Schub ert varieties and generalizations

T A Springer

Mathematisch Instituut Universiteit Utrecht

Budap estlaan CD Utrecht

the Netherlands

email springermathruunl

Abstract This contribution reviews the main results on Schub ert varieties and their gener

alizations It covers more or less the material of the lectures at the Seminar These were

partly exp ositoryintro ducing material needed by other lecturers In particular Section

reviews classical material used in several of the other contributions

Intro duction

The aim of this pap er is to give a review of the main results on Schub ert varieties and

their generalizations In the rst section Schub ert varieties over Careintro duced in the

setting of the theory of reductive groups and their Bruhat decomp osition Some geometric

results are discussed The Steinb erg variety asso ciated to a is also intro duced

Most of the material of this section is classical

In section examples are given of constructions of algebraic ob jects based on the geometry

discussed in Section For example gives an elementary geometric construction of the

Weyl group W of a reductivegroupG It uses corresp ondences on the ag variety X of G

Using machinery from algebraic top ology a calculus of corresp ondences on X pro duces the

Hecke algebra H of W This is discussed in and

Section discusses generalizations of Schub ert varieties These o ccur for example in the con

text of spherical varieties A closed subgroup H of G is spherical if a B of G

has nitely many orbits on GH Then GH is a homogeneous spherical variety The orbit

closures generalize Schub ert varieties which one recovers for H B An imp ortant sp ecial

case is the case of symmetric varieties where H is the xed p oint group of an involutorial

automorphism of G

The combinatorial prop erties of the set of orbits are discussed in A calculus of corre

sp ondences gives rise to a representation of H discussed in The last part of Section

reviews sp ecial features of the the case of symmetric varieties

I am grateful to Cathy Krilo for help in the preparation of these notes

Flag manifolds and Schub ert varieties

The origin of the Schub ert varieties lies in the Schub ert calculus devised byHSchu

th

b ert at the end of the century whichgives recip es to determine not always rigorously

numb ers of solutions of geometric problems see Sch

A simple example of such a problem determine the numb er of lines in P Cintersecting

lines in general p osition the answer is A more general example determine the number

n

of dplanes in P Cintersecting d n d planes of dimension n d in general

p osition the answer is

dd n d

n dn d n

n

Let G b e the set of ddimensional subspaces of C It is a pro jective algebraic variety

dn

n

co ordinatized by Pluc ker co ordinates It is also the varietyofd planes in P C

n

Fix a basis e e of V C and let V b e the subspace of V spanned bye e with

n i i

V fg Then F V V V V is a complete ag in V Let W be a ddimensional

n n

subspace and put

J W fj j V W V W g

j j

This is an increasing sequence of d integers the jump sequence of W It determines

the p osition of W relativetoF For example for a subspace in general p osition wehave

J W n d n Let Y b e the set of W Y G with J W J a given

J dn

sequence Then Y is the disjoint union of the Y Moreover one shows that each Y is lo cally

J J

closed in Y and is isomorphic to an ane space After ordering the set of J comp onentwise

the closures b oth in the Zariski top ology and the complex top ology are describ ed by



Y Y

J J



J J

These closures are the Schubert varieties in G

dn

In the Schub ert calculus one deals with intersections of Schub ert varieties and their multi

plicities This is b est done in terms of the ChowringofY spanned by equivalence classes of

subvarieties of Y This leads into the theory of symmetric functions See Fu I shall

not go into this

I shall concentrate on the group theoretical asp ects The group G GL C acts alge

n

braically on Y The action is transitive so Y is a homogeneous space of G and is of the form

Y GP where P is the parab olic subgroup of the g g G with g for i d and

ij ij

j d It is not hard to see that the Y are precisely the orbits of G

J

We shall consider a more general situation which will englob e the sp ecial case of Grassman

nians

Notations

The notions and results from the theory of algebraic groups whichwe use without further

reference can b e found in Bo or Hu For ro ot systems and the see Bou

G is a connected reductive linear over C one could work over an arbitrary

algebraically closed eld but I wont do this We x a maximal torus T of G and a Borel

subgroup B T AlsoN is the normalizer of T and W NT is the Weyl group Fix a

section w w of W to N

Let R b e the ro ot system of G T and let R b e the system of p ositive ro ots dened by

B For R wehave a one parameter subgroup U of G normalized by T The unip otent

radical U of B is generated bythe U with R For w W let U b e the subgroup of

w

G generated by the subgroups U with R w R

For R let s W b e the reection which it denes Let D b e the basis of R dened

by R The set S of simple reections s D generates W The corresp onding length

function on W is l Wehavedim U l w

w

Put G B wB This is a lo cally closed subset of G b eing an orbit of B B The underlying

w

top ology is the Zariski top ology One mightalsotake the complex top ology

Bruhats lemma

Prop osition Bruhats lemma i G G

w

ii u b uwb denes an isomorphism of algebraic varieties U B B wB

w

In fact G B N S make up the ingredients of a Tits system see Huno This

implies that for w Ws S

G if l sw lw

sw

G G

s w

G G if l sw lw

w sw

It follows that for s S

P G G

s e s

is a parab olic subgroup of G containing B G Wehave P B P It also follows that if

e s

s s s is a reduced decomp osition of w W where s S l l w wehave

l i

G G G G

w s s s

 

l

is the closure P P Lemma P G

s s s w

 

l

If Y and Z are varieties with a right resp ectively left B action we write Y Z for the

B

quotientofY Z bythe B action by z yb bz It is presupp osed that the quotient

exists A similar notation is used for multiple pro ducts

Put

Z P P P

s B s B B s

 

l

this is an irreducible variety The pro duct map of G induces a morphism Z G which

is prop er b ecause all quotients P B are pro jective lines Hence Im is closed and irre

s

i

ducible Moreover G is op en and dense in Z and the restriction of to this set maps it

w

bijectively onto G The lemma follows from these facts

w

Bruhat order

The closure G is a union of double cosets G Dene an order on W by x w if G G

w x x w

This is the Bruhat order originally intro duced by Chevalley

It follows from the lemma of that there is the following combinatorial description of the

Bruhat order Let s s s b e a reduced decomp osition of w W where s S l

l i

l w and let x W Then x w if and only if x is a subpro duct of s s In fact on any

l

Coxeter group there exists an order with this description see Hu

Schub ert varieties

The quotient X GB is a ag variety It is an irreducible smo oth pro jective homogeneous

space for G Let X b e the image of G in X under the canonical map this is a Bruhat cel l

w w

in X The big cel l is X where w is the longest elementofW The Bruhat cell X is a

w w



lw

lo cally closed subvarietyofX isomorphic to ane space A as a consequence of ii

The big cell is op en and dense in X

By i X is a paving of X by ane spaces or a cellular decomp osition The

w w W

X are the B orbits or U orbits on X

w

A Schubert variety is a closure S X w W It is an in general nonsmo oth irre

w w

ducible pro jectivevariety on which B acts By i and wehaveapaving S X

w x xw

n

Example Let G GL It acts on V k A ag in V of length s is a sequence of

n

distinct subspaces V i sofV with V fgV V V V The ag

i s

is complete if s n in which case dim V i for all i G acts on the set of ags and the

i

parab olic subgroups of G are the stabilizers of ags The Borel subgroups are the stabilizers

of complete ags

Let V V V b e the complete ag of Its stabilizer is the Borel group B of upp er

n

triangular matrices and GB can b e identied with the space of all complete ags Let P B

b e the stabilizer of the ag V V V Then GP is the G The canonical

d n dn

morphism GB GP maps a complete ag onto its ddimensional ingredient A classical

Schub ert variety Y as in is the image in GP of a Schub ert variety S in GB orthe

J w

closure of a B orbit in GP By Tits system theory these orbits are parametrized by the

cosets of the Weyl group of G mo dulo the Weyl group of P see Bou Ch IV p In

the present case this means that the Schub ert varieties in G are indexed by the elements

dn

of S S S ie by the delement subsets of f ng These are in bijection with

n d nd

the jump sequences of

For w W put

O fx y X X j x y G g

w w

The O are the Gorbits on X X There is a close connection with the Bruhat cells the

w

rst pro jection X X X denes a b ering O X with b ers X Similarly for the

w w

closures O It follows that

w

O O

w x

xw

The T action

The torus T acts on X and on all Schub ert varieties The xed p oints of T in X are the

images p of thew in X w W so their numb er is nite The xed p oints of T in the

w

Schub ert variety S are the p with x w

w x

Let again w b e the longest elementofW

Lemma Let x w W and assume that l x l w

i If the intersection S w S is nonempty then x w and the intersection is the point

x w w



p

w

ii X and w X intersect transversal ly at p

w w w w



iii Let xw Then uxB g B ug B u U denes a T equivariant isomorphism of

x

X w X S

x w x w



onto an open neighborhoodofp in S

x w

The intersection in i is a T stable pro jectivevariety and contains T xed p oints If p is

y

one then wehaveboth y x and w y w w ie y w This implies that y x w If

the intersection had dimension itwould contain at least two xed p oints and i follows

Part ii is proved by considering tangent spaces

Put U w U w this is the subgroup of G generated by the U with R Then

U B is op en in G by ii Hence xU BB is an op en neighborhood of p in X Then

x

iii follows by observing that

U x U xw X w

x w x



T

The variety X w X S is a transverse slice at p of X inside S Let be a

xw w x w x x w



co character of T a one parameter multiplicative subgroup such that hw i for all

R with x w R the brackets denote the pairing b etween characters and co char

acters Then contracts X to x ie if a X then lim ta x

xw xw t

Let S b e a subset of T and let H Z S b e its centralizer If S xes a p oint up u U

G w w

of X then S must centralize u and since centralizers of semisimple elements in connected

solvable groups are connected we can conclude that u lies in the connected centralizer H

Similarly the intersection of H with a Borel subgroup of G containing T is a Borel subgroup

S

of H The irreducible comp onents of the xed p oint set X of S in X are H stable and it

readily follows that each comp onent is a homogeneous space for H isomorphic to the ag

S

manifold of H Similarly one sees that the xed p ointsetsS must b e stable under the

w

Borel group B H of H and it follows that each irreducible comp onentofsuchaxedpoint

set is isomorphic to a Schub ert varietyforH Notice that if S is an algebraic subgroup of T

of co dimension d the reductive group H has semisimple rank d In particular if d

S

the irreducible comp onents of S are Schub ert varieties for SL hence are p oints or T stable

w

pro jective lines ConverselyaT stable irreducible curveinS must b e xed p ointwise bya

w

co dimension one subtorus of T and hence is a comp onent of its xed p ointset

In particular the number of T stable curves in X is nite They can b e describ ed explicitly

Let b e the set of reections in W A reection denes a dimensional subgroup G

of G generated by U and U where s s notations of For w W

put C G wB B

w

Lemma i C is a T stable curve Any T stable curve is of this form

w

ii The T xedpoints containedin C are p and p

w w w

 

iii If C C then either w w or w w

w

w

iv C S if and only if w x w x

w x

Wehave G Z Ker The observations of the previous paragraph then imply that C

G w

is a T stable curve

A T stable curve C X must contain a xed p oint p ThenwU BB wU w BB

w

is a T stable op en neighborhood of p Its intersection with C is a T stable ane curve

w

One is reduced to nding such curves or to nding T stable curves in U through the identity

element It is not hard to see that the latter are the U contained in U Then the second

part of i follows The argument also gives that C S which implies iv

w w

Geometric prop erties of Schub ert varieties

Schub ert varieties tend to b e singular There is a useful resolution of a Schub ert variety S

w

whichwenow describ e

Let s b e a reduced decomp osition of w as in With the notations of put

Z P P P B

s s B s B B s

 

l

This is a BottSamelson variety It is an iterated P bundle hence is irreducible and smo oth

From the lemma of we deduce the following

Prop osition Thereisaproper surjective birational morphism Z S

s w

The morphism is induced by the pro duct morphism in G

is not always a resolution of singularities of S in the usual sense as need not b e bijective

w

on the inverse image of the set of smo oth p oints of S For example if R is irreducible and

w

w w so S X then is bijective only if R is of typ e A

w

Criteria for smo othness and rational smo othness of a Schub ert varietywere recently given

by Kumar Ku See also Brions contribution Bri no where these matters are discussed

in the context of equivariantintersection theory Recall that an irreducible algebraic vari

ety Z is rationally smo oth at a p oint z if z has arbitarily small op en neighb orho o ds which

are homologically like op en balls A formal denition is the constant sheaf Q is its own

GrothendieckVerdier dual up to a dimension shift See also Bri

We discuss some more elementary results ab out smo othness of Schub ert varieties due to

Carrell and Peterson see Ca

Lemma Let Y bea T stable irreducible subvariety of X Ify Y is a T xedpoint the

number of T stable curves in Y passing through y is at least dim Y Equality holds if Y is

smooth at y

The pro of of the rst part is quite elementary and go es through in greater generality

Prop osition Let x w W with xwand denote by ax w the number of

with xx w

i ax w l w l x Equality holds if S is smooth at p

w x

ii If ax w l w l x then S is rational ly smooth at p

w x

i follows from the lemma The pro of of ii given in lo c cit uses a computation with

KazhdanLusztig p olynomials For another pro of see Bri

Peterson unpublished has proved that if the ro ot system R is simply laced one may replace

in ii rationally smo oth by smo oth

Example Let G SL Then W S The set S of generators of W consists of

s s s Take w s s s s x s Then l w l x

and ax w so S is not smo oth

w

Although Schub ert varieties are in general not smo oth they are always normal and Cohen

Macaulay see Ra These matters are also discussed in Littelmanns contribution Li refLittelmann

Line bundles on X

Denote by X T the character group of T Acharacter of T can b e lifted to a character of

B denoted by the same symbol

Let Y be a variety with a right B action For X T wehave a line bundle Lon YB

Namely L is the quotientofY A by the B action by a yb b a

In particular taking Y Gwehave line bundles L on the ag manifold X They are

Gequivariant and are lo cally trivial for the Zariski top ology

Some related varieties

Assume G to b e semisimple Lie algebras will b e denoted by gothic letters so g is the Lie

algebra of G Let F b e the Killing form on g a nondegenerate bilinear symmetric form

invariant under the adjoint action Ad of G on g For the matters to b e discussed in this

section see CG Ch

Consider the cotangentvariety T X of the ag variety The tangentspaceT X to X at

x

x gB is gAdgb Identifying the linear dual of g with g via F the dual of T X is the sub

x

space of g orthogonal to Adg b whichisAdg uSoT X is the set of pairs gB g X

with Adg u Let Ng be the variety of nilp otent elements of g The group G acts on

it via the adjoint action with nitely many orbits see CG

The rst pro jection induces a morphism T X N In fact can b e viewed as the

moment map for the Gaction on the symplectic variety T X see Brylinskis contribution refBrylinski

Theorem is a resolution of singularities of N

Recall that this means that T X is smo oth is prop er and that induces an isomorphism

N N where N is the op en subvarietyofsmoothpoints of N The rst two

sm sm sm

prop erties are easy Toprove the third one has to use prop erties of the regular nilp otent

elements of gAnelement x N is regular if its Gorbit is op en in N One proves that x is

regular if and only if one of the following holds

a N is smooth at x

b fxg consists of one point

The required prop erty follows For a discussion of these matters and further references see

Slo no p

We denote by Z the bre pro duct T X T X More concretely

N

Z f gB hB N X X j Adg u Adhug

This is the Steinbergvarietyof G It is clear that G acts on it

Wehave morphisms Z N and Z X X For w W put Z O where

w w

O is as in Then Z is the conormal bundle T see CG Prop

w w

O

w



Gf g Let N and let C C be two irreducible comp onents of Then Z

CC

C C is an irreducible subset of Z

Theorem Z has pure dimension dim X Its irreducible components are the closures



Z and also the closures Z

w

CC

This is proved in lo c cit Ch using symplectic geometry

For g let Z Z beitscentralizer in G and let r b e the rank of G

G

Corollary Let N The bre is connected Its irreducible components have

dimension dim Z r

See lo c cit

Let b e a set of representatives of the nilp otent orbits For N let b e the set

of irreducible comp onents of and let A b e the quotientofZ by its identity com

p onent Z This is a nite group which actson Theorem gives two descriptions

of the comp onents of Z

Corollary There is a bijection W A n

Example Let G SL C Then can b e identied with the set of partitions of n

n

ie with the set of Young diagrams with n boxes The groups Ax are all trivial If a

Young diagram then can b e viewed as the set of standard tableaux with shap e Corol

lary then leads to the RobinsonSchenstedcorrespondence between the symmetric group S

n

and pairs of standard tableaux of the same shap e See St

Corollary also givesamapofW onto the set of nilp otent orbits in g

The Steinb erg variety provides a bridge b etween Schub ert varieties and nilp otent elements

Constructions of algebraic ob jects

In the geometry of ag varieties Schub ert varieties and the Steinb erg variety alge

braic ob jects are hidden To bring these to lightvarious to ols from algebraic top ology are

used The following ob jects app ear

a The cohomology ring of a ag variety X Closely related is the Chow ring of X The

more general T equivariant cohomology and Chow ring of X are discussed in Bri

b Cohomology of the Steinberg variety Z

c Gequivariant sheaves on X intersection cohomology of Schub ert varieties

d Gequivariant coherent sheaves on X

e K theory and equivariant K theory of X and the Steinberg variety Z See CG Ch

and Gi no

I shall give some examples of constructions of algebraic ob jects

An elementary construction of the Weyl group W

Notations are as in If x y W the pro duct set G G is a union of nitely many G

x y z

Let W b e the set of these z

xy

Prop osition contains a unique minimal element for the Bruhat order namely xy

xy

Isketch a pro of If x S wehave fxy g if l xy ly and fx xy g if

xy xy

l xy ly whence the prop osition in this case and similarly in the case that y S

We use induction on l x Wemay assume that l x Cho ose s S such that l xs lx

Then G G G and

x xs s

if l sy ly

xy xssy

if l sy ly

xy xsy xssy

The prop osition will followifwe show that in the last case xsy xy Now xsy xsx xy

Let x s s s y t t t with s t S and a l xb l y s t s Then

a a b i j a

xy s s t t Ifl xsy lxy the strong exchange condition of Coxeter groups

a b

Hu p shows that x or y would have smaller length than a or bacontradiction Hence

l xy lxsy and the strong exchange condition implies that xy xsy

Remark The prop osition could b e viewed as a sp ecial case of the following result on the

Hecke algebra H of a Coxeter group WS dened in Hu Ch Let e b e the

x xW

standard basis of H denoted byT in lo c cit If x y W then all z W such that e

x z

o ccurs in e e with a nonzero co ecientare xy for the Bruhat order on W and z xy

x y

o ccurs see Sh

The prop osition can b e reformulated in terms of corresp ondences on X For x W the

Gorbit O is a lo cally closed corresp ondence on X If denotes the settheoretical comp os

x

ite of corresp ondences wehave

O O O

z x y

z

xy

It follows that O is the orbit of lowest dimension in the comp osite Also O  is the inverse

xy

x

of the corresp ondence O So the Weyl group W is pro duced by the settheoretical calculus

x

of corresp ondences

More formally the pro duct of corresp ondences is describ ed as follows Let ij

ij

b e the obvious maps X X X X X and denote byX X X

X X X X the map x y z x y y zIfS T are corresp o dences on X ie

subsets of X X their pro duct is

ST S T

A sheaftheoretical version of this formalism will pro duce the Hecke algebra of W see b e

low in Another version of the formalism in algebraic top ology leads to convolution in

BorelMo ore homologyusedinCG Ch for the Steinb erg variety Z of to construct

the group algebra QW lo c cit See also Gi

The Chow ring of X

L

r r

The Chowring A X A X whereA X is a quotient of the free group generated by

the irreducible subvarieties of X of dimension dim X r seeFu p For w W let

lw lw



A X b e the image of the Schub ert variety S

w w

Prop osition i A X is a freeabelian group with basis

w w W

ii If l x l w we have

w w x xw w

 

See De Part ii follows from lemma of It provides a duality pairing on A X

In we asso ciated to a character of T a line bundle LonX Letc A X be

its Chern class Let S b e the graded symmetric algebra of the character group X T it

is acted up on by the Weyl group W For anyroot R we dene an endomorphism d of

degree ofS by

d u u s u

If is a simple ro ot and s s we put d d

s

The maps d were intro duced in BGG BGG and in De They have the prop erties of the

following lemma

Lemma i d

ii Let s s s bea reduceddecomposition of w W Thend d d depends

r w s s

r



only on w

The lemma could b e proved algebraically but a pro of also comes out of the pro of of the next

theorem

Wehave a graded ring homomorphism c S A X Denote by I the ideal in Q S

generated by the nonconstant homogeneous W invariant elements Let S Z b e the

augmentation map

P

d u Theorem i cu

w ww

w W



ii c induces an isomorphism of graded algebras Q S I Q A X

In the pro of of i given in De rst the Chow group of the BottSamelson variety asso ciated

to a reduced decomp osition of w is determined

Wehave a cycle map from the Chowgroupof X to the cohomology of X doubling de

grees Using it one obtains Borels theorem

Corollary H X Q Q S I

By the corollary the Weyl group W acts on H X Q As a W mo dule it is the regular

representation of W

The action of W on the cohomology H X Q can also b e describ ed in a more direct manner

The canonical map GT GB X makes GT into a lo cally trivial vector bundle over X

and hence induces an isomorphism of cohomology groups

H GT Q H X Q

Now W acts on GT via right action of the normalizer N of T Hence W acts on the co

homology group of the lefthand side and hence on the one of the righthand side It can b e

shown that this action is the same as that of the theorem

Part i of the theorem solves theoretically the problem of Schub ert calculus to describ e

the intersection of Schub ert varieties ie to determine the multiplicative structure of A X

It follows from ii that cS has nite index in A X Butc need not b e surjective see De

The equivariant cohomology of X is discussed in Bri

A construction of the Hecke algebra of W

Recall that the Hecke algebra H of W is a free mo dule over Zt t with a basis e

w w W

The multiplication is determined by the rules

e e e if l sw lw

s w sw

e e t e t e if l sw lw

s w sw w

In particular e t e t Sp ecializing t we obtain the group algebra ZW see

s

s

Hu Ch

I rst sketch a sheaftheoretical construction of H via corresp ondences on X see Sp no

One works with sheaves of Qvector spaces on an algebraic variety Y which are con

structible relative to some stratication a nite decomp osition into lo cally closed irreducible

pieces Y Y ie sheaves which are lo cally constant along the strata Y and whose stalks

i i

are nitedimensional Wework with Y X X and the stratication dened by the orbits

O For w W let A b e the sheaf on X X whose restriction to O is the constant sheaf

w w w

Q and whose stalks at the p oints outside O are zero A will pro duce the basis element

w w

e of H

w

To make the calculus of corresp ondence work we need however to enlarge the category of

b

sheaves on a variety Y to the b ounded derived category D Y whose ob jects are complexes

c

i

A of Qsheaves such that the cohomology sheaves H A are constructible as b efore and

vanish if jij is large We will not go into the denition of such categories their morphisms

and the denitions of the functors b etween such categories asso ciated to morphisms see Bo

Ch V for these matters see also Gi no

i

We identify a sheaf S with the complex A such that A S A for i

b

In our situation we denote by E X X the full sub category of D X X whose ob

c

jects have cohomology sheaves which are lo cally constant and then they are in fact constant

along the Gorbits O For A E X X dene the element hA H by

w

X X

i i

dim H A t e hA

w w

w W i

i i

where H A is the stalk of H Aata pointofO these stalks are all isomorphic

w w

For example hA e Ifs S then O O O Let A b e the sheaf whichisQ on

w w s s e

s

e O and outside this set Then hA

s w

s

For A B E X X dene their convolution pro duct ABby

AB A B

Here A B is the outer tensor pro duct of A and B is the pullback morphism asso ci

ated to the morphism of and is the direct image morphism asso ciated to the prop er

morphism It is to dene such direct image functors that one needs complexes of sheaves

i

Prop osition Let s S Let A E X X be such that H A for al l even i or

for al l odd i Then A A is a complex with the same properties and hA A e hA

s

s s

This is proved by elementary means in Sp

The prop osition shows that H describ ed in terms of its the generators e s S can

s

b e constructed via the sheaf theoretic calculus of corresp ondences on the ag variety But a

cleaner construction is obtained in the context of the theory of p erverse sheaves or intersec

tion cohomology

Construction of the Hecke algebra of W in intersection cohomology

b

Let Y b e an irreducible algebraic variety In the category D Y one has a duality functor

c

D with go o d prop erties see BoVx see also Gi no

b

Y such that A perverse sheaf on Y is a complex A in D

c

i i

dim supp H A i dim supp H DA i

For the theory of p erverse sheaves we refer to BBD see also Gi no We only mention

a few essential p oints The p erverse sheaves on Y form an ab elian category P Y all of

whose ob jects have nite length The irreducible ob jects are as follows For each irreducible

subvariety Z of Y and each irreducible lo cal system on a smo oth op en piece U of Z there

is a unique irreducible p erverse sheaf I ICY L supp orted by Y such that the restriction

of I to U is the complex Ldim Z L in dimension dim Z and in the other dimensions

This is the intersection cohomology complex for Y and LWehave

i

dim supp H I i if i dim Z

If L is the constant sheaf Q then the hyp ercohomology of I is the intersection cohomology of

Z It satises Poincare duality

If Z is smo oth then ICY Q Qdim Z and intersection cohomology coincides with ordi

nary cohomology

b

A complex A D Y is semisimple if it is a direct sum of shifted irreducible p erverse

c

sheaves I

i

A I n

i i

Apowerful result is the decomposition theorem

If f Y Z is a proper morphism and if A Ob P Y is irreducible then the direct image

f A is semisimple

See BBD nos

We return to the Hecke algebra There is a ring automorphism h h of H mapping t

to t and such that e e One shows that for w W there is a unique element c H

w  w

w

with c c of the form

w w

X

lw

c t P t e

w xw x

xw

where the P b eing p olynomials with P deg P lw l xif x w The c

xw ww xw w

form a basis of HtheKazhdanLusztig basis intro duced in KL see also Hu I I

We shall now connect the KazhdanLusztig elements with the intersection cohomology com

O QoftheGorbit closures in X X For s S wehave I A plexes I IC

w s w

s

where A is as in Also h is as in

s

Theorem Let x w W

i hI c

w w

ii hI I c c

x w x w

See Sp no It follows in particular that

X

i i lw

dim H I t t P t

w x xw

from whichwe see that I satises the parity condition

w

i

H I if i l w mod

w

The theorem shows that the Hecke algebra H can b e recovered from the p erverse sheaves I

w

There is a variant of this construction whichworks with the p erverse sheaves on G Consider

the p erverse sheaves on G whichareB B equivariant the action b eing b b g bg b

Denote by K the Grothendieck group of the category of these p erverse sheaves It has as

basis the classesc of the intersection cohomology complexes I IC G Qw W To

w w w

L

I h we asso ciate the element a semisimple complex A

w w

X

h

w

hA t c Zt t K

w

The pro duct map G G induces a prop er morphism G G GLetx y W The

B

outer tensor pro duct of I and I is the pull back of a shifted p erverse sheaf A on G G

x y B

By the decomp osition theorem the direct image complex A is semisimple

Dene an algebra structure on H Zt t K by

c c h A

x y

Prop osition The Zt t algebra H is isomorphic to H the isomorphism sending c to

w

c

w

A pro of of the prop osition is contained in MS In that pro of it is shown that the

inductive formulas of KL for the KazhdanLusztig p olynomials can b e recovered from the

morphism x b eing a simple reection

The construction using G can b e generalized Instead of I one can work more generally

w

G L where L is a B B equivariant lo cal system on the smo oth with a p erverse sheaf IC

w

op en piece G of G there exist nonconstant lo cal systems of this kind this is not the case

w w

for O which is simply connected One is led to a more general kind of Hecke algebra see

w

lo c cit

The Steinb erg variety

In the Steinberg variety Z of the of the Weyl group W is hidden

In CG Ch convolution on the top BorelMo ore homology of Z is used to reveal that

representation theory This is also discussed in Gi no

Another metho d to do this uses Lusztigs observation that there is an action of the Weyl

group W on the direct image complex Q see eg Sp no where is as in It

follows that there is an action of W W on Q where Z N is the morphism of

Hence W W op erates on the cohomology with compact supp ort H N QH Z Q

c c

The irreducible comp onents of Z are parametrized by the elements of W see they all

have the same dimension d where d dimX l w The cohomology classes z which

w

d

they determine span the top cohomology group H Z Q whichthus has dimension jW j

c

Prop osition i H Z Q is isomorphic to the inducedgraded W W module

c

W W

Ind H X Qd

W

d

Z Q is isomorphic to QW under twosided ii In particular the W W module H

c

action

In i the W action on H X Qisasin For a pro of see Sp no A concrete

d

description of the W W action on the basis z of H Z Q do es not seem to b e known

w

c

Equivariant K theory on Z is used to construct the ane Hecke algebra asso ciated to W

and its representations We shall not go into this See CG Ch and Gi no

Coherent cohomology

Assume G to b e semisimple and simply connected The character group X T is the weight

lattice P of RLetP P b e the set of dominantweights For P wehave the line

bundle L of

i

Theorem Let w W P Then H S L for i

w

See Ra where more general results are discussed See also Littelmanns contribution ref Littel

mann

In the situation of the theorem H S L is a nite dimensional complex vector space

w

V on which B acts In particular T acts Let ZP b e the group ring of P It has a basis

e P with e ee and the Weyl group W acts on it

P

Put charV dim V e where V is a weight space for T This elementofZP is

P

the character of V

Let b e a simple ro ot and put s s Dene an endomorphism D of ZP by

s

u su

D u

s

e

Lemma i D D

s

s

ii Let s s s beareduceddecomposition of w W Then D D D depends

r w s s

r



only on W

This is a multiplicative analogue of the lemma of With these denitions wehave De

mazures character formula

Prop osition If P then charH S L eD e

w w

Here is half the sum of the p ositiverootsofR See De Ra p and Littelmanns

contribution ref Littel

mann

For w w wehave S X In that case H X L is Gmo dule which is irreducible

w

with highest weight w The prop osition then gives Weyls character formula

Generalizations of Schub ert varieties

Schub ert varieties asso ciated to KacMo o dy algebras

Let g b e a KacMo o dy algebra It is an innite dimensional Lie algebra asso ciated to a

generalized Cartan matrix M see Ka M denes a Coxeter group WS the Weyl group

of g

One can asso ciate to g Schub ert varieties S indexed by the elements of W They have

w

prop erties similar to those of the ordinary Schub ert varieties they are pro jective algebraic

varieties of dimension l w the length of w relativetoS and if x w then S is a closed

x

subset of S

w

In this generalitySchub ert varieties are dened and studied in Ma in arbitrary characteris

tics The denition involves representation theory of g These Schub ert varieties are normal

pro jectivevarieties In lo c cit a generalization is proved of the theorem of and of

Demazures character formula See also Littelmanns contribution refLittelmann

The Cartan matrix M denes an innite dimensional group B an analog of the Borel group

of In lo c cit analogues are constructed of the closures G of These are non

w

no etherian ane schemes B w over Cwithatwosided B B action as in the case of G

w

B op erates lo cally freely on the right and S B w B

w

I will not go further into this general case But to make things more concrete I shall briey

discuss one example in a sp ecial case The asso ciated KacMo o dy algebra would b e an ane

one however it will not app ear in the example

An example

Let A Ct b e the ring of formal p ower series and F Ct its quotient eld Let G be

a semisimple simply connected linear algebraic group over C The notations are as in

We denote by W the ane Weyl group of R the semidirect pro duct of W and the lattice of

coro ots

Put G GF the group of F valued p oints of G The obvious homomorphism A C

induces a group homomorphism of the group GAof Avalued p oints to GLetB b e the

inverse image of B under this homomorphism This is an Iwahori subgroup of G Asin

wehave subsets G B w B of G where w WWehave a Bruhat decomp osition

w

a

G G

w

w W

by the BruhatTits theory see eg Ti p

S

X Then S is the underlying set of a Schub ert Put X G B X G B and S

w w w w w

xw

variety It can b e given the structure of pro jective algebraic varietyWe shall indicate how

this can b e done for G SL following KL x

n

Let G SL soG SL F Wetake B to b e the subgroup of SL A whose elements

n n n

sp ecialize to an upp er triangular matrix for t the upp er triangular group is a Borel

subgroup of G

n n

Let V F Then A is a lattice in V over A ie a free Asubmo dule of V containing a

basis The group G op erates on the set of lattices We identify G B with the set of sequences

of lattices

t

n

such that dim j n and that G

j j

C

For i let X b e the set of such sequences with

i

i i

t t

n

Then X is an increasing sequence of subsets of X with union X to see this use that for

i

a b

any lattice there exist a b with t t

i i

Fix i and put Z t t this is a nite dimensional vector space on which t acts as

a nilp otent endomorphism Then X can b e identied with the set of sequences of tstable

i

subspaces

Z Z Z tZ

n

of Z such that dim Z Z Z tZ This set has a structure of pro jectivevariety

j j n

b eing a closed subset of a pro duct of whence such a structure on X

i

Then X is an inductive limit of pro jectivevarieties Any X is contained in an X whence a

w i

structure of pro jectivevariety on the Schub ert varieties

Spherical varieties

The notations are as in A spherical variety for G is an algebraic variety X with a

Gaction such that the Borel group B has nitely many orbits see Bri

In particular G has nitely many orbits A Gorbit is a spherical homogeneous space GH

where H is a closed subgroup such that the double coset space B nGH is nite Sucha

subgroup is called spherical It is clear that H is a spherical subgroup if and only if H acts

on X GB with nitely many orbits

Examples X GB By Bruhats lemma B is a spherical subgroup The

B orbits are indexed by W and the Schub ert varieties in X are the orbit closures

The symmetric case see Sp Let be an involution automorphism of order two of

the algebraic group G The xed p ointgroupK is reductiveandX GK is spherical It

is an ane algebraic varietycalledasymmetric variety

Assume that H G G and let b e the p ermutation automorphism x y

y xx y G The xed p ointgroupK of is Gimb edded diagonally in H and

HK G B B is a Borel subgroup of H ItactsonHK G byb b g bg b By

Bruhats lemma the B B orbits on HK are the G of Werecover example

w

The unip otent part U of B is another example of a spherical subgroup by Bruhats

lemma and GU gives an example of a spherical variety which is not symmetric as U is

not reductive Another such example is provided by H TU U where U U is the

commutator subgroup of U That H is spherical follows from Bruhats lemma and the obser

vation that T acts by conjugation on UU U with nitely many orbits the quotient b eing

Q

isomorphic to U

D

An instructive example of a symmetric varietyisG SL with the inner automor

phism Inti i Let T b e the diagonal torus and B the upp er triangular subgroup Then

K T is spherical in this case GB is the pro jectiveline P and T acts on it with three

orbits namely the closed orbits fg fg of dimension and the op en orbit P f g of

dimension

Let G PSL and let is the automorphism of G induced by The xed p oint group K

of is nonconnected it is the image in G of the normalizer N of T B denoting the image

of B wehave G B P

K has twoorbitson P viz the reducible orbit f g and its complement

The set of orbits

Let X GH b e spherical homogeneous Denote by V the set of B orbits in X This is a

nite set with combinatorial prop erties resembling those of a Weyl group

a V carries a partial order

If v w V dene v w ifv w compare with the Bruhat order of W see Since X is

irreducible V has a unique a unique maximal element v the op en B orbit

max

b The minimal elements of V are the closed orbits They are of the form BgHH where

BgH is closed in G Hence Hg BB is closed in GB which means that H g Bg is a

Borel subgroup of H This implies that all closed orbits have the same dimension We dene

a length function l on V by l v dimv dim v where v is minimal Then l is strictly

monotonic and l v if and only if v is minimal

There maybeseveral minimal orbits as example of shows

Wemay identify V with the set of K orbits in GB The order and the length of V can

also b e dened via the latter set in the same way

To analyze further the combinatorial structure of V we use the parab olic subgroups P of

s

Gaswe did in to establish the combinatorial description of the Bruhat order of W We

have to study the map P v X dened bytheGaction

s

A basic construction

Let s S b e a simple reection Wehave the parab olic subgroup

P P G G

s e s

Let v V then Pv is an irreducible lo cally closed subvariety of our variety X whichisa

nite union of B orbits There is one orbit whichisopeninPv and has maximal dimension

We denote it by msv

The P action on X denes a morphism

P v Pv

B

Lemma We have the fol lowing possibilities

I Pv v

IIa Pv v msv v is closedin Pv is generical ly bijective and dim msv dim v

IIb Pv v v v is open in Pv dim v dim v and v is in case IIa

IIIa Pv v v msv v v both v and v are closedinPv and dim msv

dim v dimv

IIIb Pv v v v both v and v are closedin Pv v v dim v dim v dim v

and v v areincase IIIa

IVa As in case IIa but is generical ly a double cover

IVb As in case IIb but v is in caseIVa

Let x v and let P b e the isotropy group of x in P Then Pv PP There is a bijection

x x

of the nite set of B orbits in Pv onto the set of P orbits on B nP P Sowe are led to

x

subgroups of the automorphism group PGL of P whichhave nitely many orbits Analysis

of the p ossible cases leads to the lemma See MS

In the case that X GB the cases I I I and IV do not o ccur which simplies matters con

siderably

For v V s S we dene sv msv in case IIa sv v in cases IIb and IIIa

and sv v in all other cases The notation is suggestiveofaW action We shall see b elow

that there is indeed a W action on V such that the simple reections act as describ ed

Denote by the canonical map B BU The restriction of to T is bijective We

identify T with BU

For x X let B b e its isotropy group It is easy to see that for all x v the image

x

B is the same Write T for this image Let s s D b e as b efore Denote by

x v

the coro ot of a homomorphism of GL into a onedimensional subgroup of T Recall that

is a nontrivial character of T

The following lemma describ es the relations b etween T and T in the cases of the pre

v

msv

vious lemma where msv v ie the cases I Ia I I Ia IVa

Lemma Assume msv v

i Case IIa T sT

v

msv

ii Case IIIa T T Ker and T T Ker

v v

msv

iii Case IVa Im T T fg and

v

msv

T Ker T Ker

v

msv

See MS The lemma implies that in case I Ia dim T dimT and that dim T

v

msv msv

dim T in the cases I I Ia and IVa

v

The monoid M W

Let M M W b e the monoid with elements mw indexed by the elements w W such

that

msmw msw if l sw lw msmw mw if l sw lw

The existence of M follows from Hu Theorem p M is generated bythemss S

sub ject to the relations ms ms and the braid relations of the Weyl group

msmtmt mtmsmt s t S

the numb er of factors on b oth sides b eing the order of st

If s s s is a reduced decomp osition of w W we put mw ms ms This

r r

is welldened

Lemma The map s v msv extends to an action of M W on V

This follows from P P P and the braid relations P P P P P P

s s s s t s t s t

Remark The D of provide another example of an M action

w

Wehave the following prop erties

Prop osition Let v w V s S

a v msv and l msv l v

b If msv msw and l v l w v w then w sv Moreover vmsv and we

have case IIIa

c If v w then msv msw

d Put V fx V j x v gIfmsv v then

v

V fmsxxsxg

msv

xv

a is a consequence of lemma of and c d follow from the equality P v Psv

s

b also follows from lemma of By a wemay assume that vmsv w msw

The lemma then shows that v P msv Lo oking at the p ossible cases b follows

s

A reduceddecomposition of v V isapairv s of a sequence v v v v of dis

r

tinct elements in V and a sequence s s s in S such that v is minimal v v and

r r

v ms v i r

i i i

In the symmetric case example of all v have a reduced decomp osition see b elowin

Examples show that this is not generally true

The symmetric case is analyzed in RS Some of the results carry over to the general case I

mention the following generalization of the combinatorial description of the Bruhat order of

Let v s b e a reduced decomp osition of v A subexpression of v s is a sequence x

x x x inV with x v such that for i r wehave one of the following alter

r

natives x x x x lx l x and x s x l x l x

i i i i i i i i i i i

and x ms x Wecall x the nal term of x

i i i r

Prop osition Let v s beareduceddecomposition of v If x v there is a subex

pression of v s with nal term x

This is a consequence of prop erty d see RS

The existence of a reduced decomp osition of v V can b e formulated as follows there

exist w W and a minimal element v V with mw v v lw l v Part i of the

following prop osition is a dual result which is generally true

Prop osition i Let v V Thereisw W with mw v v lw l v l w

max max

ii v has a reduceddecomposition

max

As in v is the maximal elementofV

max

v v Gv SinceG is generated by the parab olic sub Assume that v v Then

max max

groups P s S there must b e s S with dim P v dim v ie with msv v Then i

s s

follows by descending induction on l v Applying i with v minimal we obtain ii

Let v s b e a reduced decomp osition of v By prop osition wehave for any v V

max

a sub expression of v s with nal term v We obtain a weak sort of reduced decomp osition

of v

Finallywe mention another example of a set V with an M action where we also have

the prop erties of prop osition Namely the set V of involutions in the Weyl group W The

action of M is dened bythe msv for s S v V These are as follows msv sv s if

l sv s lv msv sv if l sv lv and sv vsand msv v in all other cases l is

the length function on W This example slightly generalized is discussed in RS Compl

no

Hecke algebra representation asso ciated to a spherical variety

The notations are as b efore The ideas of the constructions of the Hecke algebra of W with

to ols from algebraic top ology discussed in can also b e used in the context of spherical

varieties Write now Y for the ag variety and as b efore X for a spherical variety Then

G acts with nitely many orbits in Y X which are parametrized by our set V Via a

b

calculus of corresp ondences one is led to consider complexes M in D Y X whose coho

c

b

mology is lo cally constant along the Gorbits in Y X IfA D Y Y is as in a

c

complex with cohomology constant along the Gorbits O one denes as in a pro duct

w

b

Y X AM D

c

We are thus led to a representation of H on a free Zt t mo dule M which has a basis

e indexed by pairs of an orbit v V and a B equivariant lo cal system on V in the

v

sitiuation of lo cal systems did not app ear since wewere dealing with simply connected

orbits

Also in the present situation we do not haveavailable the prop osition of Tocircumvent

this diculty one passes to a situation over a nite eld where the Frob enius action can b e

exploited This was rst carried out for the case of symmetric varieties by Lusztig and Vogan

in LV The case of spherical varieties is dealt with in MS

Constructions in intersection cohomology as those of can also b e carried out in the con

text of spherical varieties see lo c cit I now describ e the structure of the representation of

H which one obtains The results are for the situation over C But as already p ointed out

they are obtained via a passage to nite elds

First some results on B equivariant lo cal systems on an orbit v These are are classied

by the character group of the nite group B B where x V Since B U is a

v x x

x

unip otent algebraic group over C it is connected and it follows that B B T T hence

x v

x v

is a nite ab elian group uniquely determined by v We describ e the connection b etween

v

and in the cases I Ia I I Ia IVaof

msv

Lemma iIncase IIa there is an isomorphism s of onto

v

msv

ii In case IIIa thereisaninjection

vs v

msv

iii In case IVathere isaninjection of in a quotient of by a subgroup

vs v

msv

msv

of order

This follows from lemma of For case I Ia this is clear In case I I Ia wehaveby part ii

of that lemma T G whence T T So

v m v

v

T T Ker T T T

v v

v v v

and there is a surjective homomorphism T T T T This implies ii The

v

msv

v

msv

pro of of iii is similar

The action of H on M is describ ed by giving the pro ducts e e v V of a

s v v

generator of H and a basis elementofM They are listed in the next prop osition The

notations are explained b elow The cases are as in lemma of

Prop osition Thereisa representation of H on M such that the products e e areas

s v

fol lows

Case I t e

v

Case IIa e

sv s

Case IIb t e t e

v sv s



Case IIIa e e

sv s

msv

   

Case IIIb t e t e e or e

v v

v v

 

Case IVa e e e

v

msv msv

  

Case IVb t e e t e or e

v v

v v

In case I I Ia wehave written and in case IVa and are the elements of

vs

msv

which pro ject onto the element of

vs

vs



In case I I Ib wehaveby the previous lemma a injections into The rst alter

v s v

v s

   

nativeprevails if and only if there are with Incase

v v v s v s

IVb and are the elements of pro jecting on an element if suchan exists



v

v s

Otherwise wehave the second alternative

The prop osition is a consequence of the results of MS There similar formulas are

established in a more general situation for more general lo cal systems on the orbits How

ever in lo c cit one works not over C but over the algebraic closure of a nite eld F over

q

whicheverything is dened The Frob enius action on cohomology comes into play In order

to deduce the prop osition one passes bya wellknown reduction pro cedure see eg BBD

no from C to a suitable nite eld F to which the results of MS can b e applied

q

They imply that wehave a representation of the sp ecialized Hecke algebra H where t is

q

replaced by q on the similarly sp ecialized mo dule M The same is true with q replaced

q

n

by q for all n Now observe that the assertion of the prop osition is equivalent to the

vanishing of a numb er of p olynomials in t which expresses the dening relations of HBy

n

lo c cit these p olynomials have innitely many zeros namely all p owers q Hence they

vanish and the prop osition follows

As a consequence of the prop osition weprove the existence of the Weyl group action on

V announced in

Corollary There exists an action of W on V such that with the notations of lemma of

we have sv msv in case IIa sv v in cases IIb and IIIa and sv v in al l other

cases

Sp ecializi ng t to in the prop osition we obtain a representation of W in a free Zmo dule F

spanned by basis elements f v V Let F b e the submo dule of F spanned by

v v r

the f with dim T r where T is as in They dene a ltration of F and it follows

v v v

from lemma of and the formulas of the prop osition that W stabilizes the ltration Let

F b e the asso ciated graded mo dule It carries a representation of W The basis f denes

v

f of F and the formulas of the prop osition show that for s S wehave a basis

v

s f f



v sv

f for some This implies that if s s s is a reduced decomp osition of w W w

l

v

is of the form

f

s s v



l

It follows that wv s s v denes an action of W on V which is as asserted

l

The W action is due to Knop Kn the results established there are more general

The pro of of the corollary also gives that for w W wf is of the form w f

v

v wv

with w indep endentof Wehavefor x w Wv V

v

xw x w

v wv v

Let W b e the isotropy group of v in W The preceding formula shows that the restriction of

v

to W isacharacter of W with values in fgIfs S W then we are in one of the cases

v v v v

IVa I I Ib IVb and by the prop osition s equals in the rst case and in the other cases

v

Symmetric varieties

From nowonwe assume that our spherical variety X is symmetric see example of

So there is an involution of G with xed p ointgroupK such that X GK By results

of Steinb erg Stpwemay assume that B and T are xed by Then op erates on R

and W

The set V of B orbits in X GK can now b e describ ed more concretely Put V fx

G j xx N g This set is acted up on by N and in particular by T on the left and by

K on the right

Lemma i The map x BxKK induces a bijection of T nV K onto V

ii The N action on V induces the W action on V of

iii The map x xx T induces a map V W whose image lies in the set of twisted

involutions I fw W j w w g

For i and ii see Sp x The niteness of V also follows For iii see RS no That

the action of W is the one of follows from RS

In the sequel we identify V and T nV K

Combinatorial questions

We describ e the cases of the lemma of in terms of ro ots Let v V let b e a simple

ro ot and put s s Put v w Consider the cases of the lemma In cases I Ia I Ib

wehave w R fg resp ectivelyw R fg We then say that is complex

relativetov In cases I I Ia and IVa w Then is imaginary In the cases I I Ib and

IVb w and is real

For R let G b e the three dimensional subgroup of G generated by U and U In cases

I I I I and IV stabilizes x G xwhere x V represents v Case I and I I I are distinguished

by the prop erty that acts trivially resp ectively nontrivially on x G x Then is com

pact imaginary resp ectively noncompact imaginary These denitions can b e given for all

ro ots of R not necessarily simple the notions are current in the theory of real Lie groups

We use the same terminology for the corresp onding reections

The R which are imaginary relativetov ie satisfying w form a closed

subsystem R of R Dene R ZZ by if and only is compact imaginary

i i

Then is a grading mo d of R ie and if R

i i

So our orbit v gives rise to algebraic ob jects an involution w of the ro ot system R and

a grading mo d on the corresp onding imaginary ro ots The connection of orbits with such

algebraic ob jects was rst p ointed out byVogan in Vo see also Sp

Let v V Then v has a reduced decomp osition see RS the pro of uses a reduced

decomp osition in W of v Let

v v s s

r r

b e one For i r wehave ms v v v see For each i wehave one of

i i i i

the cases I Ia I I Ia IVaoflemmaofLeta b c b e the resp ectivenumb ers of such iso

a b c r l v

Lemma i a l v l v

ii a b and c areindependent of the choiceofthereduceddecomposition of v

is as in lemma i shows that a is indep endent of the reduced decomp osition i follows

from RS The indep endence of b and c was proved byRichardson unpublished

Finallywe mention the following result proved in RS

Prop osition There is a bijection of the set of W orbits in V onto the set of K orbits of

stable maximal tori of G

In lo c cit no this is made more precise

Geometric questions

We shall nowwrite X for the orbit v The closures S v v V are generalizations of

v v

Schub ert varieties We briey review some facts ab out their geometry First notice that the

X need not b e isomorphic to an ane space and need not b e simply connected in contrast

v

to the case of Bruhat cells This one sees already in example of with G SL

where the op en orbit is isomorphic to C fg

The BottSamelson varieties of have an analogue in the symmetric case Let v sbe

a reduced decomp osition of v V recall that in the symmetric case all elements of V p ossess

reduced decomp ositions With notations as in dene

Z P P P X

s B s B B s B v

v s

r

r   

cv

Prop osition Thereisaproper surjective morphism Z S Itsdegreeis

v

v s

cv is the integer c of lemma of The pro of is similar to the pro of of prop ositon of

There is a bijection of the set V of B orbits on X GK onto the set of of K orbits in

X B nG K acting on the right see We write these orbits as X and their closures as

v

S The latter are pro jectivevarieties With the notations of we put C B nB wG

v w

Lemma of implies that these are the T stable curves in X T acting on the right

Now assume that is an inner automorphism Intt with t T Then T is a maximal

torus of K and T op erates on the varieties X In Sp the smo othness criterion of Carrell

v

and Peterson see the lemma of is applied in the present situation The result is as follows

Let b e a noncompact imaginary reection The group G see is acted up on

nontrivially by see Let R b e a ro ot with r Then T Im is a

maximal torus in G let n G b e an element in the normalizer of T There is x G with

xx n Then x lies in the set V of and denes an element v of V Infactv is

determined by the prop erty that S intersects C in an op en subset Let v B nBK

w

wv

this is a closed orbit

Prop osition Let v V w W be such that wv v The number of noncompact imagi

nary with wv r v is at least l v Equality holds if S is smooth in B w

v

Since S is smo oth if and only if S is the prop osition gives a smo othness criterion for the

v v

latter in the case that the involution is inner

In Bri more general results are established as an application of results from equivariant

cohomology See lo c cit

In contrast to Schub ert varieties the S are not always normal Foracounterexample in

v

Sp see BE

Let v V and let see The denition shows that in the present case we

v

have Kerw TImw T whichshows that now is an elementary ab elian

v v

group

is a one dimensional lo cal system on X LetI ICS b e the intersection

v v v v

cohomology complexes irreducible p erverse sheaves determined byv whose denition

was recalled in

Theorem I satises the parity condition ie

v

i

H I if i dim S mo d

v v

This was rst proved in LV using representation theory A geometric pro of follows from

the results of MS no where this is established over the algebraic closure of a nite

eld An imp ortant geometric ingredient is the existence in the case of symmetric varieties

of a transverse slice at a p ointofX inside S if x v see MS compare with part

x v

iii of lemma

The theorem leads to generalized KazhdanLusztig p olynomials in the present situation Let

ilv

x V x v The restriction of H I toX is a direct sum of lo cal systems in

v x x

i

For let m b e the multiplici tyof in that restriction The p olynomials are given

x

x v

by

X

i i

m t P t

x v

x v

ix

The elements

X

dim v

c t P t e

v x v x

xv

x

of the mo dule M of are analogues of the elements c of the Hecke algebra H of

w

A generalization of the Steinb erg variety

We still assume X GK to b e symmetric with G semisimple and simply connected The

involution acts on the Lie algebra gLetp be the eigenspace It is nonzero

As in let Ng b e the the variety of nilp otent elements of gWehave the resolution

T GB N theorem of Put N Np Z Z N So

Z f gB N GB j Adg ug

If G H H with permuting the factors Z is the Steinberg varietyofH

Wehave morphisms Z N and Z GB For v V put Z Kx BB

v

where x V represents v Let N and let C be a componentof Put Z

C

Kf g C Wehave the following generalization of the prop osition of

Prop osition Z is of pure dimension d dimGB Its irreducible components are

the closures Z and also the closures Z

v C

The pro of uses the following lemma due to Kostant and Rallis Ko Prop p

Lemma dim Z dim Z is independent of for p

G K

Remarks Z is isomorphic to the conormal bundle of GB along Kx BB

v

Let b e a set of representatives of the K orbits in N it is nite As in denote

by the set of irreducible comp onents of N Now let A b e the quotient

Z Z We obtain a generalized RobinsonSchensted corresp ondence a bijection

K K

a

A n V

In particular we obtain a map of V to the set of nilp otent K orbits in p

Z Q A full gen As in wehave an action of the Weyl group on the cohomology H

c

eralization of the prop osition of do es not seem to b e known Only the action of W on the

d

top cohomology H Z Q has b een describ ed in Sp via a passage to nite elds

c

This go es back to Rossmann who proved similar results by analytic metho ds in the context

d

of real Lie groups see Ro To describ e the W mo dule structure of H Z Qwe

need some more notation

Let v V and let x V representitThen x Tx is a stable maximal torus in G Put

W xK x N xK x T This is a nite subgroup of W uniquely determined by v

v

Let b e a set of representatives of the K orbits in V by the prop osition of these orbits

are in bijection with the K conjugacy classes of stable maximal tori in G For v V let

again W b e its isotropy group in W

v

Prop osition There exist characters of W with values in fg such that the W

v v

d

module H Z Q is isomorphic to

c

M

W

Ind

v

W

v

v

The are describ ed in Sp p actually they coincide with the characters intro duced

v

at the end of but this is not discussed in lo c cit

References

BE D Barbasch and S Evens K orbits on Grassmannians and a PRVconjecture for

real groups J Algebra

BBD A A Beilinson J Bernstein and P Deligne Faisceaux p ervers Asterisque no

BGG I N Bernstein I M Gelfand and S I Gelfand Schub ert cells and cohomology

of the spaces GP Russian Math Surveys

Bo A Borel et al Intersection cohomology Birkhauser

Bo A Borel Linear algebraic groups nd ed Graduate Texts in Math

SpringerVerlag

Bou N Bourbaki Group es et algebres de Lie Hermann

Bri M Brion Spherical varieties An intro duction In Top ological metho ds in alge

braic transformation groups p Birkhauser

Bri M Brion Equivariant cohomology and equivariantintersection theory these Pro

ceedings

Bri M Brion Rational smo othness and xed p oints of torus actions preprint

Bry R Brylinski these Pro ceedings

Ca J B Carrell The Bruhat graph of a Coxeter group a conjecture of Deo dhar

and rational smo othness of Schub ert varieties in Pro c Symp Pure Math p

Amer Math So c

CG N Chriss and V Ginzburg Representation theory and complex geometry

Birkhauser

De M Demazure Desingularisation des varietes de Schub ert Ann Sci Ec Norm

Sup

De M Demazure Invariants symetriques entiers des group es de Weyl et torsion Inv

Math

Fu W Fulton Intersection theory Springer Verlag

Gi V Ginzburg Geometric metho ds in representation theory of Hecke algebras and

quantum groups these Pro ceedings

Hu J E Humphreys Linear algebraic groups nd ed Graduate Texts in Math

SpringerVerlag

Hu J E Humphreys Reection groups and Coxeter groups Cambridge Univ Press

Ka V Kac Innite dimensional Lie algebras Birkhauser

KL D Kazhdan and G Lusztig Representations of Coxeter groups and Hecke alge

bras Inv Math

KL D Kazhdan and G Lusztig Schub ert varieties and Poincare duality in Pro c

Symp Pure Math p Amer Math So c

Kn F Knop On the set of orbits for a Borel subgroup Comm Math Helv

Ko B Kostant and S Rallis Orbits and representations asso ciated with symmetric

spaces Amer J Math

Ku S Kumar The nil Hecke ring and singularityofSchub ert varieties Inv Math

Li P Littelmann these Pro ceedings

LV G Lusztig and D A Vogan Singularities of closures of K orbits on ag manifolds

Inv Math

MS J G M Mars and T A Springer Character sheaves in Asterisque

MS J G M Mars and T A Springer Hecke algebra representations related to sym

metric varieties to app ear in J Repr Theory

Ma O Mathieu Formules de caracteres p our les algebres de KacMo o dy generales

Asterisque no

Ra A Ramanathan Frob enius splittings and Schub ert varieties in Pro c Hyderabad

Conf on alg groups p Madras

Ro W Rossmann Nilp otent orbital integrals in a real semisimple Lie algebra in

Actes du collo quium en lhonneur de Jacques Dixmier p Birkhaser

RS R W Richardson and T A Springer The Bruhat order on symmetric varieties

Geom Ded Complements ibid

Sch H Schub ert Kalkul der abzahlenden Geometrie reprinted Springer

Verlag

Sh Shi JianYi A result on the Bruhat order of a Coxeter group J Algebra

Slo P Slo dowy Simple singularities and simple algebraic groups Lect Notes in Math

no SpringerVerlag

Sp T A Springer Quelques applications de la cohomologie dintersection Sem Bour

baki no in Asterisque

Sp T A Springer Some results on algebraic groups with involutions in Al

gebraic groups and related topics p Adv Studies in Pure Math

KinokuniyaNorthHolland

Sp T A Springer On representations of Weyl groups in Pro c Hyderabad Conf on

alg groups p Madras

Sp T A Springer A combinatorial result on K orbits on ag manifolds in Pro c

Sophus Lie Mem Conf p Scand Univ Press

Sp T A Springer A generalization of the orthogonality relations of Green functions

Inv Math

Sp T A Springer A description of B orbits on symmetric varieties in Algebraic

groups and Lie groups memorial volume in honor of R W Richardson p

Cambridge Univ Press

St R Steinb erg Endomorphisms of linear algebraic groups Mem Amer Math So c

nr

St R Steinb erg An o ccurrence of the RobinsonSchensted corresp ondence J of Alg

Ti J Tits Reductive groups over lo cal elds in Automorphic forms representations

and Lfunctions I p Pro c Symp Pure Math vol XXI I I Amer Math

So c

Vo D A Vogan Irreducible characters of semisimple Lie groups IV Character

multiplicity dualityDuke Math J