complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621...
TRANSCRIPT
![Page 1: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/1.jpg)
Complex reflection groups in representations of finite————reflective reductive groups
Michel Broue
Institut Henri–Poincare
January 2008
Michel Broue Reflection groups and finite reductive groups
![Page 2: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/2.jpg)
FINITE COMPLEX REFLECTION GROUPS
Let K be a characteristic zero field.
A finite reflection group on K is a finite subgroup of GLK (V ) (V afinite dimensional K –vector space) generated by reflections, i.e., linearmaps represented by
ζ 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
A finite reflection group on R is called a Coxeter group.
A finite reflection group on Q is called a Weyl group.
Michel Broue Reflection groups and finite reductive groups
![Page 3: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/3.jpg)
FINITE COMPLEX REFLECTION GROUPS
Let K be a characteristic zero field.
A finite reflection group on K is a finite subgroup of GLK (V ) (V afinite dimensional K –vector space) generated by reflections, i.e., linearmaps represented by
ζ 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
A finite reflection group on R is called a Coxeter group.
A finite reflection group on Q is called a Weyl group.
Michel Broue Reflection groups and finite reductive groups
![Page 4: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/4.jpg)
FINITE COMPLEX REFLECTION GROUPS
Let K be a characteristic zero field.
A finite reflection group on K is a finite subgroup of GLK (V ) (V afinite dimensional K –vector space) generated by reflections, i.e., linearmaps represented by
ζ 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
A finite reflection group on R is called a Coxeter group.
A finite reflection group on Q is called a Weyl group.
Michel Broue Reflection groups and finite reductive groups
![Page 5: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/5.jpg)
FINITE COMPLEX REFLECTION GROUPS
Let K be a characteristic zero field.
A finite reflection group on K is a finite subgroup of GLK (V ) (V afinite dimensional K –vector space) generated by reflections, i.e., linearmaps represented by
ζ 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
A finite reflection group on R is called a Coxeter group.
A finite reflection group on Q is called a Weyl group.
Michel Broue Reflection groups and finite reductive groups
![Page 6: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/6.jpg)
FINITE COMPLEX REFLECTION GROUPS
Let K be a characteristic zero field.
A finite reflection group on K is a finite subgroup of GLK (V ) (V afinite dimensional K –vector space) generated by reflections, i.e., linearmaps represented by
ζ 0 · · · 00 1 · · · 0...
.... . .
...0 0 · · · 1
A finite reflection group on R is called a Coxeter group.
A finite reflection group on Q is called a Weyl group.
Michel Broue Reflection groups and finite reductive groups
![Page 7: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/7.jpg)
Main characterisation
Theorem (Shephard–Todd, Chevalley–Serre)
Let G be a finite subgroup of GL(V ) (V an r–dimensional vector spaceover a characteristic zero field K ). Let S(V ) denote the symmetric algebraof V , isomorphic to the polynomial ring K [X1,X2, . . . ,Xr ].The following assertions are equivalent.
1 G is generated by reflections.
2 The ring S(V )G of G –fixed polynomials is a polynomial ringK [f1, f2, . . . , fr ] in r homogeneous algebraically independant elements.
ExampleFor G = Sr , one may choose
f1 = X1 + · · ·+ Xr
f2 = X1X2 + X1X3 + · · ·+ Xr−1Xr
......
fr = X1X2 · · ·Xr
Michel Broue Reflection groups and finite reductive groups
![Page 8: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/8.jpg)
Main characterisation
Theorem (Shephard–Todd, Chevalley–Serre)
Let G be a finite subgroup of GL(V ) (V an r–dimensional vector spaceover a characteristic zero field K ). Let S(V ) denote the symmetric algebraof V , isomorphic to the polynomial ring K [X1,X2, . . . ,Xr ].The following assertions are equivalent.
1 G is generated by reflections.
2 The ring S(V )G of G –fixed polynomials is a polynomial ringK [f1, f2, . . . , fr ] in r homogeneous algebraically independant elements.
ExampleFor G = Sr , one may choose
f1 = X1 + · · ·+ Xr
f2 = X1X2 + X1X3 + · · ·+ Xr−1Xr
......
fr = X1X2 · · ·Xr
Michel Broue Reflection groups and finite reductive groups
![Page 9: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/9.jpg)
Main characterisation
Theorem (Shephard–Todd, Chevalley–Serre)
Let G be a finite subgroup of GL(V ) (V an r–dimensional vector spaceover a characteristic zero field K ). Let S(V ) denote the symmetric algebraof V , isomorphic to the polynomial ring K [X1,X2, . . . ,Xr ].The following assertions are equivalent.
1 G is generated by reflections.
2 The ring S(V )G of G –fixed polynomials is a polynomial ringK [f1, f2, . . . , fr ] in r homogeneous algebraically independant elements.
ExampleFor G = Sr , one may choose
f1 = X1 + · · ·+ Xr
f2 = X1X2 + X1X3 + · · ·+ Xr−1Xr
......
fr = X1X2 · · ·Xr
Michel Broue Reflection groups and finite reductive groups
![Page 10: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/10.jpg)
Main characterisation
Theorem (Shephard–Todd, Chevalley–Serre)
Let G be a finite subgroup of GL(V ) (V an r–dimensional vector spaceover a characteristic zero field K ). Let S(V ) denote the symmetric algebraof V , isomorphic to the polynomial ring K [X1,X2, . . . ,Xr ].The following assertions are equivalent.
1 G is generated by reflections.
2 The ring S(V )G of G –fixed polynomials is a polynomial ringK [f1, f2, . . . , fr ] in r homogeneous algebraically independant elements.
ExampleFor G = Sr , one may choose
f1 = X1 + · · ·+ Xr
f2 = X1X2 + X1X3 + · · ·+ Xr−1Xr
......
fr = X1X2 · · ·Xr
Michel Broue Reflection groups and finite reductive groups
![Page 11: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/11.jpg)
Main characterisation
Theorem (Shephard–Todd, Chevalley–Serre)
Let G be a finite subgroup of GL(V ) (V an r–dimensional vector spaceover a characteristic zero field K ). Let S(V ) denote the symmetric algebraof V , isomorphic to the polynomial ring K [X1,X2, . . . ,Xr ].The following assertions are equivalent.
1 G is generated by reflections.
2 The ring S(V )G of G –fixed polynomials is a polynomial ringK [f1, f2, . . . , fr ] in r homogeneous algebraically independant elements.
ExampleFor G = Sr , one may choose
f1 = X1 + · · ·+ Xr
f2 = X1X2 + X1X3 + · · ·+ Xr−1Xr
......
fr = X1X2 · · ·Xr
Michel Broue Reflection groups and finite reductive groups
![Page 12: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/12.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 13: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/13.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 14: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/14.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),
I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 15: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/15.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 16: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/16.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 17: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/17.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 18: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/18.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 19: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/19.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 20: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/20.jpg)
Classification
1 The finite reflection groups on C have been classified by Coxeter,Shephard and Todd.
I There is one infinite series G (de, e, r) (d ,e and r integers),I ...and 34 exceptional groups G4 , G5 , . . . , G37.
2 The group G (de, e, r) (d ,e and r integers) consists of all r × rmonomial matrices with entries in µde such that the product ofentries belongs to µd .
3 We have
G (d , 1, r) ' Cd oSr
G (e, e, 2) = D2e (dihedral group of order 2e)
G (2, 2, r) = W (Dr )
G23 = H3 , G28 = F4 , G30 = H4
G35,36,37 = E6,7,8 .
Michel Broue Reflection groups and finite reductive groups
![Page 21: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/21.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 22: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/22.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 23: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/23.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 24: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/24.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 25: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/25.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 26: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/26.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 27: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/27.jpg)
FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER
G is a connected reductive algebraic group over Fq, with Weyl groupW , endowed with a Frobenius–like endomorphism F . The groupG := GF is a finite reductive group.
Example
G = GLn(Fq) , F : (ai ,j) 7→ (aqi ,j) , G = GLn(q)
Type of G — The type G = (X ,Y ,R,R∨ ; Wφ) of G consists of theroot datum of G endowed with the outer automorphism Wφ definedby F .
Examples
GLn = (X = Y = Zn,R = R∨ = An ; φ = 1)
Un = (X = Y = Zn,R = R∨ = An ; φ = −1)
Michel Broue Reflection groups and finite reductive groups
![Page 28: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/28.jpg)
Polynomial order — There is a polynomial in Z[x ]
|G|(x) =εGxN
1
|W |∑
w∈W
1
detV (1− xwφ)
= xN∏d
Φd(x)a(d)
such that |G|(q) = |G | .
Example
|GLn|(x) = x(n2)
d=n∏d=1
(xd − 1) = x(n2)
d=n∏d=1
Φd(x)[n/d ]
|GLn|(q) = |GLn(q)| and |GLn|(−q) = ±|Un(q)|
Michel Broue Reflection groups and finite reductive groups
![Page 29: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/29.jpg)
Polynomial order — There is a polynomial in Z[x ]
|G|(x) =εGxN
1
|W |∑
w∈W
1
detV (1− xwφ)
= xN∏d
Φd(x)a(d)
such that |G|(q) = |G | .
Example
|GLn|(x) = x(n2)
d=n∏d=1
(xd − 1) = x(n2)
d=n∏d=1
Φd(x)[n/d ]
|GLn|(q) = |GLn(q)| and |GLn|(−q) = ±|Un(q)|
Michel Broue Reflection groups and finite reductive groups
![Page 30: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/30.jpg)
Polynomial order — There is a polynomial in Z[x ]
|G|(x) =εGxN
1
|W |∑
w∈W
1
detV (1− xwφ)
= xN∏d
Φd(x)a(d)
such that |G|(q) = |G | .
Example
|GLn|(x) = x(n2)
d=n∏d=1
(xd − 1) = x(n2)
d=n∏d=1
Φd(x)[n/d ]
|GLn|(q) = |GLn(q)| and |GLn|(−q) = ±|Un(q)|
Michel Broue Reflection groups and finite reductive groups
![Page 31: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/31.jpg)
Polynomial order — There is a polynomial in Z[x ]
|G|(x) =εGxN
1
|W |∑
w∈W
1
detV (1− xwφ)
= xN∏d
Φd(x)a(d)
such that |G|(q) = |G | .
Example
|GLn|(x) = x(n2)
d=n∏d=1
(xd − 1) = x(n2)
d=n∏d=1
Φd(x)[n/d ]
|GLn|(q) = |GLn(q)|
and |GLn|(−q) = ±|Un(q)|
Michel Broue Reflection groups and finite reductive groups
![Page 32: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/32.jpg)
Polynomial order — There is a polynomial in Z[x ]
|G|(x) =εGxN
1
|W |∑
w∈W
1
detV (1− xwφ)
= xN∏d
Φd(x)a(d)
such that |G|(q) = |G | .
Example
|GLn|(x) = x(n2)
d=n∏d=1
(xd − 1) = x(n2)
d=n∏d=1
Φd(x)[n/d ]
|GLn|(q) = |GLn(q)| and |GLn|(−q) = ±|Un(q)|
Michel Broue Reflection groups and finite reductive groups
![Page 33: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/33.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 34: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/34.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 35: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/35.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 36: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/36.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 37: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/37.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 38: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/38.jpg)
Admissible subgroups — The tori of G are the subgroups of the shapeTF where T is an F –stable torus (i.e., isomorphic to someF× × · · · × F× in G).
The Levi subgroups of G are the subgroups of the shape LF where Lis a centralizer of an F –stable torus in G.
Examples
The split maximal torus T1 = (F×q )n
of order (q − 1)n
The Coxeter maximal torus Tc = GL1(Fqn) of order qn − 1
Levi subgroups have shape GLn1(qa1)× · · · × GLns (qas )
Cauchy theorem
The (polynomial) order of an admissible subgroup divides the (polynomial)order of the group.
Michel Broue Reflection groups and finite reductive groups
![Page 39: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/39.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 40: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/40.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 41: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/41.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 42: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/42.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 43: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/43.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 44: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/44.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 45: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/45.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 46: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/46.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 47: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/47.jpg)
Levi subgroups and type — Let G = (X ,Y ,R,R∨ ; Wφ) be a type.
A Levi subtype of G is a type of the shape
L = (X ,Y ,R ′,R ′∨
; W ′wφ)
where
I R ′ is a parabolic system of R, with Weyl group W ′,
I w ∈W is such that wφ stablizes R ′ and R ′∨
.
There is a natural bijection between
I the set of G –conjugacy classes of Levi subgroups of G ,
and
I the set of W –conjugacy classes of Levi subtypes of G.
Michel Broue Reflection groups and finite reductive groups
![Page 48: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/48.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 49: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/49.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 50: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/50.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 51: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/51.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 52: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/52.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 53: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/53.jpg)
FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS
For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductivegroup whose (polynomial) order is a power of Φ(x). Hence such a group isa torus.
Sylow theorem
1 Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) of G have as(polynomial) order the contribution of Φ(x) to the (polynomial) orderof G .
2 Sylow Φ(x)–subgroups are all conjugate by G (i.e., their types aretransitively permuted by the Weyl group W ).
3 The (polynomial) index of the normalizer in G of a SylowΦ(x)–subgroup is congruent to 1 modulo Φ(x).
Michel Broue Reflection groups and finite reductive groups
![Page 54: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/54.jpg)
The centralizers of Φd(x)–subgroups are called the d–split Levi subgroups.
The minimal d–split Levi subgroups are the centralizers of SylowΦd(x)–subgroups. They are all conjugate under G .
Example
For each d (1 ≤ d ≤ n), GLn(q) containsa subtorus of order Φd(x)[ n
d]
Assume n = md + r with r < d . Then a minimald–split Levi subgroup has shape GL1(qd)m × GLr (q).
Michel Broue Reflection groups and finite reductive groups
![Page 55: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/55.jpg)
The centralizers of Φd(x)–subgroups are called the d–split Levi subgroups.
The minimal d–split Levi subgroups are the centralizers of SylowΦd(x)–subgroups. They are all conjugate under G .
Example
For each d (1 ≤ d ≤ n), GLn(q) containsa subtorus of order Φd(x)[ n
d]
Assume n = md + r with r < d . Then a minimald–split Levi subgroup has shape GL1(qd)m × GLr (q).
Michel Broue Reflection groups and finite reductive groups
![Page 56: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/56.jpg)
The centralizers of Φd(x)–subgroups are called the d–split Levi subgroups.
The minimal d–split Levi subgroups are the centralizers of SylowΦd(x)–subgroups. They are all conjugate under G .
Example
For each d (1 ≤ d ≤ n), GLn(q) containsa subtorus of order Φd(x)[ n
d]
Assume n = md + r with r < d . Then a minimald–split Levi subgroup has shape GL1(qd)m × GLr (q).
Michel Broue Reflection groups and finite reductive groups
![Page 57: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/57.jpg)
The centralizers of Φd(x)–subgroups are called the d–split Levi subgroups.
The minimal d–split Levi subgroups are the centralizers of SylowΦd(x)–subgroups. They are all conjugate under G .
Example
For each d (1 ≤ d ≤ n), GLn(q) containsa subtorus of order Φd(x)[ n
d]
Assume n = md + r with r < d . Then a minimald–split Levi subgroup has shape GL1(qd)m × GLr (q).
Michel Broue Reflection groups and finite reductive groups
![Page 58: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/58.jpg)
The centralizers of Φd(x)–subgroups are called the d–split Levi subgroups.
The minimal d–split Levi subgroups are the centralizers of SylowΦd(x)–subgroups. They are all conjugate under G .
Example
For each d (1 ≤ d ≤ n), GLn(q) containsa subtorus of order Φd(x)[ n
d]
Assume n = md + r with r < d . Then a minimald–split Levi subgroup has shape GL1(qd)m × GLr (q).
Michel Broue Reflection groups and finite reductive groups
![Page 59: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/59.jpg)
GENERIC AND ORDINARY SYLOW SUBGROUPS
Let ` be a prime number which does not divide |W |.
If ` divides |G | = G(q), there is a unique integer d such that ` dividesΦd(q).
Then the Sylow `–subgroups of G are nothing but the Sylow`–subgroups S` of S = SF (S a Sylow Φd(x)–subgroup of G).
We haveNG (S`) = NG (S) and CG (S`) = CG (S) .
Michel Broue Reflection groups and finite reductive groups
![Page 60: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/60.jpg)
GENERIC AND ORDINARY SYLOW SUBGROUPS
Let ` be a prime number which does not divide |W |.
If ` divides |G | = G(q), there is a unique integer d such that ` dividesΦd(q).
Then the Sylow `–subgroups of G are nothing but the Sylow`–subgroups S` of S = SF (S a Sylow Φd(x)–subgroup of G).
We haveNG (S`) = NG (S) and CG (S`) = CG (S) .
Michel Broue Reflection groups and finite reductive groups
![Page 61: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/61.jpg)
GENERIC AND ORDINARY SYLOW SUBGROUPS
Let ` be a prime number which does not divide |W |.
If ` divides |G | = G(q), there is a unique integer d such that ` dividesΦd(q).
Then the Sylow `–subgroups of G are nothing but the Sylow`–subgroups S` of S = SF (S a Sylow Φd(x)–subgroup of G).
We haveNG (S`) = NG (S) and CG (S`) = CG (S) .
Michel Broue Reflection groups and finite reductive groups
![Page 62: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/62.jpg)
GENERIC AND ORDINARY SYLOW SUBGROUPS
Let ` be a prime number which does not divide |W |.
If ` divides |G | = G(q), there is a unique integer d such that ` dividesΦd(q).
Then the Sylow `–subgroups of G are nothing but the Sylow`–subgroups S` of S = SF (S a Sylow Φd(x)–subgroup of G).
We haveNG (S`) = NG (S) and CG (S`) = CG (S) .
Michel Broue Reflection groups and finite reductive groups
![Page 63: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/63.jpg)
GENERIC AND ORDINARY SYLOW SUBGROUPS
Let ` be a prime number which does not divide |W |.
If ` divides |G | = G(q), there is a unique integer d such that ` dividesΦd(q).
Then the Sylow `–subgroups of G are nothing but the Sylow`–subgroups S` of S = SF (S a Sylow Φd(x)–subgroup of G).
We haveNG (S`) = NG (S) and CG (S`) = CG (S) .
Michel Broue Reflection groups and finite reductive groups
![Page 64: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/64.jpg)
CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM
Let L (or L, or L) be a minimal d–split Levi subgroup, the centralizerof a Sylow Φd(x)–subgroup S.
I We have
NG (L)/L ' NG (S)/CG (S) ' NW (L)/W ′
(where W ′ is the Weyl group of L).
Denote that group by WG(L).
I The “number of Sylow congruence” translates to
For ζ a primitive d-th root of the unity, we have
|WG(L)| = G(ζ)/L(ζ) .
Michel Broue Reflection groups and finite reductive groups
![Page 65: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/65.jpg)
CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM
Let L (or L, or L) be a minimal d–split Levi subgroup, the centralizerof a Sylow Φd(x)–subgroup S.
I We have
NG (L)/L ' NG (S)/CG (S) ' NW (L)/W ′
(where W ′ is the Weyl group of L).
Denote that group by WG(L).
I The “number of Sylow congruence” translates to
For ζ a primitive d-th root of the unity, we have
|WG(L)| = G(ζ)/L(ζ) .
Michel Broue Reflection groups and finite reductive groups
![Page 66: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/66.jpg)
CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM
Let L (or L, or L) be a minimal d–split Levi subgroup, the centralizerof a Sylow Φd(x)–subgroup S.
I We have
NG (L)/L ' NG (S)/CG (S) ' NW (L)/W ′
(where W ′ is the Weyl group of L).
Denote that group by WG(L).
I The “number of Sylow congruence” translates to
For ζ a primitive d-th root of the unity, we have
|WG(L)| = G(ζ)/L(ζ) .
Michel Broue Reflection groups and finite reductive groups
![Page 67: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/67.jpg)
CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM
Let L (or L, or L) be a minimal d–split Levi subgroup, the centralizerof a Sylow Φd(x)–subgroup S.
I We have
NG (L)/L ' NG (S)/CG (S) ' NW (L)/W ′
(where W ′ is the Weyl group of L).
Denote that group by WG(L).
I The “number of Sylow congruence” translates to
For ζ a primitive d-th root of the unity, we have
|WG(L)| = G(ζ)/L(ζ) .
Michel Broue Reflection groups and finite reductive groups
![Page 68: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/68.jpg)
The case d = 1 — The Sylow Φ1(x)–subgroups, as well as theminimal d–split subgroups, coincide with the split maximal tori.
In case G is split (i.e., the automorphism φ induced by F is theidentity), then the group WG(L) coincides with W .
Springer and Springer–Lehrer theorem
The group WG(L) is a complex reflection group (in its representation overthe complex vector space C⊗ X ((ZL)Φd
)).
Example
For n = mr + d (d < r), we have WG(L) ' Cd oSr
The group WG(L) is called the d–cyclotomic Weyl group.
If G is split, the 1–cyclotomic Weyl group is nothing but the ordinaryWeyl group W .
Michel Broue Reflection groups and finite reductive groups
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The case d = 1 — The Sylow Φ1(x)–subgroups, as well as theminimal d–split subgroups, coincide with the split maximal tori.
In case G is split (i.e., the automorphism φ induced by F is theidentity), then the group WG(L) coincides with W .
Springer and Springer–Lehrer theorem
The group WG(L) is a complex reflection group (in its representation overthe complex vector space C⊗ X ((ZL)Φd
)).
Example
For n = mr + d (d < r), we have WG(L) ' Cd oSr
The group WG(L) is called the d–cyclotomic Weyl group.
If G is split, the 1–cyclotomic Weyl group is nothing but the ordinaryWeyl group W .
Michel Broue Reflection groups and finite reductive groups
![Page 70: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/70.jpg)
The case d = 1 — The Sylow Φ1(x)–subgroups, as well as theminimal d–split subgroups, coincide with the split maximal tori.
In case G is split (i.e., the automorphism φ induced by F is theidentity), then the group WG(L) coincides with W .
Springer and Springer–Lehrer theorem
The group WG(L) is a complex reflection group (in its representation overthe complex vector space C⊗ X ((ZL)Φd
)).
Example
For n = mr + d (d < r), we have WG(L) ' Cd oSr
The group WG(L) is called the d–cyclotomic Weyl group.
If G is split, the 1–cyclotomic Weyl group is nothing but the ordinaryWeyl group W .
Michel Broue Reflection groups and finite reductive groups
![Page 71: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/71.jpg)
The case d = 1 — The Sylow Φ1(x)–subgroups, as well as theminimal d–split subgroups, coincide with the split maximal tori.
In case G is split (i.e., the automorphism φ induced by F is theidentity), then the group WG(L) coincides with W .
Springer and Springer–Lehrer theorem
The group WG(L) is a complex reflection group (in its representation overthe complex vector space C⊗ X ((ZL)Φd
)).
Example
For n = mr + d (d < r), we have WG(L) ' Cd oSr
The group WG(L) is called the d–cyclotomic Weyl group.
If G is split, the 1–cyclotomic Weyl group is nothing but the ordinaryWeyl group W .
Michel Broue Reflection groups and finite reductive groups
![Page 72: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/72.jpg)
The case d = 1 — The Sylow Φ1(x)–subgroups, as well as theminimal d–split subgroups, coincide with the split maximal tori.
In case G is split (i.e., the automorphism φ induced by F is theidentity), then the group WG(L) coincides with W .
Springer and Springer–Lehrer theorem
The group WG(L) is a complex reflection group (in its representation overthe complex vector space C⊗ X ((ZL)Φd
)).
Example
For n = mr + d (d < r), we have WG(L) ' Cd oSr
The group WG(L) is called the d–cyclotomic Weyl group.
If G is split, the 1–cyclotomic Weyl group is nothing but the ordinaryWeyl group W .
Michel Broue Reflection groups and finite reductive groups
![Page 73: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/73.jpg)
UNIPOTENT CHARACTERS
Generic degree –
The set Un(G ) of unipotent characters of G is naturally parametrizedby a “generic” (i.e., independant of q) set Un(G). We denote byUn(G) −→ Un(G ) , γ 7→ γq that parametrization.
Example for GLn : Un(GLn) is the set of all partitions of n.
Generic degree : For γ ∈ Un(G) there is Degγ(x) ∈ Q[x ] such that
Degγ(x)|x=q= γq(1) .
Michel Broue Reflection groups and finite reductive groups
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UNIPOTENT CHARACTERS
Generic degree –
The set Un(G ) of unipotent characters of G is naturally parametrizedby a “generic” (i.e., independant of q) set Un(G). We denote byUn(G) −→ Un(G ) , γ 7→ γq that parametrization.
Example for GLn : Un(GLn) is the set of all partitions of n.
Generic degree : For γ ∈ Un(G) there is Degγ(x) ∈ Q[x ] such that
Degγ(x)|x=q= γq(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 75: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/75.jpg)
UNIPOTENT CHARACTERS
Generic degree –
The set Un(G ) of unipotent characters of G is naturally parametrizedby a “generic” (i.e., independant of q) set Un(G). We denote byUn(G) −→ Un(G ) , γ 7→ γq that parametrization.
Example for GLn : Un(GLn) is the set of all partitions of n.
Generic degree : For γ ∈ Un(G) there is Degγ(x) ∈ Q[x ] such that
Degγ(x)|x=q= γq(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 76: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/76.jpg)
UNIPOTENT CHARACTERS
Generic degree –
The set Un(G ) of unipotent characters of G is naturally parametrizedby a “generic” (i.e., independant of q) set Un(G). We denote byUn(G) −→ Un(G ) , γ 7→ γq that parametrization.
Example for GLn : Un(GLn) is the set of all partitions of n.
Generic degree : For γ ∈ Un(G) there is Degγ(x) ∈ Q[x ] such that
Degγ(x)|x=q= γq(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 77: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/77.jpg)
UNIPOTENT CHARACTERS
Generic degree –
The set Un(G ) of unipotent characters of G is naturally parametrizedby a “generic” (i.e., independant of q) set Un(G). We denote byUn(G) −→ Un(G ) , γ 7→ γq that parametrization.
Example for GLn : Un(GLn) is the set of all partitions of n.
Generic degree : For γ ∈ Un(G) there is Degγ(x) ∈ Q[x ] such that
Degγ(x)|x=q= γq(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 78: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/78.jpg)
Example for GLn :
For λ = (λ1 ≤ · · · ≤ λm) a partition of n, we define
βi := λi + i − 1 .
Then
Degλ(x) =(x − 1) · · · (xn − 1)
∏j>i (xβj − xβi )
x(m−12 )+(m−2
2 )+... ∏i
∏βij=1(x j − 1)
The (polynomial) degree Degγ(x) of a unipotent character divides the(polynomial) order |G|(x) of G .
Note. The polynomial|G|(x)
Degγ(x)belongs to Z[x ] and is called the
(generic) Schur element of γ.
Michel Broue Reflection groups and finite reductive groups
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Example for GLn : For λ = (λ1 ≤ · · · ≤ λm) a partition of n, we define
βi := λi + i − 1 .
Then
Degλ(x) =(x − 1) · · · (xn − 1)
∏j>i (xβj − xβi )
x(m−12 )+(m−2
2 )+... ∏i
∏βij=1(x j − 1)
The (polynomial) degree Degγ(x) of a unipotent character divides the(polynomial) order |G|(x) of G .
Note. The polynomial|G|(x)
Degγ(x)belongs to Z[x ] and is called the
(generic) Schur element of γ.
Michel Broue Reflection groups and finite reductive groups
![Page 80: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/80.jpg)
Example for GLn : For λ = (λ1 ≤ · · · ≤ λm) a partition of n, we define
βi := λi + i − 1 .
Then
Degλ(x) =(x − 1) · · · (xn − 1)
∏j>i (xβj − xβi )
x(m−12 )+(m−2
2 )+... ∏i
∏βij=1(x j − 1)
The (polynomial) degree Degγ(x) of a unipotent character divides the(polynomial) order |G|(x) of G .
Note. The polynomial|G|(x)
Degγ(x)belongs to Z[x ] and is called the
(generic) Schur element of γ.
Michel Broue Reflection groups and finite reductive groups
![Page 81: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/81.jpg)
Example for GLn : For λ = (λ1 ≤ · · · ≤ λm) a partition of n, we define
βi := λi + i − 1 .
Then
Degλ(x) =(x − 1) · · · (xn − 1)
∏j>i (xβj − xβi )
x(m−12 )+(m−2
2 )+... ∏i
∏βij=1(x j − 1)
The (polynomial) degree Degγ(x) of a unipotent character divides the(polynomial) order |G|(x) of G .
Note. The polynomial|G|(x)
Degγ(x)belongs to Z[x ] and is called the
(generic) Schur element of γ.
Michel Broue Reflection groups and finite reductive groups
![Page 82: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/82.jpg)
Example for GLn : For λ = (λ1 ≤ · · · ≤ λm) a partition of n, we define
βi := λi + i − 1 .
Then
Degλ(x) =(x − 1) · · · (xn − 1)
∏j>i (xβj − xβi )
x(m−12 )+(m−2
2 )+... ∏i
∏βij=1(x j − 1)
The (polynomial) degree Degγ(x) of a unipotent character divides the(polynomial) order |G|(x) of G .
Note. The polynomial|G|(x)
Degγ(x)belongs to Z[x ] and is called the
(generic) Schur element of γ.
Michel Broue Reflection groups and finite reductive groups
![Page 83: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/83.jpg)
Deligne–Lusztig induction and restriction –
Deligne and Lusztig have defined adjoint linear maps
RGL : ZIrr(L) −→ ZIrr(G ) and ∗RG
L : ZIrr(G ) −→ ZIrr(L) .
These maps are generic :
Theorem
For any generic Levi subgroup L of G, there exist adjoint linear maps
RGL : ZUn(L) −→ ZUn(G) and ∗RG
L : ZUn(G) −→ ZUn(L) .
which specialize to Deligne–Lusztig maps for x = q.
Michel Broue Reflection groups and finite reductive groups
![Page 84: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/84.jpg)
Deligne–Lusztig induction and restriction –
Deligne and Lusztig have defined adjoint linear maps
RGL : ZIrr(L) −→ ZIrr(G ) and ∗RG
L : ZIrr(G ) −→ ZIrr(L) .
These maps are generic :
Theorem
For any generic Levi subgroup L of G, there exist adjoint linear maps
RGL : ZUn(L) −→ ZUn(G) and ∗RG
L : ZUn(G) −→ ZUn(L) .
which specialize to Deligne–Lusztig maps for x = q.
Michel Broue Reflection groups and finite reductive groups
![Page 85: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/85.jpg)
Deligne–Lusztig induction and restriction –
Deligne and Lusztig have defined adjoint linear maps
RGL : ZIrr(L) −→ ZIrr(G ) and ∗RG
L : ZIrr(G ) −→ ZIrr(L) .
These maps are generic :
Theorem
For any generic Levi subgroup L of G, there exist adjoint linear maps
RGL : ZUn(L) −→ ZUn(G) and ∗RG
L : ZUn(G) −→ ZUn(L) .
which specialize to Deligne–Lusztig maps for x = q.
Michel Broue Reflection groups and finite reductive groups
![Page 86: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/86.jpg)
Deligne–Lusztig induction and restriction –
Deligne and Lusztig have defined adjoint linear maps
RGL : ZIrr(L) −→ ZIrr(G ) and ∗RG
L : ZIrr(G ) −→ ZIrr(L) .
These maps are generic :
Theorem
For any generic Levi subgroup L of G, there exist adjoint linear maps
RGL : ZUn(L) −→ ZUn(G) and ∗RG
L : ZUn(G) −→ ZUn(L) .
which specialize to Deligne–Lusztig maps for x = q.
Michel Broue Reflection groups and finite reductive groups
![Page 87: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/87.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) where
I M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 88: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/88.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) where
I M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 89: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/89.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) where
I M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 90: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/90.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) whereI M is a d–split Levi subtype of G,
I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 91: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/91.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) whereI M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 92: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/92.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) whereI M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 93: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/93.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) whereI M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 94: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/94.jpg)
d–Harish–Chandra theories –
Let Sd(G) denote the set of all pairs (M, µ) whereI M is a d–split Levi subtype of G,I µ ∈ Un(M).
The elements of Sd(G) are called d–split pairs.
A binary relation on Sd(G) –
Definition :
(M1, µ1) ≤ (M2, µ2)
if and only if µ2 occurs in RM2M1
(µ1) .
Michel Broue Reflection groups and finite reductive groups
![Page 95: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/95.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 96: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/96.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 97: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/97.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 98: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/98.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.
I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 99: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/99.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 100: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/100.jpg)
First fundamental theorem
1 The relation ≤ is an order relation on Sd(G).
2 The minimal d–split pairs contained in a pair (G, γ) are all conjugateunder the Weyl group W .
I Such minimal pairs are called d–cuspidal.I For (L, λ) d–cuspidal, define
Un(G, (L, λ)) := {γ ∈ Un(G) | (L, λ) ≤ (G, γ)} .
3 The sets Un(G, (L, λ)), where (L, λ) runs over a system ofrepresentatives of the W –conjugacy classes of d–cuspidal pairs, forma partition of Un(G).
Michel Broue Reflection groups and finite reductive groups
![Page 101: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/101.jpg)
For (L, λ) a d–cuspidal pair, we set
WG(L, λ) := NW (L, λ)/WL = NG (L, λq)/L .
Second fundamental theorem
Whenever (L, λ) is a d–cuspidal pair, the group WG(L, λ) is (naturally) acomplex reflection group.
In the case where L is a minimal d–split Levi subtype, and λ is the trivialcharacter, the above theorem specializes onto Springer–Lehrer theorem.
Michel Broue Reflection groups and finite reductive groups
![Page 102: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/102.jpg)
For (L, λ) a d–cuspidal pair, we set
WG(L, λ) := NW (L, λ)/WL = NG (L, λq)/L .
Second fundamental theorem
Whenever (L, λ) is a d–cuspidal pair, the group WG(L, λ) is (naturally) acomplex reflection group.
In the case where L is a minimal d–split Levi subtype, and λ is the trivialcharacter, the above theorem specializes onto Springer–Lehrer theorem.
Michel Broue Reflection groups and finite reductive groups
![Page 103: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/103.jpg)
For (L, λ) a d–cuspidal pair, we set
WG(L, λ) := NW (L, λ)/WL = NG (L, λq)/L .
Second fundamental theorem
Whenever (L, λ) is a d–cuspidal pair, the group WG(L, λ) is (naturally) acomplex reflection group.
In the case where L is a minimal d–split Levi subtype, and λ is the trivialcharacter, the above theorem specializes onto Springer–Lehrer theorem.
Michel Broue Reflection groups and finite reductive groups
![Page 104: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/104.jpg)
For (L, λ) a d–cuspidal pair, we set
WG(L, λ) := NW (L, λ)/WL = NG (L, λq)/L .
Second fundamental theorem
Whenever (L, λ) is a d–cuspidal pair, the group WG(L, λ) is (naturally) acomplex reflection group.
In the case where L is a minimal d–split Levi subtype, and λ is the trivialcharacter, the above theorem specializes onto Springer–Lehrer theorem.
Michel Broue Reflection groups and finite reductive groups
![Page 105: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/105.jpg)
Third fundamental theorem : description of RGL (λ)
There exists a collection of isometries
I M(L,λ) : ZIrr(WM(L, λ)
∼−→ ZUn(M, (L, λ)) ,
such that
1 The following diagram commute :
ZIrr(WG(L, λ)IG(L,λ) // ZUn(G, (L, λ))
ZIrr(WM(L, λ)IM(L,λ) //
IndWG(L,λ)
WM(L,λ)
OO
ZUn(M, (L, λ))
RGM
OO
2 For all χ ∈ Irr(WG(L, λ)), let γχ := εχI G(L,λ)(χ). Then if ζ is a
primitive d-th root of unity, we have
Degγχ(ζ) = εχχ(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 106: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/106.jpg)
Third fundamental theorem : description of RGL (λ)
There exists a collection of isometries
I M(L,λ) : ZIrr(WM(L, λ)
∼−→ ZUn(M, (L, λ)) ,
such that
1 The following diagram commute :
ZIrr(WG(L, λ)IG(L,λ) // ZUn(G, (L, λ))
ZIrr(WM(L, λ)IM(L,λ) //
IndWG(L,λ)
WM(L,λ)
OO
ZUn(M, (L, λ))
RGM
OO
2 For all χ ∈ Irr(WG(L, λ)), let γχ := εχI G(L,λ)(χ). Then if ζ is a
primitive d-th root of unity, we have
Degγχ(ζ) = εχχ(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 107: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/107.jpg)
Third fundamental theorem : description of RGL (λ)
There exists a collection of isometries
I M(L,λ) : ZIrr(WM(L, λ)
∼−→ ZUn(M, (L, λ)) ,
such that
1 The following diagram commute :
ZIrr(WG(L, λ)IG(L,λ) // ZUn(G, (L, λ))
ZIrr(WM(L, λ)IM(L,λ) //
IndWG(L,λ)
WM(L,λ)
OO
ZUn(M, (L, λ))
RGM
OO
2 For all χ ∈ Irr(WG(L, λ)), let γχ := εχI G(L,λ)(χ). Then if ζ is a
primitive d-th root of unity, we have
Degγχ(ζ) = εχχ(1) .
Michel Broue Reflection groups and finite reductive groups
![Page 108: Complex reflection groups in representations of finite ...jessica2.msri.org/attachments/12621/12621 ppt.pdf · FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero eld](https://reader034.vdocument.in/reader034/viewer/2022042912/5f481ee5b15e36466c757e9f/html5/thumbnails/108.jpg)
Third fundamental theorem : description of RGL (λ)
There exists a collection of isometries
I M(L,λ) : ZIrr(WM(L, λ)
∼−→ ZUn(M, (L, λ)) ,
such that
1 The following diagram commute :
ZIrr(WG(L, λ)IG(L,λ) // ZUn(G, (L, λ))
ZIrr(WM(L, λ)IM(L,λ) //
IndWG(L,λ)
WM(L,λ)
OO
ZUn(M, (L, λ))
RGM
OO
2 For all χ ∈ Irr(WG(L, λ)), let γχ := εχI G(L,λ)(χ). Then if ζ is a
primitive d-th root of unity, we have
Degγχ(ζ) = εχχ(1) .
Michel Broue Reflection groups and finite reductive groups