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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

  1. 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

  2. Decomposition matrices Representations of finite groups of Lie type GL n ( q ) , Sp 2 n ( q ) , . . . , E 8 ( 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

  3. Decomposition matrices Representations of finite groups of Lie type GL n ( q ) , Sp 2 n ( q ) , . . . , E 8 ( 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

  4. Decomposition matrices Representations of finite groups of Lie type GL n ( q ) , Sp 2 n ( q ) , . . . , E 8 ( 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

  5. Decomposition matrices Representations of finite groups of Lie type GL n ( q ) , Sp 2 n ( q ) , . . . , E 8 ( 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

  6. Inductive approach M representation O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  7. Inductive approach M representation M is cuspidal O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  8. Inductive approach M representation M is cuspidal M is non-cuspidal O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  9. Inductive approach M representation M is cuspidal M is non-cuspidal M “occurs” in an induced representation R G L ( N ) with N cuspidal O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  10. Inductive approach M representation M is cuspidal M is non-cuspidal M “occurs” in an induced representation R G L ( N ) with N cuspidal O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  11. Inductive approach M representation M is cuspidal M is non-cuspidal M “occurs” in an geometric construction of M induced representation via Deligne-Lusztig varieties R G L ( N ) with N cuspidal O. Dudas (CNRS) Finding PIM’s March 2013 3 / 9

  12. Parabolic induction G reductive algebraic group over F p F : G − → G Frobenius endomorphism / F q G F = G ( q ) is a finite reductive group O. Dudas (CNRS) Finding PIM’s March 2013 4 / 9

  13. Parabolic induction G reductive algebraic group over F p F : G − → G Frobenius endomorphism / F q G F = G ( q ) is a finite reductive group → ( a q Example. G = GL n ( F p ) with F : ( a i , j ) �− i , j ) then G ( q ) = GL n ( q ) O. Dudas (CNRS) Finding PIM’s March 2013 4 / 9

  14. Parabolic induction G reductive algebraic group over F p F : G − → G Frobenius endomorphism / F q G F = G ( q ) is a finite reductive group → ( a q Example. G = GL n ( F p ) with F : ( a i , j ) �− i , j ) then G ( q ) = GL n ( q ) Parabolic induction and restriction functors, given L a standard F -stable Levi subgroup R G L : kL ( q ) -mod − → kG ( q ) -mod ∗ R G L : kG ( q ) -mod − → kL ( q ) -mod O. Dudas (CNRS) Finding PIM’s March 2013 4 / 9

  15. Parabolic induction G reductive algebraic group over F p F : G − → G Frobenius endomorphism / F q G F = G ( q ) is a finite reductive group → ( a q Example. G = GL n ( F p ) with F : ( a i , j ) �− i , j ) then G ( q ) = GL n ( q ) Parabolic induction and restriction functors, given L a standard F -stable Levi subgroup R G L : kL ( q ) -mod − → kG ( q ) -mod ∗ R G L : kG ( q ) -mod − → kL ( q ) -mod Properties of induction/restriction (i) ( R G L , ∗ R G 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

  16. Cuspidality Definition A kG ( q ) -module M is cuspidal if ∗ R G L ( M ) = 0 for all proper standard F -stable Levi subgroup. O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  17. Cuspidality Definition A kG ( q ) -module M is cuspidal if ∗ R G 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 ∗ R G L ( M ) � = 0 O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  18. Cuspidality Definition A kG ( q ) -module M is cuspidal if ∗ R G 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 ∗ R G L ( M ) � = 0 Then there exists N cuspidal kL ( q ) -module such that ◮ M is in the head of R G L ( N ) O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  19. Cuspidality Definition A kG ( q ) -module M is cuspidal if ∗ R G 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 ∗ R G L ( M ) � = 0 Then there exists N cuspidal kL ( q ) -module such that ◮ M is in the head of R G L ( N ) ◮ P M is a direct summand of R G L ( P N ) O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  20. Cuspidality Definition A kG ( q ) -module M is cuspidal if ∗ R G 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 ∗ R G L ( M ) � = 0 Then there exists N cuspidal kL ( q ) -module such that ◮ M is in the head of R G L ( N ) ◮ P M is a direct summand of R G L ( P N ) Consequence. it is enough to ◮ know the projective cover of cuspidal simple modules ◮ know how to decompose R G L ( P N ) (Howlett-Lehrer, Dipper-Du-James, Geck-Hiss. . . ) O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  21. Cuspidality know the projective cover of cuspidal simple modules ◮ O. Dudas (CNRS) Finding PIM’s March 2013 5 / 9

  22. Geometric construction of the representations W Weyl group of G O. Dudas (CNRS) Finding PIM’s March 2013 6 / 9

  23. 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

  24. 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 H i c ( X ( w ) , Q ℓ ) and H i c ( X ( w ) , F ℓ ) give f.d. representations of G ( q ) over Q ℓ or F ℓ (non-zero when i ∈ { ℓ ( w ) , . . . , 2 ℓ ( w ) } only) O. Dudas (CNRS) Finding PIM’s March 2013 6 / 9

  25. 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 H i c ( X ( w ) , Q ℓ ) and H i c ( X ( w ) , F ℓ ) give f.d. representations of G ( q ) over Q ℓ or F ℓ (non-zero when i ∈ { ℓ ( w ) , . . . , 2 ℓ ( w ) } only) Example. Drinfeld curve X = { ( x , y ) ∈ F 2 p | xy q − yx q = 1 } then H 1 c ( X ) contains the discrete series of SL 2 ( q ) (cuspidal representations) O. Dudas (CNRS) Finding PIM’s March 2013 6 / 9

  26. 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 H i c ( X ( w ) , Q ℓ ) and H i c ( X ( w ) , F ℓ ) give f.d. representations of G ( q ) over Q ℓ or F ℓ (non-zero when i ∈ { ℓ ( w ) , . . . , 2 ℓ ( w ) } only) Example. Drinfeld curve X = { ( x , y ) ∈ F 2 p | xy q − yx q = 1 } then H 1 c ( X ) contains the discrete series of SL 2 ( q ) (cuspidal representations) Problem. How to know where the representations appear? O. Dudas (CNRS) Finding PIM’s March 2013 6 / 9

  27. 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

  28. 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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