Finding PIMs for finite groups of Lie type Olivier Dudas CNRS & - - PowerPoint PPT Presentation

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Finding PIMs for finite groups of Lie type Olivier Dudas CNRS & - - PowerPoint PPT Presentation

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Finding PIMs for finite groups of Lie type Olivier Dudas CNRS & Paris-Diderot University March 2013 O. Dudas (CNRS) Finding PIMs March 2013 1 / 9 Decomposition matrices Representations of finite groups of Lie type GL n ( q ) , Sp 2

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Finding PIM’s for finite groups of Lie type

Olivier Dudas

CNRS & Paris-Diderot University

March 2013

  • O. Dudas (CNRS)

Finding PIM’s March 2013 1 / 9

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Decomposition matrices

Representations of finite groups of Lie type GLn(q), Sp2n(q), . . . , E8(q) Main goal. Extend geometric methods introduced by Deligne and Lusztig to the modular setting (representations in positive characteristic)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 2 / 9

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Decomposition matrices

Representations of finite groups of Lie type GLn(q), Sp2n(q), . . . , E8(q) Main goal. Extend geometric methods introduced by Deligne and Lusztig to the modular setting (representations in positive characteristic) Less ambitious. Determine decomposition matrices of such groups

  • O. Dudas (CNRS)

Finding PIM’s March 2013 2 / 9

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Decomposition matrices

Representations of finite groups of Lie type GLn(q), Sp2n(q), . . . , E8(q) Main goal. Extend geometric methods introduced by Deligne and Lusztig to the modular setting (representations in positive characteristic) Less ambitious. Determine decomposition matrices of such groups i.e ◮ given χ an irreducible character of G(q) (in char. 0), find the composition factors of any reduction of χ in positive characteristic

  • O. Dudas (CNRS)

Finding PIM’s March 2013 2 / 9

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Decomposition matrices

Representations of finite groups of Lie type GLn(q), Sp2n(q), . . . , E8(q) Main goal. Extend geometric methods introduced by Deligne and Lusztig to the modular setting (representations in positive characteristic) Less ambitious. Determine decomposition matrices of such groups i.e ◮ given χ an irreducible character of G(q) (in char. 0), find the composition factors of any reduction of χ in positive characteristic ◮ given a projective indecomposable module PIM (in positive characteristic), compute the character of this module (in char. 0)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 2 / 9

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Inductive approach

M representation

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Inductive approach

M representation M is cuspidal

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Inductive approach

M representation M is cuspidal M is non-cuspidal

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Inductive approach

M representation M is cuspidal M is non-cuspidal M “occurs” in an induced representation RG

L (N) with N cuspidal

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Inductive approach

M representation M is cuspidal M is non-cuspidal M “occurs” in an induced representation RG

L (N) with N cuspidal

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Inductive approach

M representation M is cuspidal M is non-cuspidal geometric construction of M via Deligne-Lusztig varieties M “occurs” in an induced representation RG

L (N) with N cuspidal

  • O. Dudas (CNRS)

Finding PIM’s March 2013 3 / 9

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Parabolic induction

G reductive algebraic group over Fp F : G − → G Frobenius endomorphism / Fq GF = G(q) is a finite reductive group

  • O. Dudas (CNRS)

Finding PIM’s March 2013 4 / 9

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Parabolic induction

G reductive algebraic group over Fp F : G − → G Frobenius endomorphism / Fq GF = G(q) is a finite reductive group

  • Example. G = GLn(Fp) with F : (ai,j) −

→ (aq

i,j) then G(q) = GLn(q)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 4 / 9

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Parabolic induction

G reductive algebraic group over Fp F : G − → G Frobenius endomorphism / Fq GF = G(q) is a finite reductive group

  • Example. G = GLn(Fp) with F : (ai,j) −

→ (aq

i,j) then G(q) = GLn(q)

Parabolic induction and restriction functors, given L a standard F-stable Levi subgroup RG

L : kL(q)-mod −

→ kG(q)-mod

∗RG L : kG(q)-mod −

→ kL(q)-mod

  • O. Dudas (CNRS)

Finding PIM’s March 2013 4 / 9

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Parabolic induction

G reductive algebraic group over Fp F : G − → G Frobenius endomorphism / Fq GF = G(q) is a finite reductive group

  • Example. G = GLn(Fp) with F : (ai,j) −

→ (aq

i,j) then G(q) = GLn(q)

Parabolic induction and restriction functors, given L a standard F-stable Levi subgroup RG

L : kL(q)-mod −

→ kG(q)-mod

∗RG L : kG(q)-mod −

→ kL(q)-mod

Properties of induction/restriction

(i) (RG

L , ∗RG L ) pair of adjoint functors

(ii) They are exact if char k = p, in particular they map projective modules to projective modules

  • O. Dudas (CNRS)

Finding PIM’s March 2013 4 / 9

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Cuspidality

Definition

A kG(q)-module M is cuspidal if ∗RG

L (M) = 0 for all proper standard

F-stable Levi subgroup.

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Cuspidality

Definition

A kG(q)-module M is cuspidal if ∗RG

L (M) = 0 for all proper standard

F-stable Levi subgroup. If M is non-cuspidal simple module, take L to be minimal s.t ∗RG

L (M) = 0

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Cuspidality

Definition

A kG(q)-module M is cuspidal if ∗RG

L (M) = 0 for all proper standard

F-stable Levi subgroup. If M is non-cuspidal simple module, take L to be minimal s.t ∗RG

L (M) = 0

Then there exists N cuspidal kL(q)-module such that ◮ M is in the head of RG

L (N)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Cuspidality

Definition

A kG(q)-module M is cuspidal if ∗RG

L (M) = 0 for all proper standard

F-stable Levi subgroup. If M is non-cuspidal simple module, take L to be minimal s.t ∗RG

L (M) = 0

Then there exists N cuspidal kL(q)-module such that ◮ M is in the head of RG

L (N)

◮ PM is a direct summand of RG

L (PN)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Cuspidality

Definition

A kG(q)-module M is cuspidal if ∗RG

L (M) = 0 for all proper standard

F-stable Levi subgroup. If M is non-cuspidal simple module, take L to be minimal s.t ∗RG

L (M) = 0

Then there exists N cuspidal kL(q)-module such that ◮ M is in the head of RG

L (N)

◮ PM is a direct summand of RG

L (PN)

  • Consequence. it is enough to

◮ know the projective cover of cuspidal simple modules ◮ know how to decompose RG

L (PN) (Howlett-Lehrer, Dipper-Du-James,

Geck-Hiss. . . )

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Cuspidality

◮ know the projective cover of cuspidal simple modules

  • O. Dudas (CNRS)

Finding PIM’s March 2013 5 / 9

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Geometric construction of the representations

W Weyl group of G

  • O. Dudas (CNRS)

Finding PIM’s March 2013 6 / 9

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Geometric construction of the representations

W Weyl group of G Given w ∈ W , Deligne-Lusztig variety X(w), quasi-projective variety of dimension ℓ(w) endowed with action of G(q)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 6 / 9

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Geometric construction of the representations

W Weyl group of G Given w ∈ W , Deligne-Lusztig variety X(w), quasi-projective variety of dimension ℓ(w) endowed with action of G(q)

  • Linearisation. ℓ-adic cohomology groups

Hi

c(X(w), Qℓ) and Hi c(X(w), Fℓ)

give f.d. representations of G(q) over Qℓ or Fℓ (non-zero when i ∈ {ℓ(w), . . . , 2ℓ(w)} only)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 6 / 9

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Geometric construction of the representations

W Weyl group of G Given w ∈ W , Deligne-Lusztig variety X(w), quasi-projective variety of dimension ℓ(w) endowed with action of G(q)

  • Linearisation. ℓ-adic cohomology groups

Hi

c(X(w), Qℓ) and Hi c(X(w), Fℓ)

give f.d. representations of G(q) over Qℓ or Fℓ (non-zero when i ∈ {ℓ(w), . . . , 2ℓ(w)} only)

  • Example. Drinfeld curve X = {(x, y) ∈ F2

p | xy q − yxq = 1} then H1 c(X)

contains the discrete series of SL2(q) (cuspidal representations)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 6 / 9

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Geometric construction of the representations

W Weyl group of G Given w ∈ W , Deligne-Lusztig variety X(w), quasi-projective variety of dimension ℓ(w) endowed with action of G(q)

  • Linearisation. ℓ-adic cohomology groups

Hi

c(X(w), Qℓ) and Hi c(X(w), Fℓ)

give f.d. representations of G(q) over Qℓ or Fℓ (non-zero when i ∈ {ℓ(w), . . . , 2ℓ(w)} only)

  • Example. Drinfeld curve X = {(x, y) ∈ F2

p | xy q − yxq = 1} then H1 c(X)

contains the discrete series of SL2(q) (cuspidal representations)

  • Problem. How to know where the representations appear?
  • O. Dudas (CNRS)

Finding PIM’s March 2013 6 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v) Hi−1

c

(Z) Hi

c(X(w))

Hi

c(X(w))

Hi

c(Z)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v) Hi−1

c

(Z) Hi

c(X(w))

Hi

c(X(w))

∼ Hi

c(Z)

H2ℓ(w)−i

c

(X(w))∗

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v) Hi−1

c

(Z) Hi

c(X(w))

Hi

c(X(w))

∼ Hi

c(Z)

H2ℓ(w)−i

c

(X(w))∗ H2ℓ(w)−i

c

(X(w))∗

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v) Hi

c(X(w))ρ ∼

Hi

c(X(w))ρ

∼ H2ℓ(w)−i

c

(X(w))∗

ρ

∼ H2ℓ(w)−i

c

(X(w))∗

ρ ∼

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. 0

Proposition (Deligne-Lusztig)

Let ρ be an ordinary character of G(q). If w is minimal such that ρ occurs in the cohomology of X(w), then ρ occurs in middle degree only

  • Proof. X(w) has a smooth compactification X(w), such that

X(w) \ X(w) = Z is a disjoint union of smaller varieties X(v) Hi

c(X(w))ρ ∼

Hi

c(X(w))ρ

∼ H2ℓ(w)−i

c

(X(w))∗

ρ

∼ H2ℓ(w)−i

c

(X(w))∗

ρ ∼

Therefore Hi

c(X(w))ρ = 0 for i = ℓ(w)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 7 / 9

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Middle degree in char. ℓ

Same result if working in the good framework Replace individual cohomology groups by a complex RΓc(X(w), Fℓ)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 8 / 9

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Middle degree in char. ℓ

Same result if working in the good framework Replace individual cohomology groups by a complex RΓc(X(w), Fℓ) The terms can be assumed to be projective modules and the character [RΓc(X(w), Fℓ)] = (−1)i[Hi

c(X(w), Fℓ)] is a virtual projective character

  • O. Dudas (CNRS)

Finding PIM’s March 2013 8 / 9

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Middle degree in char. ℓ

Same result if working in the good framework Replace individual cohomology groups by a complex RΓc(X(w), Fℓ) The terms can be assumed to be projective modules and the character [RΓc(X(w), Fℓ)] = (−1)i[Hi

c(X(w), Fℓ)] is a virtual projective character

Proposition (Bonnaf´ e-Rouquier)

Let M be a simple module and w be minimal such that (−1)i[Hi

c(X(w), Fℓ)], [M] = 0

  • O. Dudas (CNRS)

Finding PIM’s March 2013 8 / 9

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Middle degree in char. ℓ

Same result if working in the good framework Replace individual cohomology groups by a complex RΓc(X(w), Fℓ) The terms can be assumed to be projective modules and the character [RΓc(X(w), Fℓ)] = (−1)i[Hi

c(X(w), Fℓ)] is a virtual projective character

Proposition (Bonnaf´ e-Rouquier)

Let M be a simple module and w be minimal such that (−1)i[Hi

c(X(w), Fℓ)], [M] = 0

Then there exists a representative of RΓc(X(w), Fℓ) 0 − → Qℓ(w) − → Qℓ(w)+1 − → · · · − → Q2ℓ(w) − → 0 such that each Qi is a finitely generated projective module and PM is a direct summand of Qi for i = ℓ(w) only

  • O. Dudas (CNRS)

Finding PIM’s March 2013 8 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip}

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5)

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 s t st

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St s 1 + χ′ − χ − St t 1 + χ − χ′ − St st 1 + θ10 + St

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St = [P1] s 1 + χ′ − χ − St t 1 + χ − χ′ − St st 1 + θ10 + St

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

slide-45
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Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St = [P1] s 1 + χ′ − χ − St = [P1] − 2[P2] t 1 + χ − χ′ − St st 1 + θ10 + St

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

slide-46
SLIDE 46

Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St = [P1] s 1 + χ′ − χ − St = [P1] − 2[P2] t 1 + χ − χ′ − St = [P1] − 2[P3] st 1 + θ10 + St

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

slide-47
SLIDE 47

Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St = [P1] s 1 + χ′ − χ − St = [P1] − 2[P2] t 1 + χ − χ′ − St = [P1] − 2[P3] st 1 + θ10 + St =?

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9

slide-48
SLIDE 48

Application to decomposition matrices

G = Sp4(q) and 2 = ℓ|q + 1 W = s, t Weyl group of type B2 Principal ℓ-block b = {1, St, χ, χ′, θ10, non-unip} Decomposition matrix P1 P2 P3 P4 P5 1 1 · · · · χ 1 1 · · · χ′ 1 · 1 · · θ10 · · · 1 · St 1 1 1 α 1 and 2 ≤ α ≤ (q − 1)/2 (if ℓ = 5) Decomposition of virtual characters w [bRΓc(X(w))] 1 1 + χ + χ′ + St = [P1] s 1 + χ′ − χ − St = [P1] − 2[P2] t 1 + χ − χ′ − St = [P1] − 2[P3] st 1 + θ10 + St =? α ≤ 2 therefore α = 2

  • O. Dudas (CNRS)

Finding PIM’s March 2013 9 / 9