isotropy groups are just complexications of maximal compact sub

groups of real forms of G More precisely the real forms app earing

here are just all real forms of one inner class of real forms on G and

this inner class corresp onds to the involution w of our based ro ot

system where is the involution of our based ro ot system coming from

the choosen inner class of real forms on G and w is the longest element

of the Weyl group

Hence if G is endowed with the inner class of real forms containing

the compact one then w so that the corresp onding inner class

of real forms on G contains the split form In particular some com

p onents of X are isomorphic to G K G B where K is the

complexication of a maximal compact subgroup of a split form of G

Now the representations of compact forms of G corresp ond just to op en

G orbits on these G K G B X And clearly the isotropy

groups of p oints in these op en orbits are nite

SOERGEL

Tempered representations I want to suggest a link of my basic

conjecture with the results of A Guichardet Gui He considers an

Lpacket of irreducible temp ered representations and the category T of

all HarishChandra mo dules with comp osition factors only from this L

packet He then proves that this category T is equivalent to a category

S R N il for a suitable p olynomial ring S with a suitable action

R Aut S of a suitable nite group R on it

In the ABVparametrization a temp ered Lpacket corresp onds to a

collection of geometric parameters Y for a xed op en orbit Y

X If my basic conjecture is true there should b e an equivalence of