February2012 MichelBrou´e GL ( x )for x anindeterminate?

GL _n (x ) for x an indeterminate ?

Michel Brou´ e

Institut universitaire de France Universit´ e Paris-Diderot Paris 7 CNRS–Institut de Math´ ematiques de Jussieu

February 2012

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U _n (q) :=

U ∈ Mat _n ( F q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U _n (q) :=

U ∈ Mat _n ( F q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . .

denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U _n (q) :=

U ∈ Mat _n ( F q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . .

For example

U _n (q) :=

U ∈ Mat _n ( F q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U n (q) :=

U ∈ Mat n (F _q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U n (q) :=

U ∈ Mat n (F _q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

follows.

Finite Reductive Groups

Let q be a prime power. There is (up to non unique isomorphism) a single field with q elements, denoted F q .

Since Chevalley (1955), one knows how to construct Lie groups over F q , analog of the usual complex reductive Lie groups :

GL n ( F q ) , O n ( F q ) , Sp _n ( F q ) , U n ( F q ) , . . . denoted respectively

GL n (q) , O n (q) , Sp _n (q) , U n (q) , . . . For example

U n (q) :=

U ∈ Mat n (F _q

) | U. ^t U ^∗ = 1 .

There are also groups of exceptional types G ₂ , F ₄ , E ₆ , E ₇ , E ₈ over F q .

They can be viewed from the algebraic groups point of view, as

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) ,

G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) ,

G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

),

G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

),

G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G. The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G.

The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n (F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G.

The Weyl group of G is W := N _G (T)/T. Example:

For G = GL n ( F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G.

The Weyl group of G is W := N _G (T)/T.

Example:

For G = GL n ( F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F

× q





 and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G.

The Weyl group of G is W := N _G (T)/T.

Example:

For G = GL n ( F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F ^× q







and W = S _n .

Let G be a connected reductive algebraic group over F q , endowed with a Frobenius endomorphism F which defines an F q -rational structure.

Then the group G := G(q) := G ^F is a finite reductive group over F q .

Example:

Assume G = GL _n ( F q ) .

I

For F : (a

) 7→ (a

) , G = GL

(q) .

I

For F : (a

) 7→

(a

), G = U

(q) .

Let T ∼ = F ^× q × · · · × F ^× q be an F -stable maximal torus of G.

The Weyl group of G is W := N _G (T)/T.

Example:

For G = GL n ( F q ) , T =







F ^× q · · · 0 .. . . .. .. . 0 · · · F ^× q





 and W = S _n .

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ). Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ). Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ). Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object G (x) such that G (x)| _x=q = G(q).

Let Y (T) := Hom( F ^× q , T) be the group of co-characters of T, a free Z -module of finite rank.

Set V := C ⊗ _Z Y (T), a finite dimensional complex vector space.

Then W acts on V as a reflection group, and the Frobenius

endomorphism F acts on V as qϕ, where ϕ is a finite order element of N _{GL(V )} (W ).

The type of G is G := (V , W ϕ).

Example:

I

For G = GL

(q), its type is G = GL

:= ( C

, S

) .

I

For G = U

(q), its type is G = U

:= ( C

, −S

) .

Main fact

Lots of data about G = G(q) are values at x = q of polynomials in x which depend only on the type G .

As if there were an object (x) such that (x)| = G(q).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G| . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x) (well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G| . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x) (well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G | .

Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x) (well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G | . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x)

(well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G | . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x)

(well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G | . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x)

(well, precisely (−1)

_n 2

|GL

|(−x) ).

Polynomial order

Let G = (V , W ϕ) .

R. Steinberg (1967) : There is a polynomial (element of Z [x])

| G |(x) = x ^N Y

d

Φ d (x) ^a(d)

such that | G |(q) = |G(q)| = |G | . Example

|GL _n |(x) = x (

) Q d=n

d=1 (x ^d − 1) = x (

) Q d=n

d=1 Φ _d (x) ^[n/d]

|U _n |(x) = ±|GL _n |(−x) (well, precisely (−1)

_n 2

|GL

|(−x) ).

Remarks

I

The prime divisors of | G |(x) are x and cyclotomic polynomials Φ

(x).

I

N is the number of reflecting hyperplanes of the Weyl group of G.

Hence G has a trivial Weyl group, i.e., G is a torus

G ∼ = F

× · · · × F

if and only if its (polynomial) order is not divisible by x.

Remarks

I

The prime divisors of | G |(x) are x and cyclotomic polynomials Φ

(x).

I

N is the number of reflecting hyperplanes of the Weyl group of G.

Hence G has a trivial Weyl group, i.e., G is a torus

G ∼ = F

× · · · × F

if and only if its (polynomial) order is not divisible by x.

Remarks

I

The prime divisors of | G |(x) are x and cyclotomic polynomials Φ

(x).

I

N is the number of reflecting hyperplanes of the Weyl group of G.

Hence G has a trivial Weyl group, i.e., G is a torus

G ∼ = F

×

q

× · · · × F

× q

if and only if its (polynomial) order is not divisible by x.

Remarks

I

The prime divisors of | G |(x) are x and cyclotomic polynomials Φ

(x).

I

N is the number of reflecting hyperplanes of the Weyl group of G.

Hence G has a trivial Weyl group, i.e., G is a torus

G ∼ = F

×

q

× · · · × F

× q

if and only if its (polynomial) order is not divisible by x.

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Admissible subgroups

The tori of G are the subgroups of the shape T = T(q) = T ^F where T ∼ = F ^× _q × · · · × F ^× _q is an F –stable torus of G.

The Levi subgroups of G are the subgroups of the shape

L = L(q) = L ^F where L = C _G (T) is the centralizer of an F –stable torus in G.

Examples for GL n (q)

I

The split maximal torus T

= ( F

)

of order (q − 1)

T

=









F

· · · 0 . .

. . . . . . . 0 · · · F









I

The Coxeter torus T

= GL

( F

) of order q

− 1 .

I

Levi subgroups have shape GL

(q

) × · · · × GL

(q

)

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x)–group is a finite reductive group whose (polynomial) order is a power of Φ d (x). A Φ d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x)–group is a finite reductive group whose (polynomial) order is a power of Φ d (x). A Φ d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G .

Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x)–group is a finite reductive group whose (polynomial) order is a power of Φ d (x). A Φ d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x )–group is a finite reductive group whose (polynomial) order is a power of Φ _d (x).

A Φ d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x )–group is a finite reductive group whose (polynomial) order is a power of Φ _d (x). A Φ _d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x )–group is a finite reductive group whose (polynomial) order is a power of Φ _d (x). A Φ _d (x)–group is a torus.

Examples for GL _n

The split torus T ₁ is a Φ ₁ (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x )–group is a finite reductive group whose (polynomial) order is a power of Φ _d (x). A Φ _d (x)–group is a torus.

Examples for GL _n

The split torus T 1 is a Φ 1 (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Φ n (x)-subgroup.

Lagrange theorem, generic p-groups

Lagrange theorem

The (polynomial) order of an admissible subgroup (torus or Levi subgroup) divides the (polynomial) order of the group.

The following definition is the generic version of p-subgroups of G . Φ _d (x )–groups

For Φ _d (x) a cyclotomic polynomial, a Φ _d (x )–group is a finite reductive group whose (polynomial) order is a power of Φ _d (x). A Φ _d (x)–group is a torus.

Examples for GL _n

The split torus T 1 is a Φ 1 (x)-subgroup.

The Coxeter torus T _c (a cyclic group of order q ⁿ − 1) contains a

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q)| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) . Notation : Set L d := C G (S d ) and

N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Sylow theorems

1

Maximal Φ d (x)–subgroups (“Sylow Φ d (x)–subgroups”) S d of G have as (polynomial) order the contribution of Φ _d (x) to the (polynomial) order of G :

|S _d | = |S _d (q )| = Φ d (q) ^a(d) .

Notation : Set L d := C G (S d ) and N d := N G (S d ) = N G (L d )

2

Sylow Φ _d (x)–subgroups are all conjugate by G .

3

The (polynomial) index |G : N _d | is congruent to 1 modulo Φ _d (x).

4

W _d := N _d /L _d is a true finite group, a complex reflection group in its action on V _d := C ⊗ Y (S _d ).

= The group W _d is the d –cyclotomic Weyl group of the finite reductive group G .

Note that, for d = 1 and ϕ = ±1, one has W

= W .

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^] Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q ) and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^]

Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q ) and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^] Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q) and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^] Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q)

and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^] Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q) and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Example for GL _n Recall that

|GL _n |(x) = x (

) ^d=n Y

d=1

Φ _d (x) ^[n/d]

For each d (1 ≤ d ≤ n), GL _n (q) contains a subtorus of (polynomial) order Φ d (x) ^[

^] Assume n = md + r with r < d . Then

L _d = GL ₁ (q ^d ) ^m × GL _r (q) and W _d = µ _d o S _m

where µ _d denotes the cyclic group of all d -th roots of unity.

Generic and ordinary Sylow subgroups

Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d .

(M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Generic and ordinary Sylow subgroups Let ` be a prime number.

If ` divides |G | = | G |(q), let d be the order of q modulo ` (so ` divides Φ _d (q)).

Let S _d be a Sylow Φ _d (x)–subgroup of G, and let S _` be the Sylow

`–subgroup of S d (q).

Let W <sub>`</sub> be a Sylow `-subgroup of the d -cyclotomic Weyl group W _d . (M. Enguehard)

1

A Sylow `-subgroup of N d = N G (S d ) is a Sylow `-subgroup of G .

2

In general, such a Sylow is an extension of S _{`</sub> by W <sub>`} .

If “not general”, then a Sylow `-subgroup of G an extension of Z

(L

)

by W

.

Unipotent characters, generic degrees

1

The set Un(G ) of unipotent characters of G is parametrized by a

“generic” (i.e., independant of q) set Un(G). We denote by Un( G ) → Un(G ) , ρ 7→ ρ q

that parametrization.

Example for GL n : Un(GL n ) is the set of all partitions of n.

2

Generic degree : For ρ ∈ Un( G ), there exists Deg _ρ (x) ∈ Q [x] such that

Deg _ρ (x) | _x=q = ρ q (1) .

Example for GL

: For λ = (λ

≤ · · · ≤ λ

) a partition of n, let β

: λ

+ i − 1 . Then

Deg

(x) = (x − 1) · · · (x

− 1) Q

j>i

(x

− x

) x (

)

(

)

Q

i

Q

j=1

(x

− 1)

.