introduction to deligne-lusztig theory
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Introduction to Deligne-Lusztig Theory
Cedric Bonnafe
CNRS (UMR 6623) - Universite de Franche-Comte (Besancon)
Berkeley (MSRI), Feb. 2008
Γ group acting on X :
set
poset
...
Γ group acting on X :
set
poset
...
Γ group acting on X :
set
poset
...
Γ group acting on X :
set
poset
K -vector space
...
Γ group acting on X :
set
poset
K -vector space
variety
...
Γ group acting on X :
set
poset
K -vector space
variety
...
Γ group acting on X :
set
poset
K -vector space
variety
...
Γ group acting on X :
set K [X ]
poset
K -vector space
variety
...
Γ group acting on X :
set K [X ]
poset complex of chains
K -vector space
variety
...
Γ group acting on X :
set K [X ]
poset complex of chains cohomology groups
K -vector space
variety
...
Γ group acting on X :
set K [X ]
poset complex of chains cohomology groups
K -vector space
variety etale, `-adic cohomology
...
Γ group acting on X :
set K [X ]
poset complex of chains cohomology groups
K -vector space
variety etale, `-adic cohomology
...
Deligne (SGA 412, 1976). “Les exposes I a VI de SGA 4 donnent
la theorie generale des topologies de Grothendieck. Tres detailles, ilspeuvent etre precieux lors de l’etude de topologies exotiques, telle cellequi donne naissance a la topologie cristalline. Pour la topologie etale,si proche de l’intuition classique, un garde-fou si imposant n’est pasnecessaire : il suffit de connaıtre (par exemple), le livre de Godement,et d’avoir un peu de foi. (...)”
“Chapters I-VI of SGA 4 develop the general theory of Grothendiecktopologies. Very detailed, they may be a valuable tool for studyingexotic topologies, such as the one yielding the crystalline topology.For etale topology, so close to classical intuition, such imposingsafetynet is not necessary : it is sufficient to know (for instance),Godement’s book, and to have some faith. (...)”
Deligne (SGA 412, 1976). “Les exposes I a VI de SGA 4 donnent
la theorie generale des topologies de Grothendieck. Tres detailles, ilspeuvent etre precieux lors de l’etude de topologies exotiques, telle cellequi donne naissance a la topologie cristalline. Pour la topologie etale,si proche de l’intuition classique, un garde-fou si imposant n’est pasnecessaire : il suffit de connaıtre (par exemple), le livre de Godement,et d’avoir un peu de foi. (...)”
“Chapters I-VI of SGA 4 develop the general theory of Grothendiecktopologies. Very detailed, they may be a valuable tool for studyingexotic topologies, such as the one yielding the crystalline topology.For etale topology, so close to classical intuition, such imposingsafetynet is not necessary : it is sufficient to know (for instance),Godement’s book, and to have some faith. (...)”
Let p be a prime number, F = Fp
Let V be an algebraic variety over FLet Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Let p be a prime number, F = Fp
Let V be an algebraic variety over F
Let Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Let p be a prime number, F = Fp
Let V be an algebraic variety over FLet Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Let p be a prime number, F = Fp
Let V be an algebraic variety over FLet Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Let p be a prime number, F = Fp
Let V be an algebraic variety over FLet Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Let p be a prime number, F = Fp
Let V be an algebraic variety over FLet Γ be a group acting on V
Let ` be a prime number 6= p
To this datum is associated a family of finite dimensionalQ -vector spaces H i
c(V), which are acted on by Γ .
We denote by H∗c (V) the element of the Grothendieck group of the
category of finite dimensional Q Γ -modules equal to
H∗c (V) =
∑i
(−1)i [H ic(V)]
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒
Hic(P
1) ?
P1 = A1 ∪ {∞}:
rule (3) ⇒
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒0 // H0
c (A1) // H0c (P1) // H0
c ({∞})
// H1c (A1) // H1
c (P1) // H1c ({∞})
// H2c (A1) // H2
c (P1) // H2c ({∞}) // 0
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒0 // 0 // H0
c (P1) // H0c ({∞})
// 0 // H1c (P1) // H1
c ({∞})
// Q // H2c (P1) // H2
c ({∞}) // 0
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒0 // 0 // H0
c (P1) // Q
// 0 // H1c (P1) // 0
// Q // H2c (P1) // 0 // 0
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒0 // 0 // H0
c (P1) // Q
// 0 // H1c (P1) // 0
// Q // H2c (P1) // 0 // 0
H ic(P
1) =
{Q if i = 0, 2
0 otherwise
Hic(P
1) ?
P1 = A1 ∪ {∞}: rule (3) ⇒0 // 0 // H0
c (P1) // Q
// 0 // H1c (P1) // 0
// Q // H2c (P1) // 0 // 0
H ic(P
1) =
{Q if i = 0, 2
0 otherwiseas G -modules! (rule (7))
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
The general frame.
G connected reductive group /F
F : G → G Frobenius endomorphism /Fq
Let G = GF = {g ∈ G | F (g) = g }: finite reductive group
Example - G = GLn(F),
F : G −→ G(aij) 7−→ (aq
ij),
GF = GLn(Fq).
LetL : G −→ G
g 7−→ g−1F (g)
be the Lang map.
Then L is surjective and
L (g) = L (g ′) ⇐⇒ ∃ x ∈ GF , g ′ = xg .
Let Z a subvariety of G (locally closed): then GF acts (on the left)on L −1(Z).
LetL : G −→ G
g 7−→ g−1F (g)
be the Lang map.Then L is surjective and
L (g) = L (g ′) ⇐⇒ ∃ x ∈ GF , g ′ = xg .
Let Z a subvariety of G (locally closed): then GF acts (on the left)on L −1(Z).
LetL : G −→ G
g 7−→ g−1F (g)
be the Lang map.Then L is surjective and
L (g) = L (g ′) ⇐⇒ ∃ x ∈ GF , g ′ = xg .
Let Z a subvariety of G (locally closed): then GF acts (on the left)on L −1(Z).
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degreesI Algorithm for values at a semisimple elementI ...
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible characters
I Computation of their degreesI Algorithm for values at a semisimple elementI ...
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degrees
I Algorithm for values at a semisimple elementI ...
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degreesI Algorithm for values at a semisimple element
I ...
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degreesI Algorithm for values at a semisimple elementI ...
Drinfeld (1974) - Starting example: SL2(Fq) actson {(x , y) ∈ F2 | xyq − yxq = 1}
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degreesI Algorithm for values at a semisimple elementI ...
Drinfeld (1974) - Starting example: SL2(Fq) actson {(x , y) ∈ F2 | xyq − yxq = 1}
Deligne-Lusztig (1976) - Construction of a class of “interesting”varieties
Lusztig (1976-77-78-79-80-81-82-83-84) -I Parametrization of irreducible charactersI Computation of their degreesI Algorithm for values at a semisimple elementI ...
The group G × (µq+1 o 〈F 〉) acts on A2 and stabilizes
Y = {(x , y) ∈ A2 | xyq − yxq = 1}
If θ ∈ (µq+1)∧, let H i
c(Y)θ denote the θ-isotypic component ofH i
c(Y): it is a Q G -module.
Let R ′θ = −[H∗
c (Y)θ] = [H1c (Y)θ] − [H2
c (Y)θ]
This is Deligne-Lusztig induction.
The group G × (µq+1 o 〈F 〉) acts on A2 and stabilizes
Y = {(x , y) ∈ A2 | xyq − yxq = 1}
If θ ∈ (µq+1)∧, let H i
c(Y)θ denote the θ-isotypic component ofH i
c(Y)
: it is a Q G -module.
Let R ′θ = −[H∗
c (Y)θ] = [H1c (Y)θ] − [H2
c (Y)θ]
This is Deligne-Lusztig induction.
The group G × (µq+1 o 〈F 〉) acts on A2 and stabilizes
Y = {(x , y) ∈ A2 | xyq − yxq = 1}
If θ ∈ (µq+1)∧, let H i
c(Y)θ denote the θ-isotypic component ofH i
c(Y): it is a Q G -module.
Let R ′θ = −[H∗
c (Y)θ] = [H1c (Y)θ] − [H2
c (Y)θ]
This is Deligne-Lusztig induction.
The group G × (µq+1 o 〈F 〉) acts on A2 and stabilizes
Y = {(x , y) ∈ A2 | xyq − yxq = 1}
If θ ∈ (µq+1)∧, let H i
c(Y)θ denote the θ-isotypic component ofH i
c(Y): it is a Q G -module.
Let R ′θ = −[H∗
c (Y)θ] = [H1c (Y)θ] − [H2
c (Y)θ]
This is Deligne-Lusztig induction.
The group G × (µq+1 o 〈F 〉) acts on A2 and stabilizes
Y = {(x , y) ∈ A2 | xyq − yxq = 1}
If θ ∈ (µq+1)∧, let H i
c(Y)θ denote the θ-isotypic component ofH i
c(Y): it is a Q G -module.
Let R ′θ = −[H∗
c (Y)θ] = [H1c (Y)θ] − [H2
c (Y)θ]
This is Deligne-Lusztig induction.
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1 P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1 P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1 P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1 P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1 P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1
//______ P1 \ P1(Fq)
Quotient by µq+1.
The mapπ : Y −→ P1
(x , y) 7−→ [x : y ]
is G -equivariant.
The image of π is P1 \ P1(Fq).
π(x , y) = π(x ′, y ′) ⇐⇒ ∃ ξ ∈ µq+1, (x ′, y ′) = ξ(x , y).
Y
π
%%JJJJJJJJJJJJJJJJJJJJJ
��Y/µq+1
∼ //______ P1 \ P1(Fq)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq))
=︸︷︷︸rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
First decomposition of H∗c (Y)
H ic(Y) = 0 if i 6∈ {1, 2} (since Y is affine of dim. 1: rule (1aff))
H2c (Y) = 1G×T (because Y is irreducible: rule (2))
H∗c (Y)1 =︸︷︷︸
rule (5∗)
H∗c (P1 \ P1(Fq)) =︸︷︷︸
rule (3∗)
H∗c (P1) − H∗
c (P1(Fq))
= 2.1G − (1G + StG )︸ ︷︷ ︸rule(20)
= 1G − StG
⇒ H1c (Y)1 = StG .
F (H1c (Y))θ = H1
c (Y)θ−1 (as G -modules)
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U //______ A1 \ {0}
Quotient by U.
The mapυ : Y −→ A1 \ {0}
(x , y) 7−→ y
is µq+1-equivariant.
υ is surjective.
υ(x , y) = υ(x ′, y ′) ⇐⇒ ∃ u ∈ U , (x ′, y ′) = u · (x , y).
Y
υ
##GGGGGGGGGGGGGGGGGGG
��Y/U ∼ //______ A1 \ {0}
dim H1c (Y)θ?
Rule (6) ⇒ if ξ ∈ µq+1, ξ 6= 1, then
Tr(ξ, H∗c (Y)) = Tr(1, H∗
c (Yξ)).
But Yξ = ∅. So Tr(ξ, H∗c (Y)) = 0.
Therefore, as a µq+1-module, H∗c (Y) is a multiple of the regular
representation Q µq+1.So
dim R ′θ = dim R ′
1 = q − 1.
dim H1c (Y)θ?
Rule (6) ⇒ if ξ ∈ µq+1, ξ 6= 1, then
Tr(ξ, H∗c (Y)) = Tr(1, H∗
c (Yξ)).
But Yξ = ∅. So Tr(ξ, H∗c (Y)) = 0.
Therefore, as a µq+1-module, H∗c (Y) is a multiple of the regular
representation Q µq+1.So
dim R ′θ = dim R ′
1 = q − 1.
dim H1c (Y)θ?
Rule (6) ⇒ if ξ ∈ µq+1, ξ 6= 1, then
Tr(ξ, H∗c (Y)) = Tr(1, H∗
c (Yξ)).
But Yξ = ∅. So Tr(ξ, H∗c (Y)) = 0.
Therefore, as a µq+1-module, H∗c (Y) is a multiple of the regular
representation Q µq+1.So
dim R ′θ = dim R ′
1 = q − 1.
dim H1c (Y)θ?
Rule (6) ⇒ if ξ ∈ µq+1, ξ 6= 1, then
Tr(ξ, H∗c (Y)) = Tr(1, H∗
c (Yξ)).
But Yξ = ∅. So Tr(ξ, H∗c (Y)) = 0.
Therefore, as a µq+1-module, H∗c (Y) is a multiple of the regular
representation Q µq+1.
So
dim R ′θ = dim R ′
1 = q − 1.
dim H1c (Y)θ?
Rule (6) ⇒ if ξ ∈ µq+1, ξ 6= 1, then
Tr(ξ, H∗c (Y)) = Tr(1, H∗
c (Yξ)).
But Yξ = ∅. So Tr(ξ, H∗c (Y)) = 0.
Therefore, as a µq+1-module, H∗c (Y) is a multiple of the regular
representation Q µq+1.So
dim R ′θ = dim R ′
1 = q − 1.
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G //______ A1
Quotient by G .
The mapγ : Y −→ A1
(x , y) 7−→ xyq2− yxq2
is µq+1-equivariant.
γ is surjective.
γ(x , y) = γ(x ′, y ′) ⇐⇒ ∃ g ∈ G , (x ′, y ′) = u · (x , y).
Y
γ
!!CCCC
CCCC
CCCC
CCCC
C
��Y/G ∼ //______ A1
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others)
=(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
All irreducibles characters?
1G : degree 1
StG : degree q
Rα, α2 6= 1: degree q + 1 (there are (q − 3)/2 such characters)
R±α0
: degree (q + 1)/2
R ′θ, θ2 6= 1: degree q − 1 (there are (q − 1)/2 such characters)
R ′±θ0
: degree d±
So d+ + d− = q − 1 and
d2+ + d2
− 6 |G | − (sum of squares of others) =(q − 1)2
2.
So d+ = d− =q − 1
2and
Irr G = {1G , StG , Rα, R±α0
, R ′θ, R
′±θ0
}
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG
(in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori
(ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
What has been illustrated?
2 “unipotent charaters”: 1G , StG (in general, the number ofunipotent characters DOES NOT depend on q)
Mackey formula for Deligne-Lusztig induction
Parametrization of characters using character of finite subtori (ingeneral, “Jordan decomposition”)
Degrees are polynomials in q
“Cuspidal characters” appear in the cohomology associated to“non-split” tori
What has been hidden?
Etale topology...• •N
^
Character values
Weyl groups
Some references
Deligne-Lusztig, Representations of finite reductive groups,Ann. of Math. 103 (1976), 103-161
Lusztig, Representations of finite Chevalley groups, CBMSProc. Conf. (1977)
Lusztig’s orange book, Characters of finite reductivegroups, Ann. of Math. Studies (1984)
Cedric Bonnafe (CNRS, Besancon, France) Deligne-Lusztig theory Berkeley (MSRI), Feb. 2008 17 / 17
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