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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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

F stabilizes H_c¹(Y)₁ and H_c¹(Y)_θ₀

By Schur’s lemma, only one eigenvalue on H_c¹(Y)1: ρ1 ?

I LetX=Y/µq+1 =P¹\P¹(Fq)

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

q−Tr(F,H_c¹(Y)₁

| {z }

dim. q

).

⇒ ρ₁=1.

By Schur’s lemma, two eigenvalues onH_c¹(Y)_θ₀: ρ₊, ρ₋ with multiplicities(q−1)/2

I 0=|Y^F|=q−Tr(F,H_c¹(Y)) = q−Tr(F,H_c¹(Y)1) −Tr(F,H_c¹(Y)_θ₀)

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

F stabilizes H_c¹(Y)₁ and H_c¹(Y)_θ₀

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

F stabilizes H_c¹(Y)1 and H_c¹(Y)θ0

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

By Schur’s lemma, only one eigenvalue on H_c¹(Y)1: ρ1

?

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I LetX=Y/µq+1

=P¹\P¹(Fq)

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F|

|{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

By Schur’s lemma, two eigenvalues onH_c¹(Y)_θ₀: ρ₊, ρ₋

with multiplicities(q−1)/2

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

I 0=|Y^F|

=q−Tr(F,H_c¹(Y)) = q−Tr(F,H_c¹(Y)1) −Tr(F,H_c¹(Y)_θ₀)

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

I 0=|Y^F|=q−Tr(F,H_c¹(Y))

= q−Tr(F,H_c¹(Y)1) −Tr(F,H_c¹(Y)_θ₀)

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

F(H_c¹(Y)_θ) =H_c¹(Y)_θ⁻¹

I H_c¹(X) =H_c¹(Y)1

I 0=|X^F| |{z}=

Lefschetz

| {z }

dim. q

).

⇒ ρ₁=1.

⇒ ρ₊= −ρ₋.

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Eigenvalues of F

²

F² stabilizes H_cⁱ(Y)_θ and its action commutes with G Schur’s lemma ⇒F² acts by scalar mult. by λ_θ on H_c¹(Y)_θ (well..., except for θ₀ where an extra-argument is needed) Note that λ₁ =1

By Lefschetz Formula, we have, for all ξ∈µ_q+1,

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

On the other hand,|Y^ξF²|=

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

F² stabilizes H_cⁱ(Y)_θ and its action commutes with G

Schur’s lemma ⇒F² acts by scalar mult. by λ_θ on H_c¹(Y)_θ (well..., except for θ₀ where an extra-argument is needed) Note that λ₁ =1

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

F² stabilizes H_cⁱ(Y)_θ and its action commutes with G Schur’s lemma ⇒F² acts by scalar mult. by λ_θ on H_c¹(Y)_θ

(well..., except for θ₀ where an extra-argument is needed) Note that λ₁ =1

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

F² stabilizes H_cⁱ(Y)_θ and its action commutes with G Schur’s lemma ⇒F² acts by scalar mult. by λ_θ on H_c¹(Y)_θ (well..., except for θ₀ where an extra-argument is needed)

Note that λ₁ =1

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

(22)

Eigenvalues of F

²

|Y^ξF²|

= |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)

λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ

(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Eigenvalues of F

²

|Y^ξF²| = |{z}q²

action onH_c²

−qλ1−X

θ6=1

θ(ξ)λθ(q−1)

= q²−1− (q−1)X

θ

θ(ξ)λθ.

q³−q if ξ= −1

0 otherwise

⇒ Soλ_θ = −θ(−1)q if θ6=1

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Modular representations

From now on, `|q+1, ` odd LetS be the Sylow subgroup of µ_q+1. We identify S^∧ and (µq+1)^∧_` .

Blocks?

If α² 6=1, then {R_α} is a block of defect zero {R_α⁺₀}and {R_α⁻₀} are two blocks of defect zero

(1) If θis an `-regular linear character of µ_q+1 such thatθ² 6=1, then {R_θη⁰ |η∈S^∧}is a block of defect S.

(2) {R_θ⁰⁺

0,R_θ⁰⁻

0}∪{R_θ⁰₀_η | η∈S^∧, η6=1}is a block of defect S (3) {1G,StG}∪{R_η⁰ |η∈S^∧, η6=1}: principal block (defect S).

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Modular representations

From now on, `|q+1, ` odd

LetS be the Sylow subgroup of µ_q+1. We identify S^∧ and (µq+1)^∧_` .

Blocks?

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

From now on, `|q+1, ` odd LetS be the Sylow subgroup of µ_q+1.

We identify S^∧ and (µq+1)^∧_` . Blocks?

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

If α² 6=1, then {R_α} is a block of defect zero

{R_α⁺₀}and {R_α⁻₀} are two blocks of defect zero

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

If α² 6=1, then {R_α} is a block of defect zero {R_α⁺

0} and {R_α⁻

0} are two blocks of defect zero

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

0} and {R_α⁻

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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Modular representations

Blocks?

0} and {R_α⁻

(2) {R_θ⁰⁺

0,R_θ⁰⁻

0}∪{R_θ⁰₀_η | η∈S^∧, η6=1}is a block of defect S

(3) {1G,StG}∪{R_η⁰ |η∈S^∧, η6=1}: principal block (defect S).

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Modular representations

Blocks?

0} and {R_α⁻

(2) {R_θ⁰⁺

0,R_θ⁰⁻

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What has been illustrated?

Blocks are parametrized using the`⁰-part of linear characters of tori (in general, see Brou´e-Michel)

Some Morita equivalences: “Jordan decomposition” (in general, see Brou´e for tori and B.-Rouquier for a more general situation) Derived equivalences: Brou´e’s abelian defect conjecture admits a

“geometric version” (proved only for the “Coxeter” torus of GL_n(F^q) by B.-Rouquier: cyclic defect...)

Role of the Frobenius What has been omitted?

Non-abelian defect (for`=2 inG, see Gonard’s thesis) Decomposition matrices, Schur algebras

...

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...

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Some Morita equivalences: “Jordan decomposition” (in general, see Brou´e for tori and B.-Rouquier for a more general situation)

Derived equivalences: Brou´e’s abelian defect conjecture admits a

...

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“geometric version” (proved only for the “Coxeter” torus of GLn(F^q) by B.-Rouquier: cyclic defect...)

...

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Role of the Frobenius

What has been omitted?

...

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...

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Non-abelian defect (for`=2 inG, see Gonard’s thesis)

Decomposition matrices, Schur algebras ...

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...

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Curiosities

Abhyankar’s conjecture (Raynaud’s Theorem): A finite groupΓ is the Galois group of a Galois ´etale covering of A¹(F) if and only if it is generated by its Sylow p-subgroups.

Example: Γ =SL₂(Fq),