Canonical bases and affine Hecke algebras of type B

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Canonical bases and affine Hecke algebras of type B

Michela Varagnolo, Eric Vasserot

To cite this version:

Michela Varagnolo, Eric Vasserot. Canonical bases and affine Hecke algebras of type B. 2009. �hal-

00435938v3�

CANONICAL BASES AND AFFINE HECKE ALGEBRAS OF TYPE B

M. Varagnolo, E. Vasserot

Abstract. We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that we compute explicitly.

Introduction

A new family of graded algebras, called KLR algebras, has been recently intro- duced in [KL], [R]. These algebras yield a categorification of f, the negative part of the quantized enveloping algebra of any type. In particular, one can obtain a new interpretation of the canonical bases, see [VV]. In type A or A ⁽¹⁾ the KLR algebras are Morita equivalent to the affine Hecke algebras and their cyclotomic quotients.

Hence they give a new way to understand the categorification of the simple highest weight modules and the categorification of f via some Hecke algebras of type A or A ⁽¹⁾ . See [BK] for instance. One of the advantages of KLR algebras is that they are graded, while the affine Hecke algebras are not. This explain why KLR algebras are better adapted than affine Hecke algebras to describe canonical bases. Indeed one could view KLR algebras as an intermediate object between the representation theory of affine Hecke algebras and its Kazhdan-Lusztig geometric counterpart in term of perverse sheaves. This is central in [VV], where KLR algebras are proved to be isomorphic to the Ext-algebras of some complex of constructible sheaves.

In the other hand, the (branching rules for) affine Hecke algebras of type B have been investigated quite recently, see [E], [EK1,2,3], [Ka], [M]. Lusztig’s description of the canonical basis of f in type A ⁽¹⁾ in [L1] implies that this basis can be naturally identified with the set of isomorphism classes of simple objects of a category of modules of the affine Hecke algebras of type A. This identification was mentioned in [G], and was used in [A]. More precisely, there is a linear isomorphism between f and the Grothendieck group of finite dimensional modules of the affine Hecke algebras of type A, and it is proved in [A] that the induction/restriction functors for affine Hecke algebras are given by the action of the Chevalley generators and their transposed operators with respect to some symmetric bilinear form on f . In [E], [EK1,2,3] a similar behavior is conjectured and studied for affine Hecke algebras of type B. Here f is replaced by an explicit module ^θ V(λ) over an explicit algebra ^θ B.

First, it is conjectured that ^θ V(λ) admits a canonical basis. Next, it is conjectured that this basis is naturally identified with the set of isomorphism classes of simple objects of a category of modules of the affine Hecke algebras of type B. Further, in this identification the branching rules of the affine Hecke algebras of type B are

2000Mathematics Subject Classification. Primary ??; Secondary ??.

1

given by the ^θ B-action on ^θ V(λ). The first conjecture has been proved in [E] under the restrictive assumption that λ = 0. Here we prove the whole set of conjectures.

Indeed, our construction is slightly more general, see the appendix.

Roughtly speaking our argument is as follows. In [E] a geometric description of the canonical basis of ^θ V(0) was given. This description is similar to Lusztig’s description of the canonical basis of f via perverse sheaves on the moduli stack of representations of some quiver. It is given in terms of perverse sheaves on the mod- uli stack of representations of a quiver with involution. First we give a analogue of this for ^θ V(λ) for any λ. This yields the existence of a canonical basis ^θ G ^low (λ) for ^θ V(λ) for arbitrary λ. Then we compute explicitely the Ext-algebras between complexes of constructible sheaves naturally attached to quivers with involutions.

These complexes enter in a natural way in the definition of ^θ G ^low (λ). This com- putation yields a new family of graded algebras ^θ R m where m is a nonnegative integer. We prove that the algebras ^θ R m are Morita equivalent to the affine Hecke algebras of type B. Finally we describe ^θ V(λ) and the basis ^θ G ^low (λ) in terms of the Grothendieck group of ^θ R m .

The plan of the paper is the following. Section 1 contains some basic notation for Lusztig’s theory of perverse sheaves on the moduli stack of representations of quivers. Section 2 yields similar notation for the case of quivers with involutions.

Our setting is more general than in [E], where only the case λ = 0 is considered.

In Section 3 we introduce the convolution algebra associated with a quiver with involution. The main result of Section 4 is Theorem 4.17 where the polynomial representation of the Ext-algebra Z ^δ _{Λ Λ Λ,V} associated with a quiver with involution is computed. Here Λ Λ Λ is a I-graded C -vector space of dimension vector λ ∈ N I, while V is a I-graded C -vector space with a non-degenerate symmetric bilinear form of dimension vector ν ∈ N I. The polynomial representation of Z _Λ ^δ _{Λ Λ,V} is faithful. In Section 5 we give the main properties of the graded algebra ^θ R(Γ) λ,ν . In Section 6 we introduce the affine Hecke algebra of type B and we prove that it is Morita equivalent to ^θ R m , a specialization of ^θ R(Γ) λ,ν . Section 7 is a reminder on KLR al- gebras and on the main result of [VV]. In Section 8 we categorify the module ^θ V(λ) from [EK1] using the graded algebra ^θ R m . In Section 9 we prove the isomorphism

θ R(Γ) λ,ν = Z ^δ _{Λ Λ Λ,V} . This is essential to compare the construction from Section 8 with that in Section 10. In Section 10 we give a categorification of ^θ V(λ) “` a la Lusztig”

in terms of perverse sheaves on the moduli stack of representations of quivers with involution. This is essentially the same construction as in [E]. However, since we need a more general setting than in loc. cit. we have briefly reproduced the main steps of the construction. One of our initial motivations was to give a completely algebraic proof of the conjectures, without any perverse sheaves at all. We still do not know how to do this. The main result of the paper is Theorem 10.19.

The same technic yields similar results for affine Hecke algebras of any classical type. The case of type D is done in [SVV], the case of type C is done in the appendix. Note that the idea to use canonical bases technics to study affine Hecke algebras in non A type is not new, see [L3], [L4]. At the moment we do not know the precise relation between loc. cit. and our approach.

Acknowledgment. We are grateful to M. Kashiwara and G. Lusztig for some

remark on the material of this paper.

Contents 0. Notation

1. Reminder on quivers and extensions 2. Quivers with involutions

3. The convolution algebra

4. The polynomial representation of the graded algebra ^θ Z _Λ ^δ _{Λ Λ,V} 5. The graded k-algebra ^θ R(Γ) λ,ν

6. Affine Hecke algebras of type B

7. Global bases of f and projective graded modules of KLR algebras 8. Global bases of ^θ V(λ) and projective graded ^θ R-modules

9. Presentation of the graded algebra ^θ Z ^δ _{Λ Λ Λ,V}

10. Perverse sheaves on ^θ E Λ Λ Λ,V and the global bases of ^θ V(λ) A. Appendix

Index of notation

0. Notation

0.1. Combinatorics. Given a positive integer m and a tuple m = (m 1 , m 2 , . . . m r ) of positive integers we write S m for the symmetric group and S m for the group Q r

l=1 S _m

. Set

|m| = X r l=1

m l , ℓ m = X r l=1

ℓ m

, ℓ m = m(m − 1)/2.

We use the following notation for v-numbers hmi =

X m l=1

v ^m+1−2l , hmi! = Y m l=1

hli,

m + n n

= hm + ni!

hmi!hni! , hmi! = Y r l=1

hm l i!.

Given two tuples m = (m 1 , m 2 , . . . m r ), m ^′ = (m ^′ ₁ , m ^′ ₂ , . . . m ^′ _r

) we define the tuple mm ^′ = (m 1 , m 2 , . . . m r , m ^′ ₁ , m ^′ ₂ , . . . m ^′ _r

).

0.2. Graded modules over graded algebras. Let k be an algebraically closed field of caracteristic 0. Let R = L

d R d be a graded k-algebra. Unless specified otherwise the word graded we’ll always mean Z-graded. Let R-mod be the cate- gory of finitely generated graded R-modules, R-fmod be the full subcategory of finite-dimensional graded modules and R-proj be the full subcategory of projec- tive objects. Unless specified otherwise a module is always a left module. We’ll abbreviate

K(R) = [R-proj], G(R) = [R-fmod].

Here [CCC] denotes the Grothendieck group of an exact category CC C. Assume that the k-vector spaces R d are finite dimensional for each d. Then K(R) is a free Abelian group with a basis formed by the isomorphism classes of the indecompos- able objects in R-proj, and G(R) is a free Abelian group with a basis formed by the isomorphism classes of the simple objects in R-fmod. Given an object M of R-proj or R-fmod let [M ] denote its class in K(R), G(R) respectively. When there is no risk of confusion we abbreviate M = [M ]. We’ll write [M : N] for the composition multiplicity of the R-module N in the R-module M . Consider the ring A = Z [v, v ⁻¹ ]. If the grading of R is bounded below then the A-modules K(R), G(R) are free. Here A acts on G(R), K(R) as follows

vM = M [1], v ⁻¹ M = M [−1].

For any M, N in R-mod let

hom R (M, N ) = M

d

Hom R (M, N[d])

be the graded k-vector space of all R-module homomorphisms. If R = k we’ll omit the subscript R in hom’s and in tensor products. As much as possible we’ll use the following convention : graded objects are denoted by minuscules and non- graded ones by majuscules. In particular R-Mod will denote the category of finitely generated (non-graded) R-modules. We ’ll abbreviate

Hom = Hom k , ⊗ = ⊗ k , etc.

For a graded k-vector space M = L

d M d we’ll write gdim(M ) = X

d

v ^d dim(M d ), where dim is the dimension over k.

0.3. Constructible sheaves. Given an action of a complex linear algebraic group G on a quasiprojective algebraic variety X over C we write D D D G (X ) for the bounded derived category of complexes of G-equivariant sheaves of k-vector spaces on X . Objects of D D D G (X ) are referred to as complexes. If G = {e}, the trivial group, we abbreviate D D D(X ) = D D D G (X ). For each complexes L, L ^′ we’ll abbreviate

Ext ^∗ _G (L, L ^′ ) = Ext _D ^∗ _{D D}

_(X) (L, L ^′ ), Ext ^∗ (L, L ^′ ) = Ext ^∗ _{D D D(X)} (L, L ^′ )

if no confusion is possible. The constant sheaf on X with stalk k will be denoted k.

For any object L of D D D G (X ) let H _G ^∗ (X, L) be the space of G-equivariant cohomology with coefficients in L. Let D ∈ D D D G (X) be the G-equivariant dualizing complex, see [BL, def. 3.5.1]. For each L let

L ^∨ = Hom(L, D)

be its Verdier dual, where Hom is the internal Hom. Recall that

(L ^∨ ) ^∨ = L, Ext ^∗ _G (L, D) = H _G ^∗ (X, L ^∨ ), Ext ^∗ _G (k, L) = H _G ^∗ (X, L).

We define the space of G-equivariant homology by H _∗ ^G (X, k) = H _G ^∗ (X, D).

Note that D = k[2d] if X is a smooth G-variety of pure dimension d. Consider the following graded k-algebra

S G = H _G ^∗ (•, k).

The graded k-vector space H _∗ ^G (X, k) has a natural structure of a graded S G -module.

We have

H _∗ ^G (•, k) = S G

as graded S G -module. There is a canonical graded k-algebra isomorphism S G ≃ k[g] ^G .

Here the symbol g denotes the Lie algebra of G and a G-invariant homogeneous polynomial over g of degree d is given the degree 2d in S G .

Fix a morphism of quasi-projective algebraic G-varieties f : X → Y . If f is a proper map there is a direct image homomorphism

f _∗ : H _∗ ^G (X, k) → H _∗ ^G (Y, k).

If f is a smooth map of relative dimension d there is an inverse image homomor- phism

f ^∗ : H _i ^G (Y, k) → H _i−2d ^G (X, k), ∀i.

If X has pure dimension d there is a natural homomorphism H _G ⁱ (X, k) → H _i−2d ^G (X, k).

It is invertible if X is smooth. The image of the unit is called the fundamental class of X in H _∗ ^G (X, k). We denote it by [X ]. If f : X → Y is the embedding of a G-stable closed subset and X ^′ ⊂ X is the union of the irreducible components of maximal dimension then the image of [X ^′ ] by the map f ∗ is the fundamental class of X in H _∗ ^G (Y, k). It is again denoted by [X].

1. Reminder on quivers and extensions

1.1. Representations of quivers. We assume given a nonempty quiver Γ such that no arrow may join a vertex to itself. Recall that Γ is a tuple (I, H, h 7→ h ^′ , h 7→

h ^′′ ) where I is the set of vertices, H is the set of arrows, and for h ∈ H the vertices h ^′ , h ^′′ ∈ I are the origin and the goal of h respectively. Note that the set I may be infinite. For i, j ∈ I we write

H i,j = {h ∈ H; h ^′ = i, h ^′′ = j}.

We’ll abbreviate i → j for H i,j 6= ∅, i 6→ j for H i,j = ∅, and h : i → j for h ∈ H i,j . Let h i,j be the number of elements in H i,j and set

i · j = −h i,j − h j,i , i · i = 2, i 6= j.

Let V V V be the category of finite-dimensional I-graded C-vector spaces V = L

i∈I V i

with morphisms being linear maps respecting the grading. For ν = P

i ν i i in NI let V V V ν be the full subcategory of V V V whose objects are those V such that dim(V i ) = ν i

for all i. We call ν the dimension vector of V. Given an object V of V V V let E V = M

h∈H

Hom(V h

, V h

).

The algebraic group G V = Q

i GL(V i ) acts on E V by (g, x) 7→ gx = y where y h = g h

x h g ⁻¹ _h

, g = (g i ), x = (x h ), and y = (y h ).

Fix a nonzero element ν of NI. Let Y ^ν be the set of all pairs y = (i, a) where i = (i 1 , i 2 , . . . i k ) is a sequence of elements of I and a = (a 1 , a 2 , . . . a k ) is a sequence of positive integers such that P

l a l i l = ν. Note that the assignment (1.1) y 7→ (a 1 i 1 , a 2 i 2 , . . . a k i k )

identifies Y ^ν with a set of sequences

(1.2) ν ¹ , ν ² , . . . , ν ^k ∈ NI with ν = X k

l=1

ν ^l .

For each pair y = (i, a) as above we’ll call a the multiplicity of y. Let I ^ν ⊂ Y ^ν be the set of all pairs y with multiplicity (1, 1, . . . , 1). We’ll abbreviate i for a pair y = (i, a) which lies in I ^ν . Given a positive integer m we have F

ν I ^ν = I ^m , where ν runs over the set of elements ν of NI with |ν | = m. Here, we write ν = P

i∈I ν i i and |ν| = P

i ν i . In a similar way, we define Y ^m = F

ν Y ^ν .

1.2. Flags. Let ν ∈ NI, ν 6= 0, and assume that V lies in V V V ν . For each sequence y = (ν ¹ , ν ² , . . . ν ^k ) as in (1.1), (1.2), a flag of type y in V is a sequence

φ = (V = V ⁰ ⊃ V ¹ ⊃ · · · ⊃ V ^k = 0)

of I-graded subspace of V such that for any l the I-graded subspace V ^l−1 /V ^l belongs to V V V _ν

. Let F V,y be the variety of all flags of type y in V. The group G V acts transitively on F V,y in the obvious way, yielding a smooth projective G V -variety structure on F V,y .

If x ∈ E V we say that the flag φ is x-stable if x h (V ^l _h

) ⊂ V ^l _h

for all h, l. Let F e V,y be the variety of all pairs (x, φ) such that φ is x-stable. Set d y = dim( F e V,y ).

The group G V acts on F e V,y by g : (x, φ) 7→ (gx, gφ). The first projection gives a G V -equivariant proper morphism

π y : F e V,y → E V .

1.3. Ext-algebras. Let ν ∈ NI, ν 6= 0, and assume that V ∈ V V V ν . We abbreviate S V = S G

. For each sequence y ∈ Y ^ν we have the following semisimple complexes in D D D G

(E V )

L y = (π y ) ! (k), L ^∨ y = L y [2d y ], L ^δ y = L y [d y ].

For y, y ^′ in Y ^ν we consider the graded S V -module Z V,y,y

= Ext ^∗ _G

(L ^∨ y , L ^∨ y

).

For y, y ^′ , y ^′′ in Y ^ν the Yoneda composition is a homogeneous S V -bilinear map of degree zero

⋆ : Z V,y,y

× Z V,y

,y

→ Z V,y,y

. The map ⋆ equips the graded k-vector space

Z V = M

i,i

∈I

Z V,i,i

with the structure of an associative graded S V -algebra with 1. If there is no ambi- guity we’ll omit the symbol ⋆. We set

F V,y = Ext ^∗ _G

(L ^∨ _y , D), F V = M

i ∈I

F V,i .

For y, y ^′ in Y ^ν the Yoneda product gives a graded S V -bilinear map Z V,y,y

× F V,y

→ F V,y . This yields a left graded representation of Z V on F V . For each i ∈ I ^ν let 1 V,i ∈ Z V,i,i denote the identity of L i . The elements 1 V,i form a complete set of orthogonal idempotents of Z V such that

Z V,i,i

= 1 V,i ⋆ Z V ⋆ 1 V,i

, F V,i = 1 V,i ⋆ F V . We’ll change the grading of Z V in the following way. Put

Z ^δ _V,i,i

= Ext ^∗ _G

(L ^δ _i , L ^δ _i

), Z ^δ V = M

i,i

∈I

Z ^δ _V,i,i

.

The graded k-algebra Z ^δ _V depends only on the dimension vector of V. We’ll write R(Γ) ν = Z ^δ V .

This graded k-algebra has been computed explicitely in [VV]. The same result has also been anounced by R. Rouquier. See Section 7 for more details. We set also I ⁰ = {∅}, L ^δ _∅ = k (the constant sheaf over {0})

R(Γ) 0 = Z ^δ _{0} = k.

2. Quivers with involutions

In this section we introduce an analogue of the Ext-algebra R(Γ) ν . It is associ- ated with a quiver with an involution.

2.1. Representations of quivers with involution. Fix a nonempty quiver Γ such that no arrow may join a vertex to itself. An involution θ on Γ is a pair of involutions on I and H, both denoted by θ, such that the following properties hold for h ∈ H

• θ(h) ^′ = θ(h ^′′ ) and θ(h) ^′′ = θ(h ^′ ),

• θ(h ^′ ) = h ^′′ iff θ(h) = h.

We’ll always assume that θ has no fixed points in I, i.e., there is no i ∈ I such that θ(i) = i. To simplify we’ll say that θ has no fixed points.

Let ^θ V V V be the category of finite-dimensional I-graded C-vector spaces V with a non-degenerate symmetric bilinear form ̟ such that

(V i ) ^⊥ = M

j6=θ(i)

V j .

To simplify we’ll say that V belongs to ^θ V V V if there is a bilinear form ̟ such that the pair (V, ̟) lies in ^θ V V V. The morphisms in ^θ V V V are the linear maps which respect the grading and the bilinear form. Let

θ NI = {ν = P

i ν i i ∈ NI; ν _θ(i) = ν i , ∀i}.

For ν ∈ ^θ N I let ^θ V V V ν be the full subcategory of ^θ V V V consisting of the pairs (V, ̟) such that V lies in V V V ν . Note that |ν | is an even integer. We’ll usually write |ν| = 2m with m ∈ N . Given V in ^θ V V V and Λ Λ Λ in V V V we let

θ E V = {x = (x h ) ∈ E V ; x θ(h) = − ^t x h , ∀h ∈ H},

θ G V = {g ∈ G V ; g θ(i) = ^t g ⁻¹ _i , ∀i ∈ I},

θ E Λ Λ Λ,V = ^θ E V × L Λ Λ Λ,V , L Λ Λ Λ,V = Hom V V V (Λ Λ Λ, V).

The algebraic groups ^θ G V , G Λ Λ Λ act on ^θ E V , L Λ Λ Λ,V in the obvious way.

2.2. Generalities on isotropic flags. Given a finite dimensional C-vector space W with a non-degenerate symmetric bilinear form ̟, an isotropic flag in W is a sequence of subspaces

φ = (W = W ^−k ⊃ W ^1−k ⊃ · · · ⊃ W ^k = 0)

such that (W ^l ) ^⊥ = W ^−l for any l = −k, 1 − k, . . . , k − 1, k. Here the symbol ⊥

means the orthogonal relative to ̟. In particular W ⁰ is a Lagrangian subspace

of W. Let F (W) be the variety of all complete flags in W, and F (W, ̟) be

the subvariety of all complete isotropic flags, i.e., we require that φ = (W ^l ) is an

isotropic flag such that W ^l has the dimension m −l and k = m. If W has dimension

2m then F(W, ̟) has dimension 2ℓ m = m(m − 1).

2.3. Sequences. Fix a nonzero dimension vector ν in ^θ NI. Let ^θ Y ^ν be the set of all pairs y = (i, a) in Y ^ν such that

i = (i 1−k , . . . , i k−1 , i k ), a = (a 1−k , . . . , a k−1 , a k ), θ(i l ) = i 1−l , a l = a 1−l . As in (1.1) we can identify a pair y as above with a sequence

ν ^1−k , . . . , ν ^k−1 , ν ^k ∈ N I, θ(ν ^l ) = ν ^1−l , X

l

ν ^l = ν.

Let ^θ I ^ν ⊂ ^θ Y ^ν be the set of all pairs y of multiplicity (1, 1, . . . , 1). We’ll abbreviate i = (i, a) for each pair in ^θ I ^ν . Note that a sequence in ^θ I ^ν contains |ν| = 2m terms.

Unless specified otherwise the entries of a sequence i in ^θ I ^ν will always denoted by i = (i 1−m , . . . , i m−1 , i m ).

Finally, we set

θ I ^m = [

ν

θ I ^ν , ν ∈ ^θ N I, |ν| = 2m, and we define ^θ Y ^m in the same way.

2.4. Definition of the map ^θ π Λ Λ Λ,y . Fix ν ∈ ^θ N I, ν 6= 0, and λ ∈ N I. Fix an object V in ^θ V V V ν and an object Λ Λ Λ in V V V λ . For y in ^θ Y ^ν an isotropic flag of type y in V is an isotropic flag

φ = (V = V ^−k ⊃ V ^1−k ⊃ · · · ⊃ V ^k = 0)

such that V ^l−1 /V ^l lies in V V V ν

for each l. We define ^θ F V,y to be the variety of all isotropic flags of type y in V. Next, we define ^θ F e Λ Λ Λ,V,y to be the variety of all tuples (x, y, φ) satisfying the following conditions :

• x ∈ ^θ E V and φ ∈ ^θ F V,y is stable by x, i.e., x(V ^l ) ⊂ V ^l for each l,

• y ∈ L Λ Λ Λ,V and y(Λ Λ Λ) ⊂ V ⁰ . We set

d λ,y = dim( ^θ F e Λ Λ Λ,V,y ).

We have the following formulas.

2.5. Proposition. For i ∈ ^θ I ^ν we have (a) dim( ^θ F V,i ) = ℓ ν /2,

(b) d λ,i = ℓ ν /2 + P

k<l; k+l6=1 h i

,i

/2 + P

16l λ i

.

Proof : Fix a subset J ⊂ I such that I = J ⊔ θ(J ). Set V J = L

j∈J V j . The assignment (V ^k ) 7→ (V ^k ∩ V J ) takes ^θ F V,i isomorphically onto

Y

j∈J

F (V j ).

Thus we have

dim( ^θ F V,i ) = X

j∈J

ℓ ν

= X

i∈I

ℓ ν

/2.

Next, fix a sequence i as above and fix a flag φ = (V ^k ) in ^θ F V,i . Then we have d λ,i = ℓ ν /2 + dim{x ∈ ^θ E V ; x(V ^k ) ⊂ V ^k , ∀k} + dim{y ∈ L Λ Λ Λ,V ; y(Λ Λ Λ) ⊂ V ⁰ }.

Finally we have (see the discussion in Section 4.9) dim{x ∈ ^θ E V ; x(V ^k ) ⊂ V ^k , ∀k} = X

k<l; k+l6=1

h i

,i

/2,

dim{y ∈ L Λ Λ Λ,V ; y(Λ Λ Λ) ⊂ V ⁰ } = X

16l6m

λ i

.

⊓

⊔ The group ^θ G V acts transitively on ^θ F V,y . It acts also on ^θ F e Λ Λ Λ,V,y . The first projection gives a ^θ G V -equivariant proper morphism

θ π Λ Λ Λ,y : ^θ F e Λ Λ Λ,V,y → ^θ E Λ Λ Λ,V . For a future use we introduce also the obvious projection

p : ^θ F e Λ Λ Λ,V → ^θ F V , ^θ F e Λ Λ Λ,V = a

i∈

I

θ F e Λ Λ Λ,V,i , ^θ F V = a

i∈

I

θ F V,i .

2.6. Ext-algebras. Let λ, ν, Λ Λ Λ, V be as above. We abbreviate ^θ S V = S

_G

. For y ∈ ^θ Y ^ν we define the following semisimple complexes in D D D

_G

( ^θ E Λ Λ Λ,V )

θ L y = ( ^θ π Λ Λ Λ,y ) ! (k), ^θ L ^∨ _y = ^θ L y [2d λ,y ], ^θ L ^δ _y = ^θ L y [d λ,y ].

For i, i ^′ in ^θ I ^ν we consider the graded ^θ S V -module

θ Z Λ Λ Λ,V,i,i

= Ext ^∗

G

( ^θ L ^∨ _i , ^θ L ^∨ _i

).

The Yoneda composition is a homogeneous ^θ S V -bilinear map of degree zero

θ Z Λ Λ Λ,V,i,i

× ^θ Z Λ Λ Λ,V,i

,i

→ ^θ Z Λ Λ Λ,V,i,i

, i, i ^′ , i ^′′ ∈ ^θ I ^ν . It equips the k-vector space

θ Z Λ Λ Λ,V = M

i,i

∈

I

θ Z Λ Λ Λ,V,i,i

with the structure of a unital associative graded ^θ S V -algebra. For i ∈ ^θ I ^ν we have the graded ^θ S V -modules

θ F Λ Λ Λ,V,i = Ext ^∗

G

( ^θ L ^∨ _V,i , D), ^θ F Λ Λ Λ,V = M

i∈

I

θ F Λ Λ Λ,V,i .

For each i, i ^′ in ^θ I ^ν the Yoneda product gives a graded ^θ S V -bilinear map ^θ Z Λ Λ Λ,V,i,i

×

θ F Λ Λ Λ,V,i

→ ^θ F Λ Λ Λ,V,i . This yields a left graded representation of ^θ Z Λ Λ Λ,V on ^θ F Λ Λ Λ,V . Our first goal is to compute the graded algebra ^θ Z Λ Λ Λ,V and the graded representation

θ F Λ Λ Λ,V . For i ∈ ^θ I ^ν let 1 Λ Λ Λ,V,i be the identity of ^θ L i , regarded as an element of

θ Z Λ Λ Λ,V,i,i . The elements 1 Λ Λ Λ,V,i form a complete set of orthogonal idempotents of

θ Z Λ Λ Λ,V such that

θ Z Λ Λ Λ,V,i,i

= 1 Λ Λ Λ,V,i θ Z Λ Λ Λ,V 1 Λ Λ Λ,V,i

, ^θ F Λ Λ Λ,V,i = 1 Λ Λ Λ,V,i θ F Λ Λ Λ,V .

2.7. Remark. Fix a pair y in ^θ Y ^ν . Let i be the sequence of ^θ I ^ν obtained by expanding y. We have an isomorphism of complexes in the derived category

θ L ^δ _i = M

w∈S

θ L ^δ _y [ℓ b − 2ℓ(w)].

Here b = (b 1 , . . . , b m ) is a sequence such that the multiplicity of y is θ(b)b := (b m , . . . , b 2 , b 1 , b 1 , b 2 , . . . b m ).

We’ll abbreviate ^θ L ^δ _i = hbi! ^θ L ^δ _y .

2.8. Shift of the grading. Let λ, ν, Λ Λ Λ, V be as above. We define a new grading on ^θ Z Λ Λ Λ,V and ^θ F Λ Λ Λ,V by

θ Z ^δ _Λ Λ Λ,V,i,i

= Ext ^∗

G

( ^θ L ^δ _i , ^θ L ^δ _i

) = ^θ Z Λ Λ Λ,V,i,i

[d λ,i − d λ,i

],

θ Z _{Λ Λ} ^δ _Λ,V = M

i,i

∈

I

θ Z ^δ _{Λ Λ Λ,V,i,i}

,

θ L V = M

i∈

I

θ L i , ^θ L ^δ _V = M

i∈

I

θ L ^δ _i .

We set also ^θ I ⁰ = {∅}, ^θ L ^δ _∅ = k, and ^θ Z ^δ _{Λ Λ Λ,{0}} = k as a graded k-algebra. Here k is regarded as the constant sheaf over {0}.

3. The convolution algebra

Fix a quiver Γ with set of vertices I and set of arrows H. Fix an involution θ on Γ. Assume that Γ has no 1-loops and that θ has no fixed points. Fix a dimension vector ν 6= 0 in ^θ NI and a dimension vector λ in NI. Fix an object (V, ̟) in ^θ V V V ν

and an object Λ Λ Λ in V V V λ . For each sequences i, i ^′ in ^θ I ^ν we set

θ Z Λ Λ Λ,V,i,i

= ^θ F e Λ Λ Λ,V,i ×

_E

θ F e Λ Λ Λ,V,i

, ^θ Z Λ Λ Λ,V = a

i,i

∈

I

θ Z Λ Λ Λ,V,i,i

.

the reduced fiber product relative to the maps ^θ π Λ Λ Λ,i , ^θ π Λ Λ Λ,i

. Next we set

θ Z Λ Λ Λ,V = M

i,i

∈

I

θ Z Λ Λ Λ,V,i,i

, ^θ F Λ Λ Λ,V = M

i∈

I

θ F Λ Λ Λ,V,i ,

where

θ Z Λ Λ Λ,V,i,i

= H _∗

^G

( ^θ Z Λ Λ Λ,V,i,i

, k), ^θ F Λ Λ Λ,V,i = H _∗

^G

( ^θ F e Λ Λ Λ,V,i , k).

We have

θ F Λ Λ Λ,V,i = Ext ^∗

G

(k, ^θ L i ) = H

^∗ G

( ^θ E V , ^θ L i ) = H

^∗ G

( ^θ F e Λ Λ Λ,V,i , k).

We have also

(3.1) H

^∗ G

( ^θ F e Λ Λ Λ,V,i , k) = H

^∗ G

( ^θ F e Λ Λ Λ,V,i , D)[−2d λ,i ] = ^θ F Λ Λ Λ,V,i [−2d λ,i ].

This yields a graded ^θ S V -module isomorphism (3.2) ^θ F Λ Λ Λ,V,i = ^θ F Λ Λ Λ,V,i [−2d λ,i ].

We equip the ^θ S V -module ^θ Z Λ Λ Λ,V with the convolution product relative to the closed embedding of ^θ Z Λ Λ Λ,V into ^θ F e Λ Λ Λ,V × ^θ F e Λ Λ Λ,V . See [CG, sec. 8.6] for details. We obtain an associative graded ^θ S V -algebra ^θ Z Λ Λ Λ,V with 1 which acts on the graded ^θ S V -module

θ F Λ Λ Λ,V . The unit is the fundamental class of the closed subvariety ^θ Z _{Λ Λ Λ,V} ^e of ^θ Z Λ Λ Λ,V . See Section 4.6 below for the notation.

3.1. Proposition. (a) The left ^θ Z Λ Λ Λ,V -module ^θ F Λ Λ Λ,V is faithful.

(b) There is a canonical ^θ S V -algebra isomorphism ^θ Z Λ Λ Λ,V = ^θ Z Λ Λ Λ,V such that (3.2) identifies the ^θ Z Λ Λ Λ,V -action on ^θ F Λ Λ Λ,V and the ^θ Z Λ Λ Λ,V -action on ^θ F Λ Λ Λ,V .

Proof : This is standard material, see e.g., [VV]. Let us give one proof of (a). It is a consequence of the following general fact. Let G be a linear algebraic group over C and let M be a smooth quasi-projective G-variety over C. Let T ⊂ G be a maximal torus. Let Q be the fraction field of S = S T . Let Z ⊂ M × M be a closed G-stable subset (for the diagonal action on M × M ) such that p 1,3 restricts to a proper map

p ⁻¹ _1,2 (Z) ∩ p ⁻¹ _2,3 (Z) → Z,

where p i,j : M ×M ×M → M ×M is the projection along the factor not named. The convolution product equips H _∗ ^G (Z, k) with a S G -algebra structure and H _∗ ^G (M, k) with a H _∗ ^G (Z, k)-module structure, see e.g., [CG]. Assume now that the T -spaces M , Z are equivariantly formal, see e.g., [GKM, Sec. 1.2], and assume that we have the following equality of T -fixed points subsets

(3.3) Z ^T = M ^T × M ^T .

Consider the following commutative diagram of algebra homomorphisms H _∗ ^T (Z, k) ⊗ S Q ^c // End S (H _∗ ^T (M, k)) ⊗ S Q

H _∗ ^T (Z, k)

b

OO // End S (H ^T _∗ (M, k))

OO

H _∗ ^G (Z, k)

a

OO // End S

(H ^G _∗ (M, k)).

OO

The map c is invertible by (3.3) and the localization theorem in equivariant ho- mology. The map b is injective because Z is equivariantly formal. The map a is injective, compare Section 4.10 below. Thus the lower map is injective, i.e., the H _∗ ^G (Z, k)-module H _∗ ^G (M, k) is faithful.

⊓

⊔

4. The polynomial representation of the graded algebra ^θ Z _{Λ Λ} ^δ _Λ,V Fix a quiver Γ with set of vertices I and set of arrows H. Fix an involution θ on Γ. Assume that Γ has no 1-loops and that θ has no fixed points. Fix a dimension vector ν 6= 0 in ^θ N I and a dimension vector λ in N I. Set |ν | = 2m. Fix an object (V, ̟) in ^θ V V V ν and an object Λ Λ Λ in V V V λ . The main result of this section is Theorem 4.17 which yields an explicit faithful representation of the graded k-algebra ^θ Z _Λ ^δ _{Λ Λ,V} . 4.1. Notations. Let G = O(V, ̟) be the orthogonal group, and F = F (V, ̟) be the isotropic flag manifold. We can regard F as the (non connected) flag manifold of the (non connected) group G. Next, the group ^θ G V is canonically identified with a L´evi subgroup of G, i.e., with the subgroup of elements which preserve the decomposition V = L

i V i . Then ^θ F V is canonically identified with the closed subvariety of F consisting of all flags which are fixed under the action of the center of ^θ G V . Fix once for all a maximal torus T of ^θ G V . Let W V and W be the Weyl groups of the pairs ( ^θ G V , T ) and (G, T ). The canonical inclusion ^θ G V ⊂ G yields a canonical inclusion W V ⊂ W .

4.2. The root systems. Fix once for all a T -fixed flag φ V in ^θ F V . We fix once for all one-dimensional T -submodules D _1−m , . . . , D _m−1 , D m of V such that

φ V = (V ^l ), V ^l = D l+1 ⊕ · · · ⊕ D m−1 ⊕ D m .

Let χ l ∈ t ^∗ be the weight of D l . Note that D l ≃ V ^l−1 /V ^l and that the bilinear form ̟ yields a non-degenerate pairing (V ^l−1 /V ^l ) × (V ^−l /V ^1−l ) → C, because (V ^l ) ^⊥ = V ^−l . Thus we have

χ 1−l = −χ l .

Let B be the stabilizer of the flag φ V in G. Let ∆ be the set of roots of (G, T ) and let ∆ ⁺ be the subset of positive roots relative to the Borel subgroup B. We abbreviate ∆ ⁻ = −∆ ⁺ . Let Π be the set of simple roots in ∆ ⁺ . We have

∆ ⁺ = {χ k ± χ l ; 1 6 l < k 6 m}, Π = {χ l+1 − χ l , χ 2 + χ 1 ; l = 1, 2, . . . , m − 1}.

Let 6 and ℓ denote the Bruhat order and the length function on W . Note that W is an extended Weyl group of type D m . In particular we have

ℓ(w) = 0 ⇐⇒ w = e, ε 1 ,

where ε 1 is as below, and the set S of simple reflections is given by S = {s 0 , s 1 , . . . , s _m−1 },

with s k , k = 0, 1, . . . , m − 1 the reflection with respect to

α 0 = χ 2 + χ 1 , α 1 = χ 2 − χ 1 , . . . α m−1 = χ m − χ m−1 .

Note that u(∆ ⁺ ) = ∆ ⁺ if ℓ(u) = 0. Next, let ^θ ∆ V ⊂ ∆ be the set of roots of ( ^θ G V , T ). Note that ^θ G V is a product of general linear groups (this is due to the fact that θ has no fixed points). Indeed, we can (and we will) assume that

θ ∆ V ⊂ {χ l − χ k ; l 6= k, l, k = 1, 2, . . . , m}.

More precisely, given a subset J ⊂ I such that I = J ⊔ θ(J ) it is enough to choose the flag φ V such that V ⁰ = L

j∈J V j . Finally, let ^θ ∆ ⁺ _V be the subset of positive roots relative to the Borel subgroup ^θ B V = B ∩ ^θ G V . We have

θ ∆ ⁺ _V = ∆ ⁺ ∩ ^θ ∆ V .

4.3. The wreath product. Let S _m be the symmetric group, and Z ₂ = {−1, 1}.

Consider the wreath product W m = S _m ≀ Z ₂ . For l = 1, 2, . . . m let ε l ∈ ( Z ₂ ) ^m be

−1 placed at the l-th position. We’ll regard ε l as in element of W m in the obvious way. There is a unique action of W m on the set {1 − m, . . . , m − 1, m} such that S _m permutes 1, 2, . . . m and such that ε l fixes k if k 6= l, 1 − l and switches l and 1 − l. The group W m acts also on ^θ I ^ν . Indeed, view a sequence i as the map

{1 − m, . . . , m − 1, m} → I, l 7→ i l . Then we set w(i) = i ◦ w ⁻¹ for w ∈ W m .

4.4. The W -action on the set of T -fixed flags. The sets F ^T and ( ^θ F V ) ^T consisting of the flags which are fixed by the T -action are equal. The group W acts freely transitively on both. We’ll write e for the unit in W . Put

φ V,w = w(φ V ), ∀w ∈ W.

Thus we have F ^T = {φ V,w ; w ∈ W }. There is a unique group isomorphism W = W m such that

φ V,w = (V _w ^l ), V ^l _w = D w(l+1) ⊕ · · · ⊕ D _w(m−1) ⊕ D w(m) .

We’ll use this identification whenever it is convenient without recalling it explicitly.

We set also

w(χ l ) = χ _w(l) , ∀w, l.

Let ^θ B V,w be the stabilizer of the flag φ V,w under the ^θ G V -action. It is the Borel subgroup of ^θ G V containing T associated with the set of positive roots

w(∆ ⁺ ) ∩ ^θ ∆ V .

Let ^θ N V,w be the unipotent radical of ^θ B V,w . Finally, let i w be the unique sequence in ^θ I ^ν such that φ V,w lies in ^θ F V,i

. Write

(4.1) i e = (i 1−m , . . . , i m−1 , i m ).

Since φ V is a flag of type i e , we have

D l ⊂ V i

, w ⁻¹ (i e ) = i w = (i _w(1−m) , . . . , i _w(m−1) , i w(m) ).

Let W ν be the image of the group W V by the isomorphism W → W m . It is the parabolic subgroup given by

W ν = {w ∈ W m ; w(i e ) = i e }.

Note that the choices made in Section 4.2 imply that

(4.2) W ν ⊂ S _m .

There is a bijection

W ν \ W m → ^θ I ^ν , W ν w 7→ i w . For each i in ^θ I ^ν we have

( ^θ F e V,i ) ^T ≃ ( ^θ F V,i ) ^T = {φ V,w ; w ∈ W i }, W i = {w ∈ W ; i w = i}.

We’ll abbreviate

θ F V,w = ^θ F V,i

, W w = W i

, ^θ π Λ Λ Λ,w = ^θ π Λ Λ Λ,i

.

We’ll also omit the symbol w if w = e. For instance we write ^θ B V = ^θ B V,e and

θ N V = ^θ N V,e . Note that W w = W V w and that we have an isomorphism of ^θ G V - varieties

θ G V / ^θ B V,w → ^θ F V,w , g 7→ gφ V,w .

4.5. The stratification of ^θ F V × ^θ F V . The group G acts diagonally on F × F . The action of the subgroup ^θ G V preserves the subset ^θ F V × ^θ F V . For w ∈ W let

θ O ^w _V be the set of all pairs of flags in ^θ F V × ^θ F V which are in relative position w.

More precisely, we write

θ O ^w _V = ( ^θ F V × ^θ F V ) ∩ (Gφ V,e,w ), φ V,x,y = (φ V,x , φ V,y ), ∀x, y ∈ W.

Let ^θ O ¯ ^w _V be the Zariski closure of ^θ O _V ^w . For any w, x, y in W we write also

θ O ^w _V,x,y = ^θ O ^w _V ∩ ( ^θ F V,x × ^θ F V,y ), ^θ O ¯ _V,x,y ^w = ^θ O ¯ ^w _V ∩ ( ^θ F V,x × ^θ F V,y ).

We define ^θ P V,w,ws , s ∈ S , as the smallest parabolic subgroup of ^θ G V containing

θ B V,w and ^θ B V,ws .

4.6. Lemma. Let w, x, y, s, u ∈ W .

(a) The set of T -fixed elements in ^θ O ^x _V is {φ V,w,wx ; w ∈ W }.

(b) Assume that ℓ(u) = 0. We have ^θ O ¯ _V ^u = ^θ O ^u _V . It is a smooth ^θ G V -variety isomorphic to ^θ F V . We have ^θ O ^u _V,x,y = ∅ unless y = xu.

(c) Assume that ℓ(s) = 1. Set s = s ^′ u with s ^′ ∈ S and ℓ(u) = 0. We have

θ O ¯ _V ^s = ^θ O ^s _V ∪ ^θ O ^u _V . It is a smooth variety. We have ^θ O ¯ _V,x,y ^s = ∅ if y 6= xs, xu.

• If xs / ∈ W xu then

θ F V,xs 6= ^θ F V,xu , ^θ B V,xs = ^θ B V,x , ^θ O ^u _V,x,xs = ^θ O _V,x,xu ^s = ∅.

• If xs ∈ W xu then

θ F V,xs = ^θ F V,xu , ^θ B V,xs 6= ^θ B V,x ,

θ G V ×

_B

( ^θ P V,x,xs / ^θ B V,x ) → ^{∼ θ} O ¯ ^s V,x,xu = ^θ O ¯ ^s V,x,xs , (g, h) 7→ (gφ V,x , ghφ V,xu ).

Proof : The proof is standard and is left to the reader. Note that ^θ B V,x = ^θ B V,xu

and ^θ B V,xs = ^θ B V,xs

because ℓ(u) = 0. Note also that

θ B V,x = ^θ B V,xs ⇐⇒ x(α) ∈ / ^θ ∆ V ⇐⇒ xs ^′ ∈ W x ⇐⇒ xs ∈ W xu , where α is the simple root associated with s ^′ .

⊓

⊔ For a future use let us introduce the following notation. Let q be the obvious projection ^θ Z Λ Λ Λ,V → ^θ F V × ^θ F V , and, for each x ∈ W , let ^θ Z _{Λ Λ Λ,V} ^x be the Zariski closure in ^θ Z Λ Λ Λ,V of the locally closed subset q ⁻¹ ( ^θ O _V ^x ).

4.7. Euler classes in S. Consider the graded k-algebra S = S T . The weights χ 1 , χ 2 , . . . χ m are algebraically independent generators of S and they are homoge- neous of degree 2. The reflection representation on t yields a W -action on S. Recall that we have

w(χ l ) = χ w(l) , ∀l, w.

Now, let M be a finite dimensional representation of t and fix a linear form λ ∈ t ^∗ . Let M [λ] ⊂ M be the weight subspace associated with λ. The character of M is the linear form ch(M ) = P

λ dim(M [λ]) λ. Let eu(M ) be the determinant of M , viewed as an element of degree 2dim(M ) of S. We’ll call eu(M ) the Euler class of M . If M is a finite dimensional representation of T let eu(M ) be the Euler class of the differential of M , a module over t. Now, assume that X is a quasi-projective T - variety and that x ∈ X ^T is a smooth point of X. The cotangent space T _x ^∗ X at x is equipped with a natural representation of T . We’ll abbreviate eu(X, x) = eu(T _x ^∗ X ).

We’ll be particularly interested in the following elements

Λ w = eu( ^θ F e Λ Λ Λ,V , φ V,w ), Λ ^x _w,w

= eu( ^θ Z _{Λ Λ Λ,V} ^x , φ V,w,w

) ⁻¹ where ℓ(x) = 0, 1. Note that Λ w lies in S and has the degree 2d λ,w .

4.8. Description of the ^θ G V -varieties ^θ F e Λ Λ Λ,V,w . Let ^θ g V , t, ^θ n V,w , w ∈ W , be the Lie algebras of ^θ G V , T , ^θ N V,w respectively. Consider the flag

φ V,w = (V = V ^−m _w ⊃ · · · ⊃ V ^m−1 _w ⊃ V ^m _w = 0).

The ^θ G V -action on ^θ E Λ Λ Λ,V yields a representation of ^θ B V,w on the space

θ e Λ Λ Λ,V,w = {(x, y) ∈ ^θ E Λ Λ Λ,V ; x(V ^l _w ) ⊂ V ^l _w , y(Λ Λ Λ) ⊂ V ^m _w }.

There is an isomorphism of ^θ G V -varieties

θ G V ×

_B

θ e _{Λ Λ Λ,V,w} → ^θ F e Λ Λ Λ,V,w , (g, x, y) 7→ (gφ V,w , gx, gy).

Under this isomorphism the map ^θ π Λ Λ Λ,w is identified with the map

θ G V ×

B

θ e Λ Λ Λ,V,w → ^θ E Λ Λ Λ,V , (g, x, y) 7→ (gx, gy).

4.9. Character formulas. In this section we gather some character formula for a later use. For w, w ^′ ∈ W we write

θ e _{Λ Λ Λ,V,w,w}

= ^θ e _{Λ Λ Λ,V,w} ∩ ^θ e _{Λ Λ Λ,V,w}

,

θ d _Λ Λ Λ,V,w,w

= ^θ e _Λ Λ Λ,V,w / ^θ e _Λ Λ Λ,V,w,w

,

θ n _V,w,w

= ^θ n _V,w ∩ ^θ n _V,w

,

θ m V,w,w

= ^θ n V,w / ^θ n V,w,w

. We have the following T -module isomorphisms

θ n V,w = ^θ n V,w,w

⊕ ^θ m V,w,w

,

θ e _Λ Λ Λ,V,w = ^θ e _Λ Λ Λ,V,w,w

⊕ ^θ d _Λ Λ Λ,V,w,w

,

θ m V,w,w

= ( ^θ m V,w

,w ) ^∗ . Write ^θ e _V,w = ^θ e _{0},V,w . As T -modules we have

(4.3)

θ n V,w = M

α

g[α], α ∈ w(∆ ⁺ ) ∩ ^θ ∆ V ,

θ e _V,w = M

α

θ E V [α], α ∈ w(∆ ⁺ ).

Recall that V = L

l D l as I-graded T -modules, where l = 1−m, . . . , m−1, m. We’ll use the notation in (4.1). Thus i l , χ l are the dimension vector and the character of D l . Note that

h i

,i

= h i

,i

, χ l = −χ 1−l , ν θ(i) = ν i , ν = X

i

ν i i = X m l=1

(i l + i _1−l ).

Set H ⁰ = {h ∈ H; h ^′ = θ(h ^′′ )}, H ¹ = H \ H ⁰ , and λ = P

i λ i i. Note that H ⁰ = {h ∈ H ; h = θ(h)}. Decomposing a tuple x ∈ ^θ E V as the sum of (x h ) _h∈H

and (x h ) _h∈H

we get the following formula

dim( ^θ E V ) = X

h∈H

ν h

ν h

/2 + X

h∈H

ν h

(ν h

− 1)/2.

Next, the decomposition (4.3) yields the following formula ch( ^θ e V,w ) = X

χ

−χ

∈w(∆

)

h i

,i

(χ l − χ k ).

Here the sum runs over α ∈ w(∆ ⁺ ), and for each α we choose one pair (l, k) such that α = χ l − χ k . In a similar way we have also

ch( ^θ E V ) = X

χ

−χ

∈∆

h i

,i

(χ l − χ k ),

ch(L Λ Λ Λ,V ) = X

l

λ i

χ l .

Here the first sum runs over ∆. Since V ⁰ = L

l>1 D l we have also (4.4) ch( ^θ e Λ Λ Λ,V,w ) = X

χ

−χ

h i

,i

(χ l − χ k ) + X

l

λ i

χ l ,

where the first sum runs over all roots in w(∆ ⁺ ) and the second one over all l in {w(1), w(2), . . . , w(m)}. Note that (4.4) can be rewriten in the following way

ch( ^θ e _{Λ Λ Λ,V,w} ) = X

χ

−χ

∈∆

h i

,i

w(χ l − χ k ) + X

16l6m

λ i

w(χ l ).

By (4.3) the Euler class eu( ^θ n _V,w ) is the product of all roots in ^θ ∆ V ∩ w(∆ ⁺ ).

Therefore, for s ∈ S the following formulas hold

• either ws / ∈ W w and we have

eu( ^θ n _V,ws ) = eu( ^θ n _V,w ), eu( ^θ m _V,w,ws ) = eu( ^θ m _V,ws,w ) = 0,

• or ws ∈ W w and we have

eu( ^θ n _V,ws ) = −eu( ^θ n _V,w ), eu( ^θ m _V,w,ws ) = −eu( ^θ m _V,ws,w ) = w(α), where α is the simple root associated with s.

Finally, let s = s l with l = 0, 1, . . . , m − 1. Formula (4.4) yields the following.

• We have

eu( ^θ d Λ Λ Λ,V,w,wε

) = w(χ 1 ) ^λ

.

• If l 6= 0 we have

eu( ^θ d _Λ Λ Λ,V,w,ws

) = w(α l ) ^h

.

• We have

eu( ^θ d _{Λ Λ Λ,V,w,ws}

) = w(χ 1 ) ^λ

w(χ 2 ) ^λ

w(α 0 ) ^h

.

4.10. Reduction to the torus. The restriction of functions from ^θ g V to t gives an isomorphism of graded k-algebras

θ S V = k[χ 1 , χ 2 , . . . , χ m ] ^W

.

The group ^θ G V is a product of several copies of the general linear group. Hence it is connected with a simply connected derived subgroup. It is a general fact that if X is a ^θ G V -variety then the S-module H _∗ ^T (X, k) is equipped with a S-skewlinear representation of the group W V such that the forgetful map gives a ^θ S V -module isomorphism

H _∗

^G

(X, k) → H _∗ ^T (X, k) ^W

,

see e.g., [HS, thm. 2.10]. We’ll call this action on H _∗ ^T (X, k) the canonical W V -

action.

4.11. The W -action and the ^θ S V -action on ^θ F Λ Λ Λ,V . Fix a tuple i in ^θ I ^ν and an integer l = 1, 2, . . . , m. We define O Λ Λ Λ,V,i (l) to be the ^θ G V -equivariant line bundle over ^θ F e Λ Λ Λ,V,i whose fiber at the triple (x, y, φ) with

φ = (V = V ^−m ⊃ V ^1−m ⊃ · · · ⊃ V ^m = 0)

is equal to V ^l−1 /V ^l . Assigning to a formal variable x i (l) of degree 2 the first equivariant Chern class of O Λ Λ Λ,V,i (l) ⁻¹ we get a graded k-algebra isomorphism

k[x i (1), x i (2), . . . x i (m)] = H

^∗ G

( ^θ F e Λ Λ Λ,V,i , k).

So (3.1), (3.2) yield canonical isomorphisms of graded k-vector spaces (4.5) k[x i (1), x i (2), . . . x i (m)] = ^θ F Λ Λ Λ,V,i [−2d λ,i ] = ^θ F Λ Λ Λ,V,i . For a future use we set also

x i (l) = −x i (1 − l), l = 1 − m, 2 − m, . . . , 0.

For w ∈ W m we set

wf (x i (1), . . . , x i (m)) = f (x w(i) (w(1)), . . . , x w(i) (w(m))).

This yields a W m -action on ^θ F Λ Λ Λ,V such that w( ^θ F Λ Λ Λ,V,i ) = ^θ F Λ Λ Λ,V,w(i) .

The multiplication of polynomials equip both ^θ F Λ Λ Λ,V,i and ^θ F Λ Λ Λ,V with an obvious structure of graded k-algebras. For w ∈ W i the pull-back by the inclusion {φ V,w } ⊂

θ F e Λ Λ Λ,V,i yields a graded k-algebra isomorphism

(4.6) ^θ F Λ Λ Λ,V,i → S, f (−x i (1), . . . , −x i (m)) 7→ f (χ w(1) , . . . χ w(m) ).

We’ll abbreviate

w(f ) = f (χ w(1) , . . . χ w(m) ).

The isomorphism (4.6) is not canonical, because it depends on the choice of w.

Now, consider the canonical ^θ S V -action on ^θ F Λ Λ Λ,V coming from the ^θ G V -equivariant cohomology. It can be regarded as a ^θ S V -action on L

i k[x i (1), x i (2), . . . x i (m)]

which is described in the following way. The composition of the obvious projec- tion ^θ F Λ Λ Λ,V → ^θ F Λ Λ Λ,V,i with the map (4.6) identifies the graded k-algebra of the W m -invariant polynomials in the x i (l)’s, with

θ S V = S ^W

= k[χ 1 , χ 2 , . . . , χ m ] ^W

.

This isomorphism does not depend on the choice of i, w. The ^θ S V -action on ^θ F Λ Λ Λ,V

is the composition of this isomorphism and of the multiplication by W m -invariant

polynomials.

4.12. Localization and the convolution product. Let Q be the fraction field of S. Write

θ F _{Λ Λ Λ,V} ^′ = H _∗ ^T ( ^θ F e Λ Λ Λ,V , k),

θ F _{Λ Λ Λ,V} ^′′ = ^θ F _{Λ Λ Λ,V} ^′ ⊗ S Q,

θ Z _{Λ Λ Λ,V} ^′ = H _∗ ^T ( ^θ Z Λ Λ Λ,V , k),

θ Z _{Λ Λ Λ,V} ^′′ = ^θ Z _{Λ Λ Λ,V} ^′ ⊗ S Q.

Let ψ w be the fundamental class of the singleton {φ V,w } in ^θ F _{Λ Λ Λ,V} ^′ , and let ψ w,w

be the fundamental class of {φ V,w,w

} in ^θ Z _{Λ Λ Λ,V} ^′ . Let ψ w , ψ w,w

denote also the corresponding elements in the Q-vector spaces ^θ F _{Λ Λ Λ,V} ^′′ , ^θ Z _{Λ Λ Λ,V} ^′′ . Now, we consider the convolution products

θ Z _{Λ Λ Λ,V} ^′ × ^θ Z _{Λ Λ Λ,V} ^′ → ^θ Z _{Λ Λ Λ,V} ^′ , ^θ Z _{Λ Λ Λ,V} ^′ × ^θ F _{Λ Λ Λ,V} ^′ → ^θ F _{Λ Λ Λ,V} ^′

relative to the inclusion of ^θ Z Λ Λ Λ,V in the smooth scheme ^θ F e Λ Λ Λ,V × ^θ F e Λ Λ Λ,V . Both may be denoted by the symbol ⋆. We’ll use the notation in (4.1).

4.13. Proposition. (a) The S-modules ^θ F _{Λ Λ Λ,V} ^′ and ^θ Z _{Λ Λ Λ,V} ^′ are free. The canonical W V -action on the T -equivariant homology spaces ^θ F _{Λ Λ Λ,V} ^′ and ^θ Z _{Λ Λ Λ,V} ^′ is given by w(ψ x ) = ψ wx and w(ψ x,y ) = ψ wx,wy . The inclusions ^θ Z Λ Λ Λ,V ⊂ ^θ Z _{Λ Λ Λ,V} ^′ and ^θ F Λ Λ Λ,V ⊂

θ F _{Λ Λ Λ,V} ^′ commute with the convolution products.

(b) The elements ψ w , ψ w,w

yield Q-bases of ^θ F _{Λ Λ Λ,V} ^′′ , ^θ Z _{Λ Λ Λ,V} ^′′ respectively. For each i the map (4.5) yields an inclusion of k[x i (1), . . . , x i (m)] into ^θ F _{Λ Λ Λ,V,i} ^′′ such that

f (−x i (1), . . . , −x i (m)) 7→ X

w∈W

w(f )Λ ⁻¹ _w ψ w . (c) We have ψ w

,w ⋆ ψ w = Λ w ψ w

and ψ w

,w

⋆ ψ w

,w = Λ w

ψ w

,w . (d) If ℓ(s) = 0, 1 then [ ^θ Z _{Λ Λ Λ,V} ^s ] = P

w,w

Λ ^s _w,w

ψ w,w

in ^θ Z _{Λ Λ Λ,V} ^′′ . (e) We have Λ w = eu( ^θ e _{Λ Λ} ^∗ _Λ,V,w ⊕ ^θ n _V,w ).

(f ) If ℓ(u) = 0 then Λ ^u _w,w

= 0 if w ^′ 6= wu, and

Λ ^e _w,w = Λ ⁻¹ _w , Λ ^ε _w,wε

= (χ w(0) ) ^λ

Λ ⁻¹ _w = (χ w(1) ) ^λ

Λ ⁻¹ _wε

. (g) If l = 0, 1, . . . , m − 1 then

• either ws l ∈ / W w and

Λ ^s _w,ws

= Λ ^s _w,w

= eu( ^θ e ^∗ _{Λ Λ Λ,V,w,ws}

⊕ ^θ n _V,w ) ⁻¹ ,

• or ws l ∈ W w and

Λ ^s _w,w

= eu( ^θ e _Λ ^∗ _{Λ Λ,V,w,ws}

⊕ ^θ n V,w ⊕ ^θ m V,w,ws

) ⁻¹ , Λ ^s _w,ws

= eu( ^θ e ^∗ _{Λ Λ Λ,V,w,ws}

l

⊕ ^θ n _V,w ⊕ ^θ m _V,ws

_,w ) ⁻¹ .

Proof : Parts (a) to (d) are left to the reader. The fiber at φ V,w of the vector bundle

p : ^θ F e Λ Λ Λ,V → ^θ F V

is isomorphic to ^θ e _Λ Λ Λ,V,w as a T -module. Thus the cotangent space to ^θ F e Λ Λ Λ,V at the point φ V,w is isomorphic to ^θ e ^∗ _{Λ Λ Λ,V,w} ⊕ ^θ n _V,w as a T -module. This yields (e). Next, observe that the variety ^θ Z _{Λ Λ Λ,V} ^s is smooth if ℓ(s) 6 1. First, assume that ℓ(u) = 0.

The fiber at φ V,w,w

of the vector bundle

q : ^θ Z _{Λ Λ Λ,V} ^u → ^θ F V × ^θ F V

is isomorphic to ^θ e _Λ Λ Λ,V,w,wu as a T -module if w ^′ = wu and it is zero else. Thus we have

Λ ^u _w,wu = eu( ^θ d _{Λ Λ} ^∗ _Λ,V,w,wu )eu( ^θ e ^∗ _{Λ Λ Λ,V,w} ) ⁻¹ eu( ^θ F V , φ V,w ) ⁻¹ ,

= eu( ^θ d _{Λ Λ} ^∗ _Λ,V,w,wu )Λ ⁻¹ _w . Therefore, Section 4.9 yields

Λ ^u _w,wu =

( Λ ⁻¹ _w if u = e, (−χ w(1) ) ^λ

Λ ⁻¹ _w if u = ε 1 . Note that

χ _w(1−l) = −χ w(l) , ∀w, l.

This yields (f ). Finally, let us concentrate on part (g). The fiber at φ V,w,w

of the vector bundle

q : ^θ Z _{Λ Λ Λ,V} ^s

→ ^θ F V × ^θ F V

is isomorphic to ^θ e Λ Λ Λ,V,w,ws

as a T -module if w ^′ = w, ws l and it is zero else. There- fore, we have

Λ ^s _w,w

=

eu( ^θ e ^∗ _{Λ Λ Λ,V,w,ws}

l

) ⁻¹ eu( ^θ O ¯ ^s _V

, φ V,w,w

) ⁻¹ if w ^′ = w, ws l

0 else.

Next, by Lemma 4.6(c), if ws l ∈ / W w the cotangent spaces to the variety ^θ O ¯ ^s _V

at the points φ V,w,ws

and φ V,w,w are given by

T _w,ws ^∗

^θ O ¯ _V ^s

= T _w,ws ^∗

^θ O _V ^s

= ^θ n _V,w,ws

= ^θ n _V,w , T _w,w ^{∗ θ} O ¯ _V ^s

= T _w,w ^{∗ θ} O V ^e = ^θ n _V,w,w = ^θ n _V,w .

Thus they are both isomorphic to ^θ n V,w as T -modules. Similarly, if ws l ∈ W w the cotangent spaces to the variety ^θ O ¯ ^s _V

at the points φ V,w,ws

, φ V,w,w are given by

T _w,ws ^∗

^θ O ¯ _V ^s

= ^θ n V,w ⊕ ^θ m V,ws

,w , T _w,w ^{∗ θ} O ¯ _V ^s

= ^θ n V,w ⊕ ^θ m V,w,ws

,

because Lie( ^θ P V,w,ws

)/Lie( ^θ B V,w ) is dual to ^θ m V,w,ws

= ^θ n V,w / ^θ n V,w,ws

as a T - module.

⊓

⊔