Pseudo reductive groups over Fx

(1)

Pseudo reductive groups over F

^x

?

Michel Brou´ e

Institut Henri–Poincar´e

May 2009

Michel Brou´e Pseudo reductive groups overFx?

(2)

Spetses

Joint work between Michel Brou´ e, Gunter Malle, and Jean Michel,

initiated in the Greek island named SPETSES in 1993.

(3)

Spetses

Joint work between Michel Brou´ e, Gunter Malle, and Jean Michel,

initiated in the Greek island named SPETSES in 1993.

(4)

Spetses

Joint work between Michel Brou´ e, Gunter Malle, and Jean Michel,

initiated in the Greek island named SPETSES in 1993.

(5)

Unipotent characters for G

₂

Character Degree Fake degree Eigenvalue Family

φ

1,0

1 1 1 C

1

φ

_1,6

q

⁶

q

⁶

1 C

₁

φ

2,1 1

6

qΦ

²₂

Φ

3

qΦ

8

1 S

3

.(1, 1)

φ

_2,2 ¹₂

qΦ

²₂

Φ

₆

q

²

Φ

₄

1 S

₃

.(g

₂

, 1)

φ

⁰_1,3 ¹₃

qΦ

₃

Φ

₆

q

³

1 S

₃

.(g

₃

, 1)

φ

⁰⁰_1,3 ¹₃

qΦ

3

Φ

6

q

³

1 S

3

.(1, ρ)

G

2

[1]

¹₆

qΦ

²₁

Φ

6

0 1 S

3

.(1, ε)

G

₂

[−1]

¹₂

qΦ

²₁

Φ

₃

0 −1 S

₃

.(g

₂

, ε) G

₂

[ζ

₃

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃

S

₃

.(g

₃

, ζ

₃

) G

2

[ζ

₃²

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃²

S

3

.(g

3

, ζ

₃²

)

(6)

Unipotent characters for G

₂

Character Degree Fake degree Eigenvalue Family

φ

1,0

1 1 1 C

1

φ

_1,6

q

⁶

q

⁶

1 C

₁

φ

2,1 1

6

qΦ

²₂

Φ

3

qΦ

8

1 S

3

.(1, 1)

φ

_2,2 ¹₂

qΦ

²₂

Φ

₆

q

²

Φ

₄

1 S

₃

.(g

₂

, 1)

φ

⁰_1,3 ¹₃

qΦ

₃

Φ

₆

q

³

1 S

₃

.(g

₃

, 1)

φ

⁰⁰_1,3 ¹₃

qΦ

3

Φ

6

q

³

1 S

3

.(1, ρ)

G

2

[1]

¹₆

qΦ

²₁

Φ

6

0 1 S

3

.(1, ε)

G

₂

[−1]

¹₂

qΦ

²₁

Φ

₃

0 −1 S

₃

.(g

₂

, ε) G

₂

[ζ

₃

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃

S

₃

.(g

₃

, ζ

₃

) G

2

[ζ

₃²

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃²

S

3

.(g

3

, ζ

₃²

)

(7)

Unipotent characters for G

₄

3

(G

₄

= 2 × S

₃

)

Character Degree FakeDegree Eigenvalue Family

φ

_1,0

1 1 1 C

₁

φ

_2,1 ³⁻

√−3

6

qΦ

⁰₃

Φ

₄

Φ

⁰⁰₆

qΦ

₄

1 X

₃

.01

φ

2,3 3+√

−3

6

qΦ

⁰⁰₃

Φ

4

Φ

⁰₆

q

³

Φ

4

1 X

3

.02

Z

₃

: 2

√−3

3

qΦ

₁

Φ

₂

Φ

₄

0 ζ

₃²

X

₃

.12

φ

3,2

q

²

Φ

3

Φ

6

q

²

Φ

3

Φ

6

1 C

1

φ

_1,4 ⁻

√−3

6

q

⁴

Φ

⁰⁰₃

Φ

₄

Φ

⁰⁰₆

q

⁴

1 X

₅

.1

φ

_1,8

√−3

6

q

⁴

Φ

⁰₃

Φ

₄

Φ

⁰₆

q

⁸

1 X

₅

.2

φ

_2,5 ¹₂

q

⁴

Φ

²₂

Φ

₆

q

⁵

Φ

₄

1 X

₅

.3

Z

₃

: 11

√−3

3

q

⁴

Φ

₁

Φ

₂

Φ

₄

0 ζ

₃²

X

₅

.4

G

₄ ¹₂

q

⁴

Φ

²₁

Φ

₃

0 −1 X

₅

.5

Φ⁰₃,Φ⁰⁰₃ (resp. Φ⁰₆,Φ⁰⁰₆) are factors of Φ₃ (resp Φ₆) inQ(ζ₃)

(8)

Unipotent characters for G

₄

3

(G

₄

= 2 × S

₃

)

Character Degree FakeDegree Eigenvalue Family

φ

_1,0

1 1 1 C

₁

φ

_2,1 ³⁻

√−3

6

q Φ

⁰₃

Φ

₄

Φ

⁰⁰₆

qΦ

₄

1 X

₃

.01

φ

2,3 3+√

−3

6

q Φ

⁰⁰₃

Φ

4

Φ

⁰₆

q

³

Φ

4

1 X

3

.02 Z

₃

: 2

√−3

3

qΦ

₁

Φ

₂

Φ

₄

0 ζ

₃²

X

₃

.12

φ

3,2

q

²

Φ

3

Φ

6

q

²

Φ

3

Φ

6

1 C

1

φ

_1,4 ⁻

√−3

6

q

⁴

Φ

⁰⁰₃

Φ

₄

Φ

⁰⁰₆

q

⁴

1 X

₅

.1

φ

_1,8

√−3

6

q

⁴

Φ

⁰₃

Φ

₄

Φ

⁰₆

q

⁸

1 X

₅

.2

φ

_2,5 ¹₂

q

⁴

Φ

²₂

Φ

₆

q

⁵

Φ

₄

1 X

₅

.3

Z

₃

: 11

√−3

3

q

⁴

Φ

₁

Φ

₂

Φ

₄

0 ζ

₃²

X

₅

.4

G

4 1

2

q

⁴

Φ

²₁

Φ

3

0 −1 X

5

.5

Φ⁰₃,Φ⁰⁰₃ (resp. Φ⁰₆,Φ⁰⁰₆) are factors of Φ3 (resp Φ6) inQ(ζ3)

(9)

Finite reductive groups

G

a reductive group over

F_q

,

F an isogeny such that G is finite. For simplicity we assume (G, F ) split. Set G :=

G^F

.

{ G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(10)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

.

{ G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(11)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split.

Set G :=

G^F

.

{ G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(12)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

.

{ G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(13)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W }

UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(14)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W }

UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)} UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(15)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(16)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(17)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ). The blocks of S correspond to

Lusztig’s families of unipotent characters.

(18)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ).

The blocks of S correspond to

Lusztig’s families of unipotent characters.

(19)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ).

The blocks of S correspond to

Lusztig’s families of unipotent characters.

(20)

Finite reductive groups

G

a reductive group over

F_q

, F an isogeny such that G is finite.

For simplicity we assume (G, F ) split. Set G :=

G^F

. { G -conjugacy classes of F –stable maximal tori of

G

}

←→ { conjugacy classes of the Weyl group W } UnCh(G ) := {Unipotent characters}

:= {Irreducible constituents of R

_T^G

w

(1)}

UnSh(G ) := {Unipotent character sheaves}

= Another

C

–basis for the space of class functions on G generated by UnCh(G ).

Lusztig’s Fourier matrix S

:= (square) matrix between UnCh(G ) and UnSh(G ).

The blocks of S correspond to

Lusztig’s families of unipotent characters.

(21)

Lusztig’s Fourier matrix for G

₂

(1,1) (g₂,1) (g₃,1) (1, ρ) (1, ε) (g₂, ε) (g₃, ζ₃) (g₃, ζ₃²)

1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

(1,1) 0 0 ¹₆ ¹₂ ¹₃ ¹₃ ¹₆ ¹₂ ¹₃ ¹₃

(g2,1) 0 0 ¹₂ ¹₂ 0 0 −¹₂ −¹₂ 0 0

(g₃,1) 0 0 ¹₃ 0 ²₃ −¹₃ ¹₃ 0 −¹₃ −¹₃

(1, ρ) 0 0 ¹₃ 0 −¹₃ ²₃ ¹₃ 0 −¹₃ −¹₃

(1, ε) 0 0 ¹₆ −¹₂ ¹₃ ¹₃ ¹₆ −¹₂ ¹₃ ¹₃

(g2, ε) 0 0 ¹₂ −¹₂ 0 0 −¹₂ ¹₂ 0 0

(g3, ζ3) 0 0 ¹₃ 0 −¹₃ −¹₃ ¹₃ 0 ²₃ −¹₃

(g3, ζ₃²) 0 0 ¹₃ 0 −¹₃ −¹₃ ¹₃ 0 −¹₃ ²₃

(22)

Lusztig’s Fourier matrix for G

₂

(1,1) (g₂,1) (g₃,1) (1, ρ) (1, ε) (g₂, ε) (g₃, ζ₃) (g₃, ζ₃²)

1 . . . .

. 1 . . . .

(1,1) . . ¹₆ ¹₂ ¹₃ ¹₃ ¹₆ ¹₂ ¹₃ ¹₃

(g2,1) . . ¹₂ ¹₂ . . −¹₂ −¹₂ . .

(g₃,1) . . ¹₃ . ²₃ −¹₃ ¹₃ . −¹₃ −¹₃

(1, ρ) . . ¹₃ . −¹₃ ²₃ ¹₃ . −¹₃ −¹₃

(1, ε) . . ¹₆ −¹₂ ¹₃ ¹₃ ¹₆ −¹₂ ¹₃ ¹₃

(g2, ε) . . ¹₂ −¹₂ . . −¹₂ ¹₂ . .

(g3, ζ3) . . ¹₃ . −¹₃ −¹₃ ¹₃ . ²₃ −¹₃

(g3, ζ₃²) . . ¹₃ . −¹₃ −¹₃ ¹₃ . −¹₃ ²₃

(23)

Unipotent characters for G

₂

Character Degree Fake degree Eigenvalue Family

φ

1,0

1 1 1 C

1

φ

_1,6

q

⁶

q

⁶

1 C

₁

φ

2,1 1

6

qΦ

²₂

Φ

3

qΦ

8

1 S

3

.(1, 1)

φ

_2,2 ¹₂

qΦ

²₂

Φ

₆

q

²

Φ

₄

1 S

₃

.(g

₂

, 1)

φ

⁰_1,3 ¹₃

qΦ

₃

Φ

₆

q

³

1 S

₃

.(g

₃

, 1)

φ

⁰⁰_1,3 ¹₃

qΦ

3

Φ

6

q

³

1 S

3

.(1, ρ)

G

2

[1]

¹₆

qΦ

²₁

Φ

6

0 1 S

3

.(1, ε)

G

₂

[−1]

¹₂

qΦ

²₁

Φ

₃

0 −1 S

₃

.(g

₂

, ε) G

₂

[ζ

₃

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃

S

₃

.(g

₃

, ζ

₃

) G

2

[ζ

₃²

]

¹₃

qΦ

²₁

Φ

²₂

0 ζ

₃²

S

3

.(g

3

, ζ

₃²

)

(24)

The Fourier matrix for G

₄

01 02 12 1 2 3 4 5

1 0 0 0 0 0 0 0 0 0

01 0

³⁻

√−3 6

3+√

−3 6

√−3

3

0 0 0 0 0 0

02 0

³⁺

√−3 6

3−√

−3

6

−

√−3

3

0 0 0 0 0 0

12 0

√−3

3

−

√−3 3

√−3

3

0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

1 0 0 0 0 0 −

√−3 6

1 2

√−3 3

1 2

2 0 0 0 0 0

√−3

6

−

√−3 6

1

2

−

√−3 3

1 2

3 0 0 0 0 0

¹₂ ¹₂ ¹₂

0 −

¹₂

4 0 0 0 0 0

√−3

3

−

√−3

3

0

√−3

3

0 5 0 0 0 0 0

¹₂ ¹₂

−

¹₂

0

¹₂

(25)

The Fourier matrix for G

₄

01 02 12 1 2 3 4 5

1 . . . . . . . . .

01 .

³⁻

√−3 6

3+√

−3 6

√−3

3

. . . . . .

02 .

³⁺

√−3 6

3−√

−3

6

−

√−3

3

. . . . . .

12 .

√−3

3

−

√−3 3

√−3

3

. . . . . .

. . . . 1 . . . . .

1 . . . . . −

√−3 6

1 2

√−3 3

1 2

2 . . . . .

√−3

6

−

√−3 6

1

2

−

√−3 3

1 2

3 . . . . .

¹₂ ¹₂ ¹₂

. −

¹₂

4 . . . . .

√−3

3

−

√−3

3

.

√−3

3

.

5 . . . . .

¹₂ ¹₂

−

¹₂

.

¹₂

(26)

Unipotent characters for G

₄

3

Character Degree FakeDegree Eigenvalue Family

φ

1,0

1 1 1 C

1

φ

2,1 3−√

−3

6

q Φ

⁰₃

Φ

4

Φ

⁰⁰₆

qΦ

4

1 X

3

.01

φ

_2,3 ³⁺

√−3

6

q Φ

⁰⁰₃

Φ

₄

Φ

⁰₆

q

³

Φ

₄

1 X

₃

.02 Z

3

: 2

√−3

3

qΦ

1

Φ

2

Φ

4

0 ζ

₃²

X

3

.12

φ

_3,2

q

²

Φ

₃

Φ

₆

q

²

Φ

₃

Φ

₆

1 C

₁

φ

1,4 −√

−3

6

q

⁴

Φ

⁰⁰₃

Φ

4

Φ

⁰⁰₆

q

⁴

1 X

5

.1

φ

_1,8

√−3

6

q

⁴

Φ

⁰₃

Φ

₄

Φ

⁰₆

q

⁸

1 X

₅

.2

φ

2,5 1

2

q

⁴

Φ

²₂

Φ

6

q

⁵

Φ

4

1 X

5

.3

Z

3

: 11

√−3

3

q

⁴

Φ

1

Φ

2

Φ

4

0 ζ

₃²

X

5

.4

G

4 1

2

q

⁴

Φ

²₁

Φ

3

0 −1 X

5

.5

(27)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P

F

AA

AA ^N^G

F

(L)

V

F

||

L

F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

I Harish–Chandra inductionR_L^G := Ind^G_P ·Res_P→L

:_CLmod→_CGmod

I and restriction ^∗R_L^G :_CGmod→_CLmod

I which do not depend on the choice ofP.

(28)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P

F

AA

AA ^N^G

F

(L)

V

F

||

L

F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

:_CLmod→_CGmod

(29)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P

F

AA

AA ^N^G

F

(L)

V

F

||

L

F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

:_CLmod→_CGmod

(30)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

:_CLmod→_CGmod

(31)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

:_CLmod→_CGmod

(32)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

:_C_Lmod→_C_Gmod

(33)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

I Harish–Chandra inductionR_L^G := Ind^G_P ·Res_P→L:_C_Lmod→_C_Gmod

(34)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

(35)

Harish-Chandra series

A Levi subgroup of an F –stable parabolic subgroup

P

of

G

is called a 1–Levi subgroup of

G.

It gives rise to

P^F AA

AA ^N^G^F^(L) V^F

||

L^F

W_G(L)

vv vv v 1

DDDDD {{{{{

and to

(36)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal

if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(37)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal

if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(38)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(39)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(40)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(41)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(42)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(43)

Definition : cuspidal character

An irreducible character γ of G is said to be 1–cuspidal if, whenever L is a proper 1–Levi subgroup, we have

^∗

R

_L^G

(γ) = 0 .

A pair (L, λ) is called 1–cuspidal if L is a 1–Levi subgroup of G and λ a 1–cuspidal irreducible character of L.

Main theorem

1

Partition :

Irr

_K

(G ) =

•

[

(L,λ) cuspidal/G

Irr R

_L^G

(λ)

2

For (L, λ) 1–cuspidal, the relative Weyl group W

_G

(L, λ) := N

_G

(L, λ)/L is a finite Coxeter group.

3

The commuting algebra End

_KG

R

_L^G

(λ) is a Hecke algebra for W

G

(L, λ).

(44)

Degrees and Eigenvalues

Degrees

The elements of UnCh(G ) and UnSh(G ) are parametrized by finite sets which are independent of q

(depend only on the “type” of

G).

The degrees of elements of UnCh(G ) and UnSh(G ) are polynomials evaluated at q .

I Among the unipotent character sheaves are the functions

R_χ= 1

|W| X

w∈W

χ(w)R_T^G

w

and DegR_χ is the graded multiplicity ofχ in the coinvariant algebra of W, evaluated atq(thefake degree ofχ).

I The degrees of the irreducible constituents ofR_L^G(λ) are given by the

“generic degrees” for the Hecke algebra ofWG(L, λ).

The Fourier matrix is independent of q.

(45)

Degrees and Eigenvalues

Degrees

The elements of UnCh(G ) and UnSh(G ) are parametrized by finite sets which are independent of q

(depend only on the “type” of

G).

The degrees of elements of UnCh(G ) and UnSh(G ) are polynomials evaluated at q .

R_χ= 1

|W| X

w∈W

χ(w)R_T^G

w

The Fourier matrix is independent of q.

(46)

Degrees and Eigenvalues

Degrees

The elements of UnCh(G ) and UnSh(G ) are parametrized by finite sets which are independent of q (depend only on the “type” of

G).

The degrees of elements of UnCh(G ) and UnSh(G ) are polynomials evaluated at q.

R_χ= 1

|W| X

w∈W

χ(w)R_T^G

w

The Fourier matrix is independent of q.

(47)

Degrees and Eigenvalues

Degrees

The elements of UnCh(G ) and UnSh(G ) are parametrized by finite sets which are independent of q (depend only on the “type” of

G).

The degrees of elements of UnCh(G ) and UnSh(G ) are polynomials evaluated at q.

R_χ= 1

|W| X

w∈W

χ(w)R_T^G

w