EQUIVARIANT COHOMOLOGY AND CHERN CLASSES OF

SYMMETRIC VARIETIES OF MINIMAL RANK

M. BRION AND R. JOSHUA

Abstract. We describe the equivariant Chow ring of the wonderful compactiﬁ-

cation X of a symmetric space of minimal rank, via restriction to the associated

toric varietyY. Also, we show that the restrictions toY of the tangent bundleTX

and its logarithmic analogueS decompose into a direct sum of line bundles. ThisX

yields closed formulae for the equivariant Chern classes ofT andS , and, in turn,X X

for the Chern classes of reductive groups considered by Kiritchenko.

0. Introduction

The purpose of this article is to describe the equivariant intersection ring and

equivariantChernclassesofaclassofalmosthomogeneousvarieties,namely,complete

symmetric varieties of minimal rank. These include those (complete, nonsingular)

equivariant compactiﬁcations of a connected reductive group, that are regular in the

sense of [BDP90].

The main motivation for this work comes from questions of enumerative geometry

on a complex algebraic varietyM. IfM is a spherical homogeneous spherical under a

reductive group G, these questions ﬁnd their proper setting in the ring of conditions

∗C (M), isomorphic to the direct limit of cohomology rings of G-equivariant com-

pactiﬁcations X of M (see [DP83, DP85]). In particular, the Euler characteristic of

any complete intersection of hypersurfaces in M has been expressed by Kiritchenko

[Ki06], in terms of the Chern classes of the logarithmic tangent bundle S of anyX

∗regular compactiﬁcation X. As shown in [Ki06], these elements of C (M) are in-

dependent of the choice of X, and their determination may be reduced to the case

where X is a “wonderful variety”.

In fact, it is more convenient to work with the rational equivariant cohomology

∗ ∗ring H (X), from which the ordinary rational cohomology ring H (X) is obtainedG

∗by killing the action of generators of the polynomial ring H (pt); the Chern classes

G

∗of S have natural representatives in H (X), the equivariant Chern classes. WhenX G

∗X is a complete symmetric variety, the ring H (X) admits algebraic descriptions byG

work of Bifet, De Concini, Littelman, and Procesi (see [BDP90, LP90]).

Here we consider the case where X is a wonderful symmetric variety of minimal

rank,thatis,thewonderfulcompactiﬁcationofasymmetricspaceG/K ofrankequal

to rk(G)−rk(K). Moreover, we adopt a purely algebraic approach: we work over

an arbitrary algebraically closed ﬁeld, and replace the equivariant cohomology ring

The second author was supported by a grant from the NSA and by MPI(Bonn).

16

2 M. BRION AND R. JOSHUA

∗with the equivariant intersection ring A (X) of [EG98] (for wonderful varieties overG

the complex numbers, both rings are isomorphic over the rationals).

We show in Theorem 2.2.1 that a natural pull-back map

∗ ∗ WKr :A (X)→A (Y)G T

is an isomorphism over the rationals, where T ⊂ G denotes a maximal torus con-

taining a maximal torus T ⊂ K with Weyl group W , and Y denotes the closureK K

in X of T/T ⊂G/K. Furthermore, Y is the toric variety associated with the WeylK

chambers of the restricted root system of G/K.

We also determine the images under r of the equivariant Chern classes of the

tangent bundle T and its logarithmic analogue S . For this, we show in TheoremX X

3.1.1 that the normal bundle N decomposes (as a T-linearized bundle) into aY/X

direct sum of line bundles indexed by certain roots of K; moreover, any such line

1bundle is the pull-back of O 1(1) under a certain T-equivariant morphism Y →P .P

By Proposition 1.1.1, the product of these morphisms yields a closed immersion of

the toric variety Y into a product of projective lines, indexed by the restricted roots.

In the case of regular compactiﬁcations of reductive groups, Theorem 2.2.1 is due

to Littelmann and Procesi for equivariant cohomology rings (see [LP90]); it has been

adapted to equivariant Chow ring in [Br98], and to equivariant Grothendieck rings

by Uma in [Um05]. Here we adapt the approach of [Br98], based on a precise version

of the localization theorem in equivariant intersection theory (inspired, in turn, by

a similar result in equivariant cohomology, see [GKM99]). The main ingredient is

that X contains only ﬁnitely many T-stable points and curves. This fact also plays

an essential role in Tchoudjem’s description of the cohomology of line bundles on

wonderful varieties of minimal rank, see [Tc05].

Theorem3.1.1seemstobenew,alreadyinthegroupcase; ityieldsaclosedformula

fortheimageunderr oftheequivariantToddclassofX,analogoustothewell-known

formula expressing the Todd class of a toric variety in terms of boundary divisors.

ThetoricvarietyY associatedtoWeylchambersisconsideredin[Pr90,DL94], where

its cohomology is described as a graded representation of the Weyl group; however,

its simple realization as a general orbit closure in a product of projective lines seems

to have been unnoticed.

This article is organized as follows. Section 1 gathers preliminary notions and

results on symmetric spaces, their wonderful compactiﬁcations, and the associated

toric varieties. In particular, for a symmetric space G/K of minimal rank, we study

therelationsbetweentherootsystemsandWeylgroupsofG,K, andG/K; theseare

our main combinatorial tools. In Section 2, we ﬁrst describe the T-invariant points

and curves in a wonderful symmetric variety X of minimal rank; then we obtain our

∗main structure result for A (X), and some useful complements as well. Section 3G

contains the decompositions of N and of the restrictions T | , S | , togetherY/X X Y X Y

with their applications to equivariant Chern and Todd classes.

Throughoutthis article, we consider algebraic varieties over an algebraically closed

ﬁeld k of characteristic = 2; by a point of such a variety, we mean a closed point.SYMMETRIC VARIETIES OF MINIMAL RANK 3

As general references, we use [Ha77] for algebraic geometry, and [Sp98] for algebraic

groups.

1. Preliminaries

1.1. ThetoricvarietyassociatedwithWeylchambers. LetΦbearootsystem

in a real vector space V (we follow the conventions of [Bo81] for root systems; in

particular, Φ is ﬁnite but not necessarily reduced). Let W be the Weyl group, Q the

∨root lattice in V, and Q the dual lattice (the co-weight lattice) in the dual vector

∗ ∗space V . The Weyl chambers form a subdivision of V into rational polyhedral

∗convex cones; let Σ be the fan of V consisting of all Weyl chambers and their faces.

∨The pair (Q ,Σ) corresponds to a toric variety

Y =Y(Φ)

∨equippedwithanactionofW viaitsactiononQ whichpermutestheWeylchambers.

The group W acts compatibly on the associated torus

∨T := Hom(Q,G ) =Q ⊗ G .m Z m

Thus, Y is equipped with an action of the semi-direct product T ·W. Note that the

character group X(T) is identiﬁed with Q; in particular, we may regard each α∈ Φ

as a homomorphism α :T →G .m

The choice of a basis of Φ,

Δ ={α ,...,α },1 r

deﬁnesapositiveWeylchamber,thedualconetoΔ. LetY ⊂Y bethecorresponding0

rT-stable open aﬃne subset. Then Y is isomorphic to the aﬃne space A on which0

T acts linearly with weights −α ,...,−α . Moreover, the translates w·Y , where1 r 0

w∈W, form an open covering of Y.

Inparticular,thevarietyY isnonsingular. Also,Y isprojective,asΣisthenormal

fan to the convex polytope with vertices w·v (w ∈ W), where v is any prescribed

regular element of V. The following result yields an explicit projective embedding

of Y:

Proposition 1.1.1. (i) For any α ∈ Φ, the morphism α : T → G extends to am

morphism

1f :Y →P .α

1 1 −1Moreover, f and f diﬀer by the inverse mapP →P , z7→z .α −α

(ii) The product morphism

Y Y

1f := f :Y → Pα

α∈Φ α∈Φ

is a closed immersion. It is equivariant under T·W, where T acts on the right-hand

1side via its action on each factor P through the character α, and W acts via itsα

natural action on the set Φ of indices.

Q

1(iii) Conversely, the T-orbit closure of any point of (P \{0,∞}) is isomorphic

α∈Φ

to Y.6

6

4 M. BRION AND R. JOSHUA

(iv) Any non-constant morphismF :Y →C, whereC is an irreducible curve, factors

1through f :Y →P where f is an indivisible root, unique up to sign. Thenα α

(1.1.1) (f ) O =O 1.α ∗ Y P

Proof. (i) Since α has a constant sign on each Weyl chamber, it deﬁnes a morphism

1of fans from Σ to the fan ofP , consisting of two opposite half-lines and the origin.

This implies our statement.

(ii) The equivariance property of f is readily veriﬁed. Moreover, the product map

rY

1 rf :Y → (P )αi

i=1

1 rrestricts to an isomorphism Y → (P \{∞}) , since each f restricts to the i-th0 αi

r∼coordinate function on Y = A . Since Y = W · Y , it follows that f is a closed0 0

immersion.

(iii) follows from (ii) by using the action of tuples (t ) of non-zero scalars, viaα α∈Φ

componentwise multiplication.

(iv) Taking the Stein factorization, we may assume that F O =O . Then C is∗ Y C

normal, and hence nonsingular. Moreover, the action ofT onY descends to a unique

action on C such that F is equivariant (indeed, F equals the canonical morphism

∗ ∗Y → ProjR(Y,F L), where L is any ample invertible sheaf on C, and R(Y,F L)L

∗ n ∗denotesthesectionring Γ(Y,F L ). Furthermore,F LadmitsaT-linearization,n

1∼and hence T acts on R(Y,L)). It follows that C = P where T acts through a

character χ, uniquely deﬁned up to sign. Thus, F induces a morphism from the fan

1 ∗of Y to the fan ofP ; this morphism is given by the linear map χ :V →R. In other

words, χ has a constant sign on each Weyl chamber. Thus, χ is an integral multiple

of an indivisible root α, uniquely deﬁned up to sign. Since F has connected ﬁbers,

then χ =±α.

Conversely, ifα is an indivisible root, then the ﬁbers of the morphismα :T →Gm

are irreducible. This implies (1.1.1).

Next, for later use, we determine the divisor of each f regarded as a rationalα

function on Y. Since f is a T-eigenvector, its divisor is a linear combination of theα

T-stable prime divisors Y ,...,Y of the toric variety Y, also called its boundary1 m

divisors. Recall that Y ,...,Y correspond bijectively to the rays of the Weyl cham-1 m

∨ ∨bers, i.e., to the W-translates of the fundamental co-weights ω ,...,ω (which form1 r

∨the dual basis of the basis of simple roots). The isotropy group of each ω in W isi

the maximal parabolic subgroup W generated by the reﬂections associated with thei

∨∼simple roots α , j =i. Thus, the orbit Wω =W/W is in bijection with the subsetj ii

i +W :={w∈W | wα ∈ Φ for all j =i}j

of minimal representatives for the coset space W/W . So the boundary divisors arei

indexed by the set

i ∨∼E(Φ) :={(i,w)| 1≤i≤r, w∈W } {wω | 1≤i≤r, w∈W}.= iSYMMETRIC VARIETIES OF MINIMAL RANK 5

Furthermore, we have

X

∨(1.1.2) div(f ) = hα,wω iYα i,wi

(i,w)∈E(Φ)

by Proposition 1.1.1 and the classical formula for the divisor of a character in a toric

∨variety (see e.g. [Od88, Prop. 2.1]). Also, note that hα,wω i is the i-th coordinatei

−1of w α in the basis of simple roots.

1.2. Symmetric spaces. Let G be a connected reductive algebraic group, and

θ :G→G

an involutive automorphism. Denote by

θK =G ⊂G

thesubgroupofﬁxedpoints; thenthehomogeneousspaceG/K isasymmetricspace.

We now collect some results on the structure of symmetric spaces, referring to

0[Ri82, Sp85] for details and proofs. The identity component K is reductive, and

non-trivial unless G is a θ-split torus, i.e., a torus where θ acts via the inverse map

−1g7→g .

A parabolic subgroup P ⊆G is said to be θ-split if the parabolic subgroup θ(P) is

0opposite to P. The minimal θ-split parabolic subgroups are all conjugate under K ;

we choose such a subgroup P and put

L :=P ∩θ(P),

aθ-stable Levi subgroup ofP. The intersectionL∩K contains the derived subgroup

[L,L]; thus, every maximal torus of L is θ-stable. We choose such a torus T, so that

θ −θ θ −θ(1.2.1) T =T T and T ∩T is ﬁnite.

Moreover, the identity component

−θ,0A :=T

0is a maximal θ-split subtorus of G. All such subtori are conjugate in K ; their

common dimension is the rank of the symmetric space G/K, denoted by rk(G/K).

Moreover,

C (A) =L = (L∩K)AG

(where C (A) denotes the centralizer of A in G), and (L∩K)∩A =A∩K consistsG

of all elements of order 2 of A.

0The product PK ⊆G is open, and equals PK; thus, PK/K is an open subset of

G/K, isomorphic to P/P ∩K =P/L∩K. Let P be the unipotent radical of P, sou

that P =P ·L. Then the mapu

(1.2.2) ι :P ×A/A∩K →PK/K, (g,x)7→g·xu

is an isomorphism.

The character group X(A/A∩K) may be identiﬁed with the subgroup 2X(A)⊂

∼X(A). On the other hand, A/A∩K T/T ∩K and hence X(A/A∩K) may be=6 M. BRION AND R. JOSHUA

identiﬁed with the subgroup of X(T) consisting of those characters that vanish on

T ∩K, i.e.,

(1.2.3) X(A/A∩K) ={χ−θ(χ)| χ∈X(T)}

(where θ acts onX(T) via its action on T).

Denote by

Φ ⊂X(T)G

the root system of (G,T), with Weyl group

W =N (T)/T.G G

+ChooseabasisΔ consistingofrootsofP. LetΦ ⊂ Φ bethecorrespondingsubsetG GG

of positive roots and let Δ ⊂ Δ be the subset of simple roots of L. The naturalL G

action of the involution θ on Φ ﬁxes pointwise the sub-root system Φ . Moreover, θL

+ + − −exchangesthesubsetsΦ \Φ andΦ \Φ (thesetsofrootsofP andθ(P ) =θ(P) ).u u uG L G L

Also, denote by

p :X(T)→X(A)

the restriction map from the character group ofT to that ofA. Thenp(Φ )\{0} is aG

(possibly non-reduced) root system called the restricted root system, that we denote

by Φ . Moreover,G/K

Δ :=p(Δ \Δ )G/K G L

is a basis of Φ . The corresponding Weyl group isG/K

∼(1.2.4) W =N (A)/C (A) =N 0(A)/C 0(A).G/K G G K K

θ∼Also, W N (A)/C (A), and N (A) = W whereas C (A) = W . This=G/K W W W W LG

yields an exact sequence

θ(1.2.5) 1→W →W →W → 1.L G/KG

1.3. The wonderful compactiﬁcation of an adjoint symmetric space. We

keep the notation and assumptions of Subsec. 1.2 and we assume, in addition, thatG

is semi-simple and adjoint; equivalently, Δ is a basis ofX(T). Then the symmetricG

space G/K is said to be adjoint as well.

By [DP83, DS99], G/K admits a canonical compactiﬁcation: the wonderful com-

pactiﬁcation X, which satisﬁes the following properties.

(i) X is a nonsingular projective variety.

(ii) G acts on X with an open orbit isomorphic to G/K.

(iii) The complement of the open orbit is the union of r = rk(G/K) nonsingular

prime divisors X ,...,X with normal crossings.1 r

(iv) The G-orbit closures in X are exactly the partial intersections

\

X := XI i

i∈I

where I runs over the subsets of{1,...,r}.

∼(v) The unique closed orbit, X ∩···∩X , is isomorphic to G/P G/θ(P).=1 r

We say that X is a wonderful symmetric variety, and X ,...,X are its boundary1 r

divisors. By (iii) and (iv), each orbit closure X is nonsingular.ISYMMETRIC VARIETIES OF MINIMAL RANK 7

Let Y be the closure in X of the subset

∼A/A∩K AK/K =LK/K ⊆G/K.=

Then Y is invariant under the action of the subgroup LN (A) ⊆ G. Since L∩K

∼N (A) = L∩K = C (A), and N (A)/C (A) W by (1.2.4), we obtain an=K K K K G/K

exact sequence

1→L→LN (A)→W → 1.K G/K

Moreover, sinceY is ﬁxed pointwise byL∩K, the action ofLN (A) factors throughK

an action of the semi-direct product

∼(L/L∩K)·W (A/A∩K)·W .=G/K G/K

We identify the group X(A/A∩K) = 2X(A) with X(A). Then each p(χ), where

χ∈X(T), is identiﬁed with χ−θ(χ). Then the adjointness of G and (1.2.3) imply

thatX(A/A∩K) is the restricted root lattice, with basis

Δ ={α−θ(α)| α∈ Δ \Δ }.G/K G L

Moreover, Y is the toric variety associated with the Weyl chambers of the restricted

root system Φ as in Subsec. 1.1. This deﬁnes the open aﬃne toric subvarietyG/K

Y ⊂Y associated with the positive Weyl chamber dual to Δ . Note that0 G/K

(1.3.1) Y =W ·Y .G/K 0

Also, recall the local structure of the wonderful symmetric variety X: the subset

X :=P ·Y =P ·Y0 0 u 0

is open in X, and the map

(1.3.2) ι :P ×Y →X , (g,x)7→g·xu 0 0

isaP-equivariantisomorphism. Moreover,anyG-orbitinX meetsX alongaunique0

orbit of P, and meets transversally Y along a unique orbit of A/A∩K.0

It follows that the G-orbit structure of X is determined by that of the associated

toric variety Y: any G in X meets transversally Y along a disjoint union of

orbit closures of A/A∩K, permuted transitively by W . As another consequence,G/K

X ∩G/K =PK/K and Y ∩G/K =AK/K, so that ι restricts to the isomorphism0 0

(1.2.2).

Finally, the closed G-orbit meets Y transversally at a unique point z. Then the0

isotropy group G equals θ(P), and the normal space to G·z at z is identiﬁed withz

the tangent space to Y at that point. Hence the weights of T in the tangent space

+ +to X at z are the positive roots α∈ Φ \Φ (the contribution of the tangent space

G L

to G·z), and the simple restricted roots γ = α−θ(α), where α ∈ Δ \ Δ (theG L

contribution of the tangent space to Y).8 M. BRION AND R. JOSHUA

1.4. Symmetricspacesofminimalrank. WereturntothesettingofSubsect.1.2.

Inparticular, weconsideraconnectedreductivegroupGequippedwithaninvolutive

θautomorphism θ, and the ﬁxed point subgroup K =G .

Let T be any θ-stable maximal torus of G. Then (1.2.1) implies that

rk(G)≥ rk(K)+rk(G/K)

θ,0 0with equality if and only if the identity component T is a maximal torus of K ,

−θ,0andT is a maximalθ-split subtorus. We then say that the symmetric spaceG/K

0is of minimal rank; equivalently, all θ-stable maximal tori of G are conjugate in K .

We refer to [Br04, Subsec. 3.2] for the proof of the following auxiliary result, where

we put

θ,0 0T :=T = (T ∩K) .K

0Lemma 1.4.1. (i) The roots of (K ,T ) are exactly the restrictions to T of theK K

roots of (G,T).

0 θ(ii) The Weyl group of (K ,T ) may be identiﬁed with W .K G

In particular, C (T ) =T by (i) (this may also be seen directly). We putG K

θW :=W and N :=N 0(T) =N 0(T ).K K K K KG

By (ii), this yields an exact sequence

(1.4.1) 1→T →N →W → 1.K K K

Moreover, by (1.2.5), W ﬁts into an exact sequenceK

(1.4.2) 1→W →W →W → 1.L K G/K

θThe group W acts on Φ and stabilizes the subset Φ = Φ ; the restriction mapK G LG

q :X(T)→X(T )K

0is W -equivariant and θ-stable. Denoting by Φ the root system of (K ,T ), (i) isK K K

equivalent to the equality

Φ =q(Φ ).K G

We now obtain two additional auxiliary results:

Lemma 1.4.2. Let β∈ Φ . Then one of the following cases occurs:K

−1(a) q (β) consists of a unique root, α∈ Φ .L

+ +−1(b) q (β) c of two strongly orthogonal roots α, θ(α), where α∈ Φ \Φ and

G L

− −θ(α)∈ Φ \Φ .

G L

−1 ∼In particular, q induces a bijection q q(Φ ) = Φ = q(Φ ). Moreover, α andL L L

θθ(α) are strongly orthogonal for any α ∈ Φ \ Φ ; then s s ∈ W = W is aG L α θ(α) KG

representative of the reﬂection of W associated with the restricted root α−θ(α).G/K

Proof. Let S ⊆ T be the identity component of the kernel of β. Then the central-K

izer C (S) is a connected reductive θ-stable subgroup of G containing T, and theG

symmetric space C (S)/C (S) is of minimal rank. Moreover, θ yields an involutionG K

of the quotient group C (S)/S, and the corresponding symmetric space is still ofG

minimal rank. So we may reduce to the case where S is trivial, i.e., K has rank 1.

0 0 ∼Since β is a root of K , it follows that K SL(2) or PSL(2). Together with the=SYMMETRIC VARIETIES OF MINIMAL RANK 9

minimal rank assumption, it follows that one of the following cases occurs, up to an

isogeny of G:

0(a) K =G = SL(2); then θ is trivial.

0(b) K = SL(2) and G = SL(2)×SL(2); then θ exchanges both factors.

This implies our assertions.

We will identify q(Φ ) with Φ in view of Lemma 1.4.2.L L

Lemma 1.4.3. (i) q induces bijections

+ −∼ ∼Φ \Φ Φ \Φ Φ \Φ .= =L K L LG G

(ii) Φ \Φ is a sub-root system of Φ , invariant under W .K L K K

(iii) The restricted root system Φ is reduced.G/K

Proof. (i) follows readily from Lemma 1.4.2.

(ii) Since Φ is invariant under W , then so is Φ \Φ . In particular, the latter isL K K L

invariant under any reﬂection s , where β∈ Φ \Φ . It follows that Φ \Φ ⊂ Φβ K L K L K

is a sub-root system.

(iii)ArguingasintheproofofLemma1.4.2,wereducetothecasewhererk(G/K) =

1. Also, we have to check the non-existence of roots α,β ∈ Φ \ Φ such thatG L

β−θ(β) = 2(α−θ(α)). Considering the identity component of the intersection of

kernels of α, β and θ(α), we may also reduce to the case where rk(G)≤ 3. Then the

result follows by inspection.

2. Equivariant Chow ring

2.1. Wonderful symmetric varieties of minimal rank. From now on, we con-

sider an adjoint semi-simple group G equipped with an involutive automorphism θ

such that the corresponding symmetric space G/K is of minimal rank. Then the

group K is connected, semi-simple and adjoint, by [Br04, Lem. 5].

−θ,0Wechooseaθ-stablemaximaltorusT ⊆G, sothatA :=T isamaximalθ-split

θsubtorus. Also, we put T := T ; this group is connected by [Br04, Lem. 5] again.K

Thus, T is a maximal torus of K. In agreeement with the notation of Subsec. 1.4,K

we denote by N the normalizer of T in K, and by W the Weyl group of (K,T ).K K K K

As in Subsec. 1.3, we denote by X the wonderful compactiﬁcation of G/K, also

called a wonderful symmetric variety of minimal rank. The associated toric variety

∼Y is the closure in X of T/T = A/A∩K. Recall that Y is invariant under theK

subgroup LN ⊆ G, and ﬁxed pointwise by L∩K. Thus, LN acts on Y via itsK K

quotient group

∼ ∼LN /(L∩K) =T/T ·W =TN /T .K K G/K K K

We will mostly consider Y as a TN -variety.K

By [Tc05, Sec. 10], X contains only ﬁnitely many T-stable curves. We now obtain

a precise description of all these curves, and of those that lie in Y. This may be

deduced from the results of [loc. cit.], which hold in the more general setting of

wonderfulvarietiesofminimalrank,butweprefertoprovidedirect,somewhatsimpler

arguments.10 M. BRION AND R. JOSHUA

Lemma2.1.1. (i) TheT-ﬁxed points inX (resp.Y) are exactly the pointsw·z, where

∼w∈W (resp. W ). They are parametrized by W /W (resp. W /W W ).=K G L K L G/K

+ +(ii) For any α∈ Φ \Φ , there exists a unique irreducible T-stable curve C whichz,αG L

contains z and on which T acts through its character α. The T-ﬁxed points in Cz,α

are exactly z and s ·z.α

(iii) For any γ = α−θ(α)∈ Δ , there exists a unique irreducible T-stable curveG/K

C which containsz and on whichT acts through its characterγ. TheT-ﬁxed pointsz,γ

in C are exactly z and s s ·z.z,γ α θ(α)

(iv) The irreducible T-stable curves in X are the W -translates of the curves CG z,α

1and C . They are all isomorphic toP .z,γ

(v) The irreducible T-stable curves in Y are the W -translates of the curves C .G/K z,γ

Proof. The assertions on the T-ﬁxed points in X are proved in [Br04, Lem. 6]. And

since Y is the toric variety associated with the Weyl chambers of Φ , the groupG/K

W acts simply transitively on its T-ﬁxed points. This proves (i).G/K

LetC ⊂X be an irreducibleT-stable curve. ReplacingC with aW -translate, weG

may assume that it contains z. Then C∩X is an irreducible T-stable curve in X ,0 0

+ +an aﬃne space where T acts linearly with weights the positive roots α ∈ Φ \Φ ,G L

and the simple restricted roots γ = α−θ(α), α∈ Δ \Δ . Since these weights allG L

have multiplicity 1, it follows that C ∩X is a coordinate line in X . Thus, C is0 0

1isomorphic toP where T acts through α or γ. In the former case, C is contained in

the closed G-orbit G·z; it follows that its other T-ﬁxed point is s ·z. In the latterα

case, C is contained in Y, and hence its other T-ﬁxed point corresponds to a simple

reﬂection in W . By considering the weight of the T-action on C, this simpleG/K must be the image in W of s s ∈ W . This implies the remainingG/K α θ(α) K

assertions (ii)-(v).

2.2. Structure of the equivariant Chow ring. We will obtain a description of

the G-equivariant Chow ring of X with rational coeﬃcients. For this, we brieﬂy

recall some properties of equivariant intersection theory, referring to [Br97, EG98]

for details.

To any nonsingular varietyZ carrying an action of a linear algebraic groupH, one

∗associates an equivariant Chow ring A (Z). This is a positively graded ring withH

degree-0 partZ, and degree-1 part the equivariant Picard group Pic (Z) consistingH

of isomorphism classes of H-linearized invertible sheaves on Z.

Every closed H-stable subvariety Y ⊆ Z of codimension n yields an equivariant

class

n[Y] ∈A (Z).H H

∗The class [Z] is the unit element of A (Z).H H

0 0Any equivariant morphism f :Z →Z , where Z is a nonsingular H-variety, yields

a pull-back homomorphism

∗ ∗ 0 ∗f :A (Z )→A (Z).H H

∗ ∗In particular, A (Z) is an algebra over the equivariant ring of the point, A (pt).H H