Parabolic Deligne-Lusztig varieties and Brou´ e’s conjectures for reductive groups Jean Michel (joint work with F. Digne) University Paris VII Nagoya, March 2012 Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 1 / 19 Motivation To simplify we consider just the case of the principal block. Conjecture (Brou´ e) If the ℓ -Sylow S of the finite group G is abelian, then the principal ℓ -block B of Z ℓ G is derived-equivalent to the principal block b of Z ℓ N G ( S ) . By Rickard’s theorem there exists then a tilting complex T , a complex in D b ( B ) of finitely generated and projective B -modules, such that Hom D b ( B ) ( T , T [ k ]) = 0 for k � = 0. End D b ( B ) ( T ) ≃ b . For a reductive group in characteristic p � = ℓ , T should be the ℓ -adic cohomology complex of some Deligne-Lusztig variety, and End D b ( B )( T ) ≃ b should come through the action on that cohomology of a cyclotomic Hecke algebra associated to N G ( S ) / C G ( S ), which is a complex reflection group. Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 2 / 19
Finite reductive groups Let G be a connected reductive algebraic group over the algebraic closure of a finite field of characteristic p , and F an isogeny such that some power F δ is a split Frobenius for an F q δ -structure on G ; this defines δ and a positive real number q which is a power of √ p if G is simple. We choose an F -stable pair T ⊂ B of a maximal torus and a Borel subgroup; the Weyl group W = N G ( T ) / T acts as a reflection group on the complex vector space V = X ( T ) ⊗ C , The action of F on V is of the form q φ where q is a power of √ p and φ ∈ N GL ( V ) ( W ) is of finite order. The action of W φ is defined on X ( T ) ⊗ Q W φ where Q W φ = Q except for the Suzuki and Ree √ √ groups where Q W φ = Q ( 2) or Q ( 3) (we denote Z W φ the ring of integers of Q W φ ). The polynomial invariants S ( V ) W are a polynomial algebra C [ f 1 , . . . , f n ] where n = dim V . We denote by d i the degree of f i ; they can be chosen eigenvectors of φ , and we denote ε i the corresponding eigenvalues. Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 3 / 19 Polynomial order � n i =1 ( d i − 1) � n � n i =1 ( d i − 1) � We have | G F | = q i =1 ( q d i − ε i ) = q Φ Φ( q ) | a (Φ) | where Φ runs over the irreducible cyclotomic polynomials over Q W φ , and a (Φ) = a ( ζ ) = { d i | ζ d i = ε i } , where ζ is a root of Φ . √ √ √ 2)[ x ] we have Φ 8 = x 4 + 1 = ( x 2 − 2 q + 1)( x 2 + In Q ( 2 q + 1). We have the following theorem for an arbitrary complex reflection group W (Springer) | a ( ζ ) | is the dimension of a maximal ζ -eigenspace of an element of W φ ⊂ GL ( V ) . Two maximal ζ -eigenspaces are W -conjugate. For a maximal ζ -eigenspace V ζ , the group W ζ := N W ( V ζ ) / C W ( V ζ ) , acting on V ζ , is a complex reflection group with reflection degrees a ( ζ ) . (Lehrer-Springer) Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 4 / 19
Geometric Sylows Assume that w φ ∈ W φ has an eigenspace of dimension a = | a ( ζ ) | . Let ( T w , F ) be G -conjugate to ( T , wF ). The factor Φ a of the characteristic polynomial of w φ defines a wF -stable sublattice of X ( T ) ⊗ Z W φ , thus an F -stable subtorus S of T w such that | S F | = Φ( q ) a . In the previous situation We say that S is a Φ -Sylow. From the Springer theorem they form a single orbit under G F -conjugacy. For ℓ � = p assume that a ℓ -Sylow S of G F is abelian. Then | S | divides a unique factor Φ( q ) | a (Φ) | of G F , and there is a unique Φ -Sylow S ⊃ S. It follows that C G ( S ) = C G ( S ) is a Levi subgroup L . N G F ( S ) / C G F ( S ) = N W ( V ζ ) / C W ( V ζ ) = W ζ , attached to the ζ -eigenspace V ζ of some element of W φ . The principal block b of N G F ( S ) is isomorphic to Z ℓ ( S ⋊ W ζ ). Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 5 / 19 Deligne-Lusztig varieties Let P be a parabolic subgroup with F -stable Levi L and unipotent radical V . The Deligne-Lusztig variety Y V = { g V ∈ G / V | g V ∩ F ( g V ) � = ∅} has a left action of G F and a right action of L F . The (virtual) G F -module- L F given by � i ( − 1) i H i c ( Y V , Z ℓ ) defines the Deligne-Lusztig induction R G F L F . If F ( V ) = V the variety X V reduces to the discrete variety G F / V F and the alternating sum reduces to Z ℓ [ G F / V F ], giving Harish-Chandra induction . Conjecture (Geometric version) There exists P of Levi L = C G ( S ) such that R Γ c ( Y V , Z ℓ ) considered as an object of D b ( Z ℓ G F ⊗ ( Z ℓ L F ) opp ) , and restricted to B, is tilting between B and Z ℓ [ S ⋊ W ζ ] . Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 6 / 19
Shadow on unipotent characters The map g V �→ g P makes Y V an L F -torsor over the variety X P = { g P ∈ G / P | g P ∩ F ( g P ) � = ∅} For any λ ∈ Irr ( L F ) we have H i c ( X V , Z ℓ ) χ = H i c ( X P , F λ ). The conjecture can be mostly reduced to the study of X P with sheaves F λ associated to unipotent characters. We will look at the case χ = Id , and further, discard any torsion by going from Z ℓ to Q ℓ . Conjecture (Restricted) c ( X P , Q ℓ ) � G F = 0 for i � = j. 1 � H i c ( X P , Q ℓ ) , H i 2 End G F ⊕ i H i c ( X P , Q ℓ ) ≃ Q ℓ W ζ . A braid monoid attached to the complex reflection group W ζ acts on X P as G F -endomorphisms, such that on the cohomology the action factors through a cyclotomic Hecke algebra for W ζ . Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 7 / 19 Choice of P Let ( W , S ) be the Coxeter system associated to the BN-pair ( B , N G ( T )). We can conjugate P to a standard parabolic subgroup P I . This conjugates the ζ -eigenspace to a V ζ such that C W ( V ζ ) = W I , the Weyl group of L I . The w φ ∈ W φ with ζ -eigenspace V ζ form a class W I w φ . We choose w φ to be I -reduced. I , w , φ I X P is isomorphic to { P | P − − − − → F P } which means that ( P , F P ) ∼ G ( P I , w P φ I ) (we have w φ I = I). w → φ I ). We denote this variety X ( I − The choice of a parabolic subgroup with Levi C G ( S ) corresponds to the choice of a class W I w φ up to W -conjugacy, or to the choice of an I -reduced element w such that w φ I = I up to W -conjugacy of such pairs w → φ I ) = l ( w ). ( w , I ); for such an element we have dim X ( I − Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 8 / 19
Craven’s formula Block theory and the work of Rouquier and Craven in constructing “perverse equivalences” led to a very specific conjecture for the cohomology of the variety X P we are looking for. Let ρ be a unipotent character which occurs in H i ( X P , F λ ), where P is the “right” parabolic subgroup of Levi C G ( S ), where S is a Φ-Sylow. Choose ζ as the root of Φ with minimal argument and write ζ = e 2 ik π/ d . Let P = deg ρ/ deg λ , a polynomial in q . Then Conjecture (Craven) i = k / d ( degree ( P ) + valuation ( P ))+ { number of roots of P of argument less than that of ζ } − 1 / 2 { number of times 1 is a root of P } Further, we should have dim X P = 2 k / d ( l ( w 0 ) − l ( w I )) where w 0 (resp. w I ) is the longest element of W (resp W I ). Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 9 / 19 w G F -endomorphisms of X ( I → φ I ) − w The idea for constructing G F -endomorphisms of X ( I → φ I ) is: − If w = xy with l ( w ) = l ( x ) + l ( y ), and I x = J ⊂ S , I , w , φ I → P ′ J , y , φ I → F P there is a unique P ′ such that P I , x , J → F P . when P − − − − − − − − − − If we have also l ( y φ ( x )) = l ( y ) + l ( φ ( x )), then since P ′ J , y , φ I φ I ,φ ( x ) , φ J → F ( P ′ ), we have P ′ ∈ X ( J y φ ( x ) → φ J ), − − − → F ( P ) − − − − − − − − − thus P �→ P ′ defines a map X ( I w → φ I ) D x y φ ( x ) → φ J ) which is − − → X ( J − − − G F -equivariant. If in addition I x = I and x commutes to w φ we get an endomorphism. There are too many conditions so this do not construct enough endomorphisms. Jean Michel (joint work with F. Digne) () Parabolic Deligne-Lusztig varieties Nagoya, March 2012 10 / 19
Recommend
More recommend
