Introduction to Deligne-Lusztig Theory

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Introduction to Deligne-Lusztig Theory

C´edric Bonnaf´e

CNRS (UMR 6623) - Universit´e de Franche-Comt´e (Besan¸con)

Berkeley (MSRI), Feb. 2008

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Γ group acting onX :

set poset

...

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Γ group acting onX : set

poset

...

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poset

...

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poset

K-vector space

...

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poset

K-vector space variety

...

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poset

...

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poset

...

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Γ group acting onX : set K[X] poset

...

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Γ group acting onX : set K[X]

poset complex of chains K-vector space

variety ...

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poset complex of chains cohomology groups K-vector space

variety ...

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variety ´etale, `-adic cohomology ...

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variety ´etale, `-adic cohomology ...

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Deligne (SGA 4¹₂, 1976). “Les exposés I à VI de SGA 4 donnent la théorie générale des topologies de Grothendieck. Très détaillés, ils peuvent être précieux lors de l’étude de topologies exotiques, telle celle qui donne naissance à la topologie cristalline. Pour la topologie étale, si proche de l’intuition classique, un garde-fou si imposant n’est pas nécessaire : il suffit de connaˆıtre (par exemple), le livre de Godement, et d’avoir un peu de foi. (...)”

“Chapters I-VI of SGA 4 develop the general theory of Grothendieck topologies. Very detailed, they may be a valuable tool for studying exotic topologies, such as the one yielding the crystalline topology. For ´etale topology, so close to classical intuition, such imposing safetynet is not necessary : it is sufficient to know (for instance), Godement’s book, and to have some faith. (...)”

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Deligne (SGA 4¹₂, 1976). “Les exposés I à VI de SGA 4 donnent la théorie générale des topologies de Grothendieck. Très détaillés, ils peuvent être précieux lors de l’étude de topologies exotiques, telle celle qui donne naissance à la topologie cristalline. Pour la topologie étale, si proche de l’intuition classique, un garde-fou si imposant n’est pas nécessaire : il suffit de connaˆıtre (par exemple), le livre de Godement, et d’avoir un peu de foi. (...)”

“Chapters I-VI of SGA 4 develop the general theory of Grothendieck topologies. Very detailed, they may be a valuable tool for studying exotic topologies, such as the one yielding the crystalline topology.

For ´etale topology, so close to classical intuition, such imposing safetynet is not necessary : it is sufficient to know (for instance), Godement’s book, and to have some faith. (...)”

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Letp be a prime number,F=F^p

LetV be an algebraic variety over F LetΓ be a group acting onV Let` be a prime number 6=p

To this datum is associated a family of finite dimensional Q`-vector spaces H_cⁱ(V), which are acted on by Γ.

We denote byH_c^∗(V)the element of the Grothendieck group of the category of finite dimensionalQ`Γ-modules equal to

H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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Letp be a prime number,F=F^p LetV be an algebraic variety over F

LetΓ be a group acting onV Let` be a prime number 6=p

H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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Letp be a prime number,F=F^p LetV be an algebraic variety over F LetΓ be a group acting onV

Let` be a prime number 6=p

H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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Letp be a prime number,F=F^p LetV be an algebraic variety over F LetΓ be a group acting onV Let` be a prime number 6=p

H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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H_c^∗(V) =X

i

(−1)ⁱ [H_cⁱ(V)]

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}:

rule (3) ⇒

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

0 ^//H_c⁰(A¹) ^//H_c⁰(P¹) ^//H_c⁰({∞})

//H¹_c(A¹) ^//H¹_c(P¹) ^//H¹_c({∞})

//H²_c(A¹) ^//H²_c(P¹) ^//H²_c({∞}) ^//0

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

0 ^//0 ^//H_c⁰(P¹) ^//H_c⁰({∞})

//0 ^//H¹_c(P¹) ^//H¹_c({∞})

// Q` //H²_c(P¹) ^//H²_c({∞}) ^//0

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

0 ^//0 ^//H_c⁰(P¹) ^//Q`

//0 ^//H¹_c(P¹) ^//0

// Q` //H²_c(P¹) ^// 0 ^//0

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

0 ^//0 ^//H_c⁰(P¹) ^//Q`

//0 ^//H¹_c(P¹) ^//0

// Q` //H²_c(P¹) ^// 0 ^//0

H_cⁱ(P¹) =

Q` if i =0,2 0 otherwise

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H

ⁱ_c

(P

¹

) ?

P¹ =A¹∪{∞}: rule (3)⇒

0 ^//0 ^//H_c⁰(P¹) ^//Q`

//0 ^//H¹_c(P¹) ^//0

// Q` //H²_c(P¹) ^// 0 ^//0

H_cⁱ(P¹) =

Q` if i =0,2

0 otherwise as G-modules! (rule (7))

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The general frame.

Gconnected reductive group /F

F :G→GFrobenius endomorphism /F^q

LetG =G^F ={g ∈G |F(g) =g}: finite reductive group Example - G=GL_n(F),

F : G −→ G (a_ij) 7−→ (a^q_ij), G^F =GL_n(Fq).

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The general frame.

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The general frame.

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The general frame.

LetG =G^F ={g ∈G |F(g) =g}: finite reductive group

Example - G=GL_n(F),

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The general frame.

F : G −→ G (a_ij) 7−→ (a_ij^q),

G^F =GL_n(Fq).

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The general frame.

F : G −→ G (a_ij) 7−→ (a_ij^q), G^F =GL_n(Fq).

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Let L : G −→ G g 7−→ g⁻¹F(g) be the Lang map.

Then L is surjective and

L(g) =L(g⁰)⇐⇒∃ x ∈G^F,g⁰ =xg.

LetZ a subvariety of G (locally closed): thenG^F acts (on the left) onL⁻¹(Z).

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Let L : G −→ G g 7−→ g⁻¹F(g) be the Lang map.Then L is surjective and

L(g) =L(g⁰)⇐⇒∃x ∈G^F,g⁰ =xg.

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Let L : G −→ G g 7−→ g⁻¹F(g) be the Lang map.Then L is surjective and

L(g) =L(g⁰)⇐⇒∃x ∈G^F,g⁰ =xg.

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Deligne-Lusztig (1976) - Construction of a class of “interesting”

varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I Parametrization of irreducible characters

I Computation of their degrees

I Algorithm for values at a semisimple element

I ...

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varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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Drinfeld (1974) - Starting example: SL₂(Fq) acts on{(x,y)∈F² | xy^q−yx^q =1}

varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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Drinfeld (1974) - Starting example: SL₂(Fq)acts on {(x,y)∈F² |xy^q−yx^q=1}

varieties

Lusztig (1976-77-78-79-80-81-82-83-84) -

I ...

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The group G ×(µ_q+1ohFi) acts on A² and stabilizes Y ={(x,y)∈A² |xy^q−yx^q =1}

If θ∈(µ_q+1)^∧, let H_cⁱ(Y)_θ denote the θ-isotypic component of H_cⁱ(Y): it is a Q`G-module.

LetR_θ⁰ = −[H_c^∗(Y)θ] = [H_c¹(Y)θ] − [H_c²(Y)θ] This isDeligne-Lusztig induction.

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If θ∈(µ_q+1)^∧, let H_cⁱ(Y)_θ denote theθ-isotypic component of H_cⁱ(Y)

: it is a Q`G-module.

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If θ∈(µ_q+1)^∧, let H_cⁱ(Y)_θ denote theθ-isotypic component of H_cⁱ(Y): it is a Q`G-module.

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LetR_θ⁰ = −[H_c^∗(Y)θ] = [H_c¹(Y)θ] − [H_c²(Y)θ]

This isDeligne-Lusztig induction.

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

(x,y) 7−→ [x :y] isG-equivariant.

The image ofπ is P¹ \P¹(Fq).

π(x,y) =π(x⁰,y⁰)⇐⇒∃ ξ∈µ_q+1, (x⁰,y⁰) =ξ(x,y). Y

π

%%J

JJ JJ JJ JJ JJ JJ JJ JJ JJ JJ

Y/µq+1 P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π

%%J

Y/µq+1 P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π

%%J

Y/µq+1 P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π(x,y) =π(x⁰,y⁰)⇐⇒∃ ξ∈µ_q+1, (x⁰,y⁰) =ξ(x,y).

Y

π

%%J

Y/µq+1 P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π(x,y) =π(x⁰,y⁰)⇐⇒∃ ξ∈µ_q+1, (x⁰,y⁰) =ξ(x,y).

Y

π

%%J

Y/µq+1 P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π(x,y) =π(x⁰,y⁰)⇐⇒∃ ξ∈µ_q+1, (x⁰,y⁰) =ξ(x,y).

Y

π

%%J

Y/µq+1_ _ _ _ _ _//P¹\P¹(Fq)

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Quotient by µ

_q+1

.

The map

π: Y −→ P¹

π(x,y) =π(x⁰,y⁰)⇐⇒∃ ξ∈µ_q+1, (x⁰,y⁰) =ξ(x,y).

Y

π

%%J

Y/µq+1^_ ^_

∼

^_ ^_ ^_ ^_^//P¹\P¹(Fq)

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First decomposition of H

_c^∗

(Y)

H_cⁱ(Y) =0if i 6∈{1,2}(since Y is affine of dim. 1: rule (1aff)) H_c²(Y) =1G×T (because Y is irreducible: rule (2))

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

F(H_c¹(Y))_θ =H_c¹(Y)_θ⁻¹ (as G-modules)

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First decomposition of H

_c^∗

(Y)

H_cⁱ(Y) =0if i 6∈{1,2}(since Y is affine of dim. 1: rule (1aff))

H_c²(Y) =1G×T (because Y is irreducible: rule (2)) H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq))

|{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

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First decomposition of H

_c^∗

(Y)

H_c^∗(Y)₁ |{z}=

rule (5^∗)

H_c^∗(P¹\P¹(Fq)) |{z}=

rule (3^∗)

H_c^∗(P¹) −H_c^∗(P¹(Fq))

=2.1_G − (1_G +St_G)

| {z }

rule(20)

=1_G −St_G

⇒ H_c¹(Y)₁ =St_G.

F(H_c¹(Y))_θ=H_c¹(Y)_θ⁻¹ (as G-modules)

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Quotient by U .

The map

υ: Y −→ A¹\ {0} (x,y) 7−→ y isµ_q+1-equivariant.

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃ u∈U, (x⁰,y⁰) =u·(x,y). Y

υ

##G

GG GG GG GG GG GG GG GG GG

Y/U A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃ u∈U, (x⁰,y⁰) =u·(x,y). Y

υ

##G

Y/U A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃ u∈U, (x⁰,y⁰) =u·(x,y). Y

υ

##G

Y/U A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃u ∈U, (x⁰,y⁰) =u·(x,y).

Y

υ

##G

Y/U A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃u ∈U, (x⁰,y⁰) =u·(x,y).

Y

υ

##G

Y/U A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃u ∈U, (x⁰,y⁰) =u·(x,y).

Y

υ

##G

Y/U ^_ ^_ ^_ ^_ ^_ ^_^//A¹\ {0}

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Quotient by U .

The map

υ is surjective.

υ(x,y) =υ(x⁰,y⁰)⇐⇒∃u ∈U, (x⁰,y⁰) =u·(x,y).

Y

υ

##G

Y/U ^_ ^_

∼

^_ ^_ ^_ ^_^//A¹\ {0}

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dim H

_c¹

(Y)

_θ

?

Rule (6)⇒ if ξ∈µ_q+1,ξ6=1, then

Tr(ξ,H_c^∗(Y)) =Tr(1,H_c^∗(Y^ξ)).

ButY^ξ =∅. So Tr(ξ,H_c^∗(Y)) = 0.

Therefore, as a µ_q+1-module, H_c^∗(Y) is a multiple of the regular representationQ`µq+1.So

dimR_θ⁰ =dimR₁⁰ =q−1.

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dim H

_c¹

(Y)

_θ

?

Tr(ξ,H_c^∗(Y)) =Tr(1,H_c^∗(Y^ξ)).

ButY^ξ =∅. So Tr(ξ,H_c^∗(Y)) = 0.

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dim H

_c¹

(Y)

_θ

?

Tr(ξ,H_c^∗(Y)) =Tr(1,H_c^∗(Y^ξ)).

ButY^ξ =∅. So Tr(ξ,H_c^∗(Y)) = 0.

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dim H

_c¹

(Y)

_θ

?

Tr(ξ,H_c^∗(Y)) =Tr(1,H_c^∗(Y^ξ)).

ButY^ξ =∅. So Tr(ξ,H_c^∗(Y)) = 0.

Therefore, as a µ_q+1-module, H_c^∗(Y) is a multiple of the regular representationQ`µq+1.

So

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dim H

_c¹

(Y)

_θ

?

Tr(ξ,H_c^∗(Y)) =Tr(1,H_c^∗(Y^ξ)).

ButY^ξ =∅. So Tr(ξ,H_c^∗(Y)) = 0.

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Quotient by G .

The map

γ: Y −→ A¹ (x,y) 7−→ xy^q²−yx^q² isµ_q+1-equivariant.

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y). Y

γ

!!C

CC CC CC CC CC CC CC CC

Y/G A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y). Y

γ

!!C

Y/G A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y). Y

γ

!!C

Y/G A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y).

Y

γ

!!C

Y/G A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y).

Y

γ

!!C

Y/G A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y).

Y

γ

!!C

Y/G ^_ ^_ ^_ ^_ ^_ ^_^//A¹

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Quotient by G .

The map

γis surjective.

γ(x,y) =γ(x⁰,y⁰)⇐⇒∃g ∈G, (x⁰,y⁰) =u·(x,y).

Y

γ

!!C

Y/G ^_

∼

^_ ^_ ^_ ^_ ^_^//_A¹

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All irreducibles characters?

1_G: degree 1 St_G: degree q

R_α, α² 6=1: degree q+1 (there are (q−3)/2 such characters) R_α^±₀: degree(q+1)/2

R_θ⁰, θ² 6=1: degree q−1 (there are (q−1)/2 such characters) R_θ^0±

0: degreed±

Sod₊+d₋ =q−1and

d₊² +d₋² 6|G|− (sum of squares of others) = (q−1)²

2 .

Sod₊ =d₋ = q−1 2 and

IrrG ={1_G,St_G,R_α,R_α^±

0,R_θ⁰,R_θ^0±

0}

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All irreducibles characters?

0: degreed±

2 .

0,R_θ⁰,R_θ^0±

0}

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All irreducibles characters?

0: degreed±

2 .

0,R_θ⁰,R_θ^0±

0}

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All irreducibles characters?

0: degreed±

2 .

0,R_θ⁰,R_θ^0±

0}

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All irreducibles characters?

0: degreed±

d₊² +d₋² 6|G|− (sum of squares of others)

= (q−1)²

2 .

0,R_θ⁰,R_θ^0±

0}

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All irreducibles characters?

0: degreed±

2 .

0,R_θ⁰,R_θ^0±

0}