Geometric Satake equivalence and intersection form over intersection homology groups

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Geometric Satake equivalence and intersection form over intersection homology groups

Arnaud Demarais

To cite this version:

Arnaud Demarais. Geometric Satake equivalence and intersection form over intersection homology

groups. 2016. �hal-01406557�

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INTERSECTION HOMOLOGY GROUPS

A. DEMARAIS

Abstract. The geometric Satake correspondence is an equivalence between the category of finite dimensional representations of the Langlands dualGˆof a complex connected reductive algebraic groupGand the category ofG(C[[z]])-equivariant perverse sheaves on the aﬃne GrassmannianGr=G(C((z)))/G(C[[z]]).

Through this correspondence, the highest weight module L( ) can be seen as the intersection homology group of a Schubert varietyGr . The purpose of this paper is to present the precise relationship between the intersection formh., .ion

IH

_•

(Gr , C )

and the contravariant form onL( ).

1. Notations

Let G be a complex connected reductive algebraic group. We denote by G ˇ the Langlands dual group of G. This is the complex connected reductive group whose root datum is obtained from that of G by switching roots and coroots, characters and cocharacters. We write O for the formal power series ring C [[z]] and K for its fraction field. The aﬃne Grassmannian Gr is defined as the quotient G( K )/G( O ).

Let us fix a maximal torus T ⇢ G. Let W denote the Weyl group and X

_⇤

(T ) = Hom( C

^⇤

, T ) the cocharacter lattice. We choose a Borel subgroup B T. We denote its unipotent radical by N. We use the convention that the roots in B are the positive ones. For and µ in X

_⇤

(T ), we say that  µ when µ is a sum of positive roots. Let ⇢ denote half the sum of the positive roots.

Each coweight 2 X

_⇤

(T ) gives an homomorphism K

^⇤

! T ( K ), considering the image of (t) in the quotient G( K )/G( O ), we get an element L of Gr. We denote by Gr the set G( O ).L . Every orbit under the action by left multiplication of G( O ) is of the form Gr where is a dominant coweight. It is known that Gr is simply connected and is an algebraic variety of finite dimension 2⇢( ). We denote by B the Borel subgroup opposite to B with respect to T and by N its unipotent radical. For each ⌫ in X

_⇤

(T) we consider the sets T

⌫

= N ( K ).L

⌫

and S

⌫

= N( K ).L

⌫

.

We consider the stratification of Gr by the G( O ) -orbits. This is a Whitney stratification that we denote by S. We denote by D

_S

(Gr) the bounded derived category of S-constructible sheaves with coeﬃcients in C and by P

_S

(Gr) the full subcategory of perverse sheaves.

Lastly, we denote the inclusion Gr , ! Gr by j and the trivial local system on Gr by 1. We denote by I the intersection complex j

!

IC (Gr , 1)[dim(Gr )]. The sheaves I are the simple objects of the category P

_S

(Gr).

The geometric Satake correspondence is the following equivalence of categories (see [8] Theorem 7.3) : Theorem 1. The category P

_S

(Gr) and the category of finite dimensional rational representations of G ˇ are equivalent. The underlying vector space of the representation associated to a sheaf F is the cohomology group H

^•

(Gr, F ) .

For a dominant coweight 2 X

_⇤

(T ), we denote by L( ) the simple G-module with highest weight . ˇ Then our functor sends I to L( ).

1

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GEOMETRIC SATAKE EQUIVALENCE AND INTERSECTION FORM OVER INTERSECTION HOMOLOGY GROUPS 2

2. Verdier’s duality

As Gr is a projective variety, we have H

_c^p

(Gr , A ) = H

^p

(Gr , A ) for all p and A 2 D

_S

(Gr ) . We denote by C the constant sheaf on Gr and we make identification H

^p

(Gr , I ) = Hom

_D

S(Gr )

( C , I [p]) for all p 2 Z. We denote by D the dualising complex of Gr . It represents the functor

F : A 7! (Hom

_D

S(Gr )

( C , A ))

^⇤

from D

_S

(Gr ) to V ect

_C

; in other words there is a natural isomorphism



_A

: Hom

_D

S(Gr )

( A , D ) ! F( A ) (see [3] theorem V.2.1). Let ´

2 F ( D ) denote the image of the identity id

_D

by 

_D

. We regard ´ as a linear form on L

k2Z

H

^k

(Gr , D ) that is non-zero only in degree 0. Yoneda’s lemma ensures that 

_A

sends f 2 Hom

_D

S(Gr )

( A , D ) to the linear form g 7! ´

f g on Hom

_D

S(Gr )

( C , A ).

The dualising complex D allows us to define the Verdier duality functor D = RH om(., D ) in the category D

_S

(Gr ). Equality 8.6 in [8] says that there exists an isomorphism ⌘ : I ! D (I ).

Let x and y be elements of H

^•

(Gr , I ) , homogeneous of degrees respectively k and k

⁰

. So x 2 Hom

_D

S(Gr )

( C , I [k]) and y 2 Hom

_D

S(Gr )

( C , I [k

⁰

]). We want to define the cup product x [ y.

Proposition 2.6.1 in [6] assures that we have a natural isomorphism of functors Hom

_D

S(Gr )

( A ⌦

^L

B , C ) ⇠ = Hom

_D

S(Gr )

( A , RH om( B , C )) for all A , B , C 2 D

_S

(Gr ) . In particular, there is a natural transformation

T : Hom

_D

S(Gr )

(., D (I )) !

^⇠⁼

Hom

_D

S(Gr )

(. ⌦

^L

I , D ).

We denote by 2 Hom

_D

S(Gr )

(I ⌦

^L

I , D ) the image by T

I

of the isomorphism ⌘.

So we can compose and x ⌦

^L

y 2 Hom

_D_S_(Gr ₎

( C ⌦

^L

C , I ⌦

^L

I [k + k

⁰

]) .

For all A 2 D

S

(Gr), let r

_A

: A ⌦

^L

C ! A be the right contraction quasi-isomorphim and l

_A

: C ⌦

^L

A ! A be the left contraction quasi-isomorphism. Both are natural transformations.

We define the cup product x [ y to be the element (x ⌦

^L

y) l

_C¹

2 Hom

_D

S(Gr )

( C , D [k + k

⁰

]). With the identification we made at the beginning of this section, x [ y 2 H

^k+k⁰

(Gr , D ) and we define the intersection form on H

^•

(Gr , I ) ⇥ H

^•

(Gr , I ) to be

h x, y i = ˆ

x [ y = ˆ

(x ⌦

^L

y) l

_C¹

.

Theorem 2. The intersection form is a non degenerate bilinear form on H

^•

(Gr , I ).

Proof. We have to show that the map x 7! h x, . i from H

^•

(Gr , I ) to its dual is bijective.

We know that l is a natural transformation, so the following diagram is commutative :

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C

^y

// I [ p]

C ⌦

L

C

_L

1⌦y

//

l_C

OO

C ⌦

^L

I [ p]

lI

OO

Thus we have h x, y i = ´

(x ⌦

^L

y) l

_C¹

= ´

(x ⌦

^L

id

I

) l

_I¹

y.

Left composition with ⌘ and with l

_I¹

are isomorphisms. The natural transformation T

_C

, which maps ⌘ x to (x ⌦

^L

id

I

) is an isomorphism. The natural transformation 

I [ p]

, which maps f 2 Hom

_D_S_(Gr ₎

(I , D [p]) to the map g 7! ´

f g from Hom

_D

S(Gr )

( C , I [ p]) to C is also an isomorphism.

Consequently x 7! h x, . i = x 7! 

I [ p]

(T

_C

(⌘ x) l

_I¹

) and it is obviously an isomorphism. ⇤ 3. Properties of the intersection form

So far we have defined a non degenerate form on the cohomology of the perverse sheaf I . The subspace L( )

⌫

of all vectors of weight ⌫ in L( ) identifies with the cohomology in degree 2⇢(⌫) with support in T

⌫

, which we denote by H

_T^2⇢(⌫)

⌫

(Gr, I ). We investigate the behaviour of the intersection form regarding the weight decomposition (see [8], Theorem 3.6)

H

^•

(Gr, I ) = L

⌫2X⇤(T)

H

^2⇢(⌫)

T⌫

(Gr, I ).

Let w

0

be the longest element in the Weyl group.

Lemma 3. Let x and y be vectors in H

^•

(Gr, I ) of weights µ and ⌫ respectively. If h x, y i 6 = 0, then ⌫ = w

0

(µ).

Proof. Let us suppose that h x, y i 6 = 0. As stated in [8] theorem 3.6, the independence of the spaces H

_T^2⇢(⌫)

⌫

(Gr, I ) on the pair (T, B) gives us H

_T^2⇢(µ)

µ

(Gr, I ) ⇠ = H

_S^2⇢(µ)

w0 (µ)

(Gr, I ).

As h x, y i = ´

x [ y we must have x [ y 6 = 0. Equality II.10.1 in [3] tells us that x [ y 2 H

2⇢(µ)+2⇢(⌫)

Sw0 (µ)\T⌫

(Gr, D ).

But Lemma 2.1 in [5] with w = e and v = w

0

tells us that S

w0(µ)

\ T

⌫

is empty unless ⌫  w

0

(µ). Moreover, if ´

x [ y 6 = 0 , then x [ y must lie in degree 0 and so ⇢(⌫ w

0

(µ)) = 0 . Combining these two results we get

⌫ = w

0

(µ) . ⇤

Let e 2 H

²

(Gr, C ) be any cohomological class of degree 2 of Gr. We have the following crucial lemma : Lemma 4. For all x and y elements of H

^•

(Gr, I ), we have h x [ e, y i = h x, y [ e i .

Proof. Let x and y be two homogeneous elements in H

^•

(Gr, I ) of degree say k and k

⁰

. Let us recall that H

^p

(Gr, I ) ⇠ = Hom

_D

S(Gr )

( C , I [p]) for all p.

The cup product with e is defined as z [ e = r (z ⌦

^L

e) r

¹

for all z 2 H

^•

(Gr, I ) where r is the right contraction isomorphism defined above. From equality II.10.3 in [3] and the fact that e lies in even degree, we also have z [ e = l (e ⌦

^L

z) l

¹

for all z 2 H

^•

(Gr, I ) where l is the left contraction isomorphism.

So h x [ e, y i = ´

(r ⌦

^L

id

I

) (x ⌦

^L

e ⌦

^L

y) (r

¹

⌦

^L

id

_C

) l

_C¹

.

From proposition 2.2.3 in [1], we have r

_A

⌦

^L

id

_B

= id

_A

⌦

^L

l

_B

for all A , B 2 D

_S

(Gr ).

Combining all these results we have the following commutative diagram :

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GEOMETRIC SATAKE EQUIVALENCE AND INTERSECTION FORM OVER INTERSECTION HOMOLOGY GROUPS 4

C ⌦

^L

C ⌦

^L

C

^x

⌦eL ⌦y^L

//

idCL

⌦CL

⌦C

✏✏

I [k] ⌦

^L

C [2] ⌦

^L

I [k

⁰

]

idI

⌦LlI

))

id

I [k]L

⌦C[2]L

⌦I [k0]

✏✏

C ⌦

^L

C

id_C⌦l^L_C¹

55

r_C¹⌦^Lid_C

))

I ⌦

^L

I [k + k

⁰

+ 2].

C ⌦

^L

C ⌦

^L

C

x⌦e^L ⌦y^L

// I [k]

^L

⌦ C [2]

^L

⌦ I [k

⁰

]

rI

⌦LidI

55 Thus we have

h x [ e, y i = ˆ

(r

I

⌦

L

id

I

) (x ⌦

^L

e ⌦

^L

y) (r

_C¹

⌦

^L

id

_C

) l

_C¹

= ˆ

(id

I

⌦

L

l

I

) (x ⌦

^L

e ⌦

^L

y) (id

_C

⌦

^L

l

_C¹

) l

_C¹

= h x, y [ e i

and the lemma is proved. ⇤

4. Contravariant form

Through geometric Satake correspondence, H

^•

(Gr, I ) is identified with L( ) and H

^2⇢(µ)

Tµ

(Gr, I ) with L( )

µ

. We see the intersection form as a bilinear form on L( ), which we also denote by h ., . i .

Let us recall the definition of the contravariant form. For details, the reader is refered to [4], Kapitel 1. Let ˇ g be the Lie algebra of G. Let ˇ ˇ h be a Cartan subalgebra. We denote by n its dimension. Let (e

i

, f

i

, h

i

)

1in

denote the Chevalley generators and (↵

i

)

1in

the simple roots of ˇ g. Let ! denote the involutive antiautomorphism of U(ˇ g) defined by !(e

i

) = f

i

and !(h) = h for all Chevalley’s generators (e

i

, f

i

) and h 2 ˇ h. A contravariant form on the highest weight module L( ) is a bilinear, symmetric, non degenerate form that verifies (z.x, y) = (x, !(z).y) for all x, y 2 L( ) and z 2 U (ˇ g) . Such a form exists and is unique up to a multiplicative constant. Let v

high

be an highest weight vector of L( ).

Let i

1

, ..., i

p

be such that s

i1

...s

ip

is a reduced word for w

0

. We pose k

j

= s

ij+1

...s

ip

( )(h

ij

) for all 1  j < p and k

p

= (h

ip

). Proposition 28.1.2 in [7] ensures that

^f

k1 i1

k1!

...

^f

kp ip

kp!

does not depend on the choice of i

1

, ..., i

p

and proposition 28.1.4 in [7] tells us that

^f

k1 i1

k1!

...

^f

kp ip

kp!

.v

high

is a lowest weight vector. We denote it by v

low

. We normalize the contravariant form so that (v

low

, v

low

) = h v

low

, v

high

i.

Let us recall the definition of the Schützenberger involution. For details of the constructions stated below, we refer to [9], section 3.3.2.

For all i 2 { 1, ..., n } , we define i

^⇤

to be the unique index such that w

0

(↵

i

) = ↵

i^⇤

and we define the automorphism ⇣ of U (g) on the generators : ⇣(e

i

) = f

i^⇤

, ⇣(f

i

) = e

i^⇤

, ⇣(h

i

) = h

i^⇤

.

We define the Schützenberger involution S of L( ) by putting S(v

high

) =v

low

and S(a.x) = ⇣(a).S(x) for all a 2 U (g) and x 2 L( ).

We can now state the main theorem of this paper:

Theorem 5. For all (x, y) 2 L( )

²

, we have h x, y i = (x, S (y)) .

Proof. We choose a W -invariant scalar product on X

_⇤

(T ) and we define q(↵

i

) to be the square of the length

of the root ↵

i

. We consider e = P q(↵

i

)e

i

. The action of this principal nilpotent element can be identified

with the cup product with the first Chern class e of a line bundle on Gr (see [10] and [2], theorem 1.7.6).

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The highest and lowest weight subspaces of L( ) both have dimension 1. By definition we have h v

low

, v

high

i = (v

low

, v

low

) = (v

low

, S(v

high

))

so by linearity and orthogonality of the weight subspaces, the formula of the theorem is true for all x 2 L( )

w0( )

and y 2 L( ).

We proceed by induction on the height of x and prove that the following property is true for all possible heights.

P (n) : For each weight ⌫ of height n and each x 2 L( )

⌫

, we have h x, z i = (x, S(z)) for all z 2 L( ) . The only weight of lowest possible height is w

0

( ) so P(ht(w

0

( )) is true.

Suppose that P (n) is true.

Let ⌫ be a weight of height n + 1 and let x 2 L( )

⌫

. We can write x = P

q(↵

j

)e

j

x

j

where x

j

is a vector of weight ⌫ ↵

j

. By induction, we have h x

j

, z i = (x

j

, S(z)) for all z 2 L( ).

If y is a vector of weight diﬀerent from w

0

(⌫), then h x, y i = 0 and (x, S(y)) = 0 for weight reasons.

If y 2 L( )

w0(⌫)

, then h (q(↵

j

)e

j

.x

j

, y i = h e.x

j

, y i and (q(↵

j

)e

j

.x

j

, S(y)) = (e.x

j

, S(y)) for all j. From Lemma 5 and the induction hypothesis, we obtain:

h x, y i = X

h e.x

j

, y i = X

h x

j

, e.y i = X

(x

j

, S(e.y)) = X

(x

j

, ⇣(e).S(y)) = X

(!(⇣(e)).x

j

, S(y)).

But as we chose a W -invariant scalar product we have that q(↵

j

) = q(↵

j^⇤

) so !(⇣(e)) = e.

Thus

X (!(⇣(e)).x

j

, S(y)) = X

(e.x

j

, S(y)) = X

(q(↵

j

)e

j

.x

j

, S(y))

so h x, y i = (x, S(y)) for all y 2 L( ) and P(n + 1) is true. ⇤

One can easily check that the Schützenberger involution is an isometry for the contravariant form.

From Theorem 6, it follows that the intersection form on H

^•

(Gr, I ) is symmetric, which can also be estab- lished by geometric methods.

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