aa r X i v : . [ m a t h . R T ] J un COUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPSOLIVIER BRUNATAbstra t. This arti le is on erned with the relative M Kay onje ture for(cid:28)nite redu tive groups. Let G be a onne ted redu tive group de(cid:28)ned overthe (cid:28)nite (cid:28)eld F q of hara teristi p > with orresponding Frobenius map F . We prove that if the F - oinvariants of the omponent group of the enterof G has prime order and if p is a good prime for G , then the relative M Kay onje ture holds for G F at the prime p . In parti ular, this onje ture is truefor G F in de(cid:28)ning hara teristi for a simple and simply- onne ted group G of type B n , C n , E and E . Our main tools are the theory of Gelfand-Graev hara ters for onne ted redu tive groups with dis onne ted enter developedby Digne-Lehrer-Mi hel and the theory of uspidal Levi subgroups. We alsoexpli itly ompute the number of semisimple lasses of G F for any simplealgebrai group G . 1. Introdu tionLet G be a (cid:28)nite group and p be a prime divisor of | G | . As usually, we denoteby Irr( G ) the set of irredu ible hara ters of G and by Irr p ′ ( G ) the subset of irre-du ible hara ters with degree prime to p . For any (cid:28)xed p -Sylow subgroup P of G , John M Kay has onje tured that | Irr p ′ ( G ) | = | Irr p ′ (N G ( P )) | . This is a tuallyproved for some groups but remains open in general. Re ently, Isaa s, Malle andNavarro redu ed this onje ture to a new question, the so- alled indu tive M Kay ondition, whi h on erns properties of perfe t entral extensions of (cid:28)nite simplegroups; see [8℄.In this arti le, we are interested in the relative M Kay onje ture, asserting thatfor every linear hara ter ν of the enter Z of G , if Irr p ′ ( G | ν ) denotes the subset of hara ters χ ∈ Irr p ′ ( G ) lying over ν (i.e. satisfying h χ, Ind GZ ( ν ) i G = 0 ), then onehas the equality | Irr p ′ ( G | ν ) | = | Irr p ′ (N G ( P ) | ν ) | . In order to prove the indu tiveM Kay ondition, we in parti ular have to show that the relative M Kay onje tureholds for some perfe t entral extensions of (cid:28)nite simple groups. This is one of themotivations to onsider this question in this work.Let G be a onne ted redu tive group de(cid:28)ned over a (cid:28)nite (cid:28)eld with q elements F q of hara teristi p > with orresponding Frobenius map F : G → G . Throughoutthis paper, we will always assume that p is a good prime for G , that is p does notdivide the oe(cid:30) ients of the highest root of the root system asso iated to G (see [4,1.14℄). Let T be a maximal F -stable torus of G ontained in an F -stable Borelsubgroup B of G and let U denote the unipotent radi al of B (whi h is F -stable).Note that, if U is not trivial, then the prime p divides the order of the (cid:28)nite(cid:28)xed-point subgroup G F and the subgroup U F ⊆ G F is a p -Sylow subgroup of G F . Moreover, one has N G F ( U F ) = B F . If the enter of G is onne ted, thenthe M Kay onje ture is true for the group G F at the prime p . We will see in the1991 Mathemati s Subje t Classi(cid:28) ation. 20C15, 20C33.1 OLIVIER BRUNATfollowing that the relative M Kay onje ture holds in this ase (see Proposition 6.5).This question is more di(cid:30) ult when the enter of G is dis onne ted. In this arti le,we will solve it in a spe ial situation. Denote by Z ( G ) = Z( G ) / Z( G ) ◦ the groupof omponents of the enter of G and by H ( F, Z ( G )) the set of the F - lasses of Z ( G ) . Then our main result is the following.Theorem 1.1. Let G be a onne ted redu tive group de(cid:28)ned over the (cid:28)nite (cid:28)eld F q of hara teristi p > and let F : G → G denote the orresponding Frobeniusmap. Let T be a maximal F -stable torus ontained in an F -stable Borel subgroup B of G . If p is a good prime for G and if the group H ( F, Z ( G )) is trivial or hasprime order, then for every linear hara ter ν of Z( G F ) , one has | Irr p ′ ( G F | ν ) | = | Irr p ′ ( B F | ν ) | . As onsequen e, this proves the relative M Kay onje ture in de(cid:28)ning hara -teristi for G F with G a simple group given in Table 4.This paper is organized as follows. In Se tion 2, we re all some results fromBonnafé [1℄ on the uspidal Levi subgroups of onne ted redu tive groups. We willneed this theory (cid:28)rst, in order to asso iate to every linear hara ter of Z ( G ) a uspidal Levi subgroup of G ( orresponding to a uspidal lo al system in Lusztigtheory), and se ondly to ontrol the dis onne ted part of the inertial subgroup oflinear hara ters of U F . In Se tion 3, we apply the theory of Gelfand-Graev hara -ters of G F for onne ted redu tive group G with dis onne ted enter, developed byDigne-Lehrer-Mi hel in [5℄. Note that we need here that p is a good prime for G . Inparti ular, we give a formula to ompute the s alar produ t of two Gelfand-Graev hara ters; see Proposition 3.2. As onsequen e, we obtain an expli it formula forthe number of semisimple lasses of G F (see Theorem 3.5) and ompute this num-ber for G F with G any simple algebrai group; see Corollary 3.6. Re all that the onstituents of the duals of Gelfand-Graev hara ters (for the Alvis-Curtis dualityfun tor) are the so- alled semisimple hara ters of G F . When p is a good primefor G , the semisimple hara ters are the p ′ - hara ters of G F (that is, the elementsof Irr p ′ ( G F ) ). In Se tion 4, using the results of Se tion 3, we ompute the numberof semisimple hara ters of G F when H ( F, Z ( G )) has prime order; see Proposi-tion 4.2. In Se tion 5, we give a formula for the number of p ′ - hara ters of B F depending on the uspidal Levi subgroups of G ; see Proposition 5.6. Finally, inSe tion 6, we show that if the enter of G is onne ted or if H ( F, Z ( G )) has primeorder, then for a linear hara ter ν of Z( G F ) the number of semisimple hara tersof G F lying over ν does not depend on ν ; see Proposition 6.5 and Proposition 6.6.We an then prove Theorem 1.1; see Remark 6.7.2. Cuspidal Levi subgroups and entral hara tersLet G be a onne ted redu tive group de(cid:28)ned over F q with orresponding Frobe-nius map F : G → G . As above, we denote by T a maximal F -stable torus of G ontained in an F -stable Borel subgroup B of G . Write Φ for the root system of G and Φ + for the set of positive roots with respe t to B . Denote by ∆ the setof orresponding simple roots and by W the Weyl group of G with respe t to T ,identi(cid:28)ed with the quotient N( T ) / T . Moreover, we asso iate to every α ∈ Φ are(cid:29)e tion w α ∈ W and for any subset I of ∆ , we denote by W I the subgroup of W generated by w α for α ∈ I . The subgroup P I = B W I B is a standard paraboli OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 3subgroup of G (relative to B ). We denote by L I the Levi subgroup of P I ontaining T . Note that every Levi subgroup L of G is onjugate in G to a Levi L I for somesubset I of ∆ .Let L be a Levi subgroup of G . Then the in lusion Z( G ) ⊆ Z( L ) indu es asurje tive map h L : Z ( G ) → Z ( L ) , where Z ( G ) = Z( G ) / Z( G ) ◦ . We re all that G is uspidal if ker( h L ) = { } forevery proper Levi L of G . Moreover, a linear hara ter ζ of Z ( G ) is uspidal if, forevery Levi subgroup L of G , the subgroup ker( h L ) is not ontained in ker( ζ ) .Let ζ be a linear hara ter of Z ( G ) . Then there is a Levi subgroup L (whi h is uspidal) and a uspidal hara ter ζ L of Z ( L ) su h that ζ = ζ L ◦ h L . More pre isely, for a subgroup K of Z ( G ) , denote by L ( K ) the set of Levi sub-groups L of G su h that ker( h L ) ⊆ K and by L min ( K ) the subset of minimal ele-ments of L ( K ) . In [1, 2.16℄, Bonnafé proves that the Levi subgroups of L min ( K ) are uspidal and G - onjugate. Therefore, we asso iate to the linear hara ter ζ of Z ( G ) a standard Levi L I of L min (ker( ζ )) . Note that all Levi subgroups in L min ( K ) have the same semisimple rank.Let H ( F, Z ( G )) be the set of F - lasses of Z ( G ) . Sin e Z ( G ) is abelian, theLang map L : Z ( G ) → Z ( G ) : g g − F ( g ) is a morphism of groups and we have H ( F, Z ( G )) = Z ( G ) / L ( Z ( G )) . In parti ular, a hara ter ζ of H ( F, Z ( G )) anbe seen as a hara ter of Z ( G ) with L ( Z ( G )) in its kernel. Hen e, we an asso iateto every hara ter ζ of H ( F, Z ( G )) a uspidal Levi L of G and a uspidal ζ L of Z ( L ) . Note that L an be hosen F -stable and with this hoi e, ζ L is F -stable.In the following, we write H ( F, Z ( G )) ∧ for the set of irredu ible hara ters of H ( F, Z ( G )) . 3. Number of semisimple lasses3.1. Gelfand-Graev hara ters. Let G be a onne ted redu tive group de(cid:28)nedover F q with Frobenius map F : G → G . We denote by T a maximal F -stabletorus of G ontained in an F -stable Borel subgroup B of G . We write U for theunipotent radi al of B . We re all that p is supposed to be a good prime for G .As above, we denote by Φ the root system of G , by Φ + the set of positive rootswith respe t to B and by ∆ the set of orresponding simple roots. We write X α for the non-trivial minimal losed unipotent subgroup of U normalized by T and orresponding to the root α ∈ Φ + . Re all that the Frobenius map F indu es apermutation on Φ su h that F (Φ + ) = Φ + and F (∆) = ∆ . Put U = Y α ∈ Φ + \ ∆ X α . Denote by U the quotient U / U and write π U : U → U for the anoni alproje tion map. Then we have U ≃ Q α ∈ ∆ X α and(1) U F = Y ω ∈O X Fω , where O is the set of F -orbits on ∆ and X ω = Q α ∈ ω X α . Re all that an elementof G is regular if its entralizer has a minimal possible dimension. By [6, 14.14℄ theregular unipotent elements of U are the elements u ∈ U su h that for every α ∈ ∆ , OLIVIER BRUNAT π U ( u ) α = 1 . Moreover by [6, 14.25℄, the set of regular unipotent lasses of G F areparametrized by H ( F, Z ( G )) . For z ∈ H ( F, Z ( G )) , denote by U z the onjuga y lass of unipotent elements orresponding to z and put γ z : G F → C , g (cid:26) |U z | / | G F | if g ∈ U z otherwiseRe all that a linear hara ter ψ of U F is a regular hara ter if it has U F in itskernel and if the indu ed linear hara ter on U F (always denoted by ψ ) satis(cid:28)es Res U F X Fω ( ψ ) = 1 X Fω for every ω ∈ O . By [6, 14.28℄, the set of T F -orbits of regular hara ters of U F is parametrized by H ( F, Z ( G )) as follows.Fix ψ a regular linear hara ter of U F and z ∈ H ( F, Z ( G )) . Choose t z ∈ T su h that t − z F ( t z ) Z( G F ) = z . Then the T F -orbit of the regular hara ters of U F orresponding to z has ψ z = t z ψ for representative.We now an de(cid:28)ne the Gelfand-Graev hara ters of G F by setting for every z ∈ H ( F, Z ( G )) Γ z = Ind G F U F ( φ z ) . Denote by D G the Alvis-Curtis duality map. For z ∈ H ( F, Z ( G )) , there is avirtual hara ter ϕ z of U F (see the proof of [6, 14.33℄) with U F in its kernel, whi his zero outside regular unipotent elements and satisfying D G (Γ z ) = Ind G F U F ( ϕ z ) . In parti ular, D G (Γ z ) is onstant on U z and there are omplex numbers c z,z ′ (for z ′ ∈ H ( F, Z ( G )) ) with(2) D G (Γ z ) = X z ′ ∈ H ( F, Z ( G )) c z,z ′ γ z ′ . Following [5℄, we now re all how to ompute the oe(cid:30) ients c z,z ′ . For this, weneed some notations. For z ∈ H ( F, Z ( G )) , put σ z = X ψ ∈ Ψ z − ψ ( u ) , where u ∈ U and Ψ z denotes the T F -orbit of ψ z . Moreover, for any hara ter ζ of H ( F, Z ( G )) , we de(cid:28)ne σ ζ = X z ∈ H ( F, Z ( G )) ζ ( z ) σ z . In [5, 2.3, 2.5℄, the following result is proven.Proposition 3.1. With the above notation, if p is a good prime for G , then the ma-trix ( c z,z ′ ) z, z ′ ∈ H ( F, Z ( G )) is invertible and its inverse is ( η G σ z ( z ′ ) − ) z,z ′ ∈ H ( F, Z ( G )) ,where η G = ( − F q -rk( G ) . Moreover, we have c z,z ′ = c z ( z ′ ) − , and if we put c ζ = P z ∈ H ( F, Z ( G )) ζ ( z ) c z, for any hara ter ζ of H ( F, Z ( G )) , then there isa fourth root of unity ξ ζ su h that c ζ = η G η L q − (ss-rk( L ζ )) ξ ζ , where L ζ is the uspidal Levi of G asso iated to the hara ter ζ as explained inSe tion 2.OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 5Proposition 3.2. With the notation as above, if p is a good prime for G , then for z , z ∈ H ( F, Z ( G )) , one has h Γ z , Γ z i G F = | Z( G ) ◦ F | X ζ ∈ H ( F, Z ( G )) ∧ ζ ( z ) ζ ( z ) q l − (ss-rk( L ζ )) , where L ζ is the uspidal Levi of G asso iated to the hara ter ζ of H ( F, Z ( G )) and l is the semisimple rank of G .Proof. Fix z and z in H ( F, Z ( G )) and put I = h Γ z , Γ z i G F . Sin e the dualityfun tor D G is an isometry, one has I = h D G (Γ z ) , D G (Γ z ) i G F . Furthermore,thanks to Equation (2), we dedu e h D G (Γ z ) , D G (Γ z ) i G F = X z,z ′ ∈ H ( F, Z ( G )) c z ,z c z ,z ′ h γ z , γ z ′ i G F . Note that, if z ′ = z , then h γ z , γ z ′ i G F = 0 . Moreover, h γ z , γ z i G F = | C G F ( u z ) | for u z ∈ U z . We dedu e(3) I = X z ∈ H ( F, Z ( G )) c z ,z c z ,z | C G F ( u z ) | , However, the group C G ( u ) is abelian (be ause the hara teristi is good for G ).It then follows that | C G F ( u z ) | = | C G F ( u ) | for every z ∈ H ( F, Z ( G )) ; see [6,14.22℄. Moreover, [6, 14.23℄ implies | H ( F, Z ( G )) | | G F || C G F ( u z ) | = | G F || Z( G ) ◦ F | q l . Sin e | H ( F, Z ( G )) | = | Z( G ) F | / | Z( G ) ◦ F | , we dedu e(4) | C G F ( u z ) | = | Z( G ) F | q l . For every ζ ∈ H ( F, Z ( G )) ∧ , we have c ζ = P z ∈ H ( F, Z ( G )) ζ ( z ) c z, . Denote by T the hara ter table of H ( F, Z ( G )) (identi(cid:28)ed with the quotient group Z ( G ) / L ( Z ( G )) as above). Write m = | H ( F, Z ( G )) | . Sin e T is the hara ter table of a (cid:28)niteabelian group, it follows that T is invertible and T − = m t T . We then dedu ethat, for every z ∈ H ( F, Z ( G )) (5) c z = 1 m X ζ ∈ H ( F, Z ( G )) ∧ ζ ( z ) c ζ . Furthermore, by Proposition 3.1 one has c z i ,z = c z i ( z ) − , . Then Equations (3), (4)and (5) imply I = X z ∈ H ( F, Z ( G )) m X ζ,ζ ′ ∈ H ( F, Z ( G )) ∧ ζ ( z z − ) ζ ′ ( z z − ) | C G F ( u z ) | c ζ c ζ ′ = | Z( G ) F | q l m X ζ,ζ ′ ∈ H ( F, Z ( G )) ∧ ζ ( z ) ζ ′ ( z ) h ζ, ζ ′ i H ( F, Z ( G )) c ζ c ζ ′ = | Z( G ) F | q l m X ζ ∈ H ( F, Z ( G )) ∧ ζ ( z ) ζ ( z ) | c ζ | . OLIVIER BRUNATNow, Proposition 3.1 implies c ζ = η G η L q − (ss-rk( L ζ )) ξ ζ . Thus | c ζ | = q − (ss-rk( L ζ )) | ξ ζ | = q − (ss-rk( L ζ )) . Moreover, | Z( G ) F | m = | Z( G ) ◦ F | . This proves the laim. (cid:3)
Remark 3.3. Note that h Γ z , Γ z ′ i G F does not depend on the fourth roots of unity ξ ζ asso iated to ζ ∈ H ( F, Z ( G )) ∧ as in Proposition 3.1.Remark 3.4. If the enter of G is onne ted, there is only one Gelfand-Graev hara ter Γ and the uspidal Levi subgroup asso iated to the trivial hara ter of H ( F, Z ( G )) is a maximal torus, whi h has semisimple rank equal to zero. Thus,we obtain h Γ , Γ i G F = | Z( G ) F | q l , whi h is a well-known result [4, 8.3.1℄.3.2. Number of semisimple lasses.Theorem 3.5. Let G be a onne ted redu tive group de(cid:28)ned over a (cid:28)nite (cid:28)eld of hara teristi p > with q elements F q and let F : G → G denote the orrespondingFrobenius map. Write S for a set of representatives of semisimple lasses of G F .Denote by ( G ∗ , F ∗ ) a dual pair of ( G , F ) . With the above notation, if p is a goodprime for G , then we have |S| = | Z( G ) ◦ F | X ζ ∈ H ( F ∗ , Z ( G ∗ )) ∧ q l − (ss-rk( L ∗ ζ )) , where l is the semisimple rank of G and L ∗ ζ is a uspidal Levi subgroup of G ∗ asso iated to ζ ∈ H ( F ∗ , Z ( G ∗ )) ∧ as explained in Se tion 2.Proof. Denote by ( G ∗ , F ∗ ) a pair dual to ( G , F ) . As explained in Se tion 3.1, we an asso iate to every z ∈ H ( F ∗ , Z ( G ∗ )) a Gelfand-Graev hara ter Γ z of G ∗ F ∗ .Re all that Γ z is multipli ity free. We an des ribe more pre isely the onstituentsof Γ z as follows. Fix s ∈ S . Using Deligne-Lusztig hara ters, Digne-Mi hel de(cid:28)nedin [6, 14.40℄ a lass fun tion χ s and proved that for every z ∈ H ( F ∗ , Z ( G ∗ )) , thereis exa tly one irredu ible hara ter of G F , denoted by χ s,z , whi h is a ommon onstituent of χ s and Γ z and satisfying (see [6, 14.49℄):(6) Γ z = X s ∈S χ s,z . Equation (6) implies |S| = h Γ , Γ i G ∗ F ∗ . Now, thanks to Proposition 3.2, theresult follows. (cid:3) We now will pre ise some notations. For a simple algebrai group G de(cid:28)nedover F q , if the orresponding Frobenius map is split, then we denote it by F + .Otherwise, if the F q -stru ture is given by a non-split Frobenius, we denote it by F − . Moreover, if G is of type X and has split and non-split Frobenius map F + and F − , then we put ǫ X ( q ) = G F ǫ for ǫ ∈ {− , } .Fix some positive integer n and denote by G sc a simple simply- onne ted alge-brai group of type A n . For any divisor r of n + 1 , there is a simple algebrai group G r of type A n and a surje tive morphism π r : G sc → G r satisfying ker( π r ) equalsOUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 7Type |S| ǫ A rn ( q ) r | ( n + 1) m = gcd( r, q − ǫ ) P d/m φ ( d ) q n +1 d − B n ( q ) adjoint q = 0 mod 2 q = 1 mod 2 q n q n + q n − C n ( q ) adjoint q = 0 mod 2 q = 1 mod 2 q n q n + q ⌊ n/ ⌋ ǫ D n +1 ( q ) adjoint q = 0 , q = ǫ mod 4 q = − ǫ mod 4 q n +1 q n +1 + 2 q n − + q n − q n +1 + q n − SO ǫ n +2 ( q ) q = 0 mod 2 q = 1 mod 2 q n +1 q n +1 + q n − ǫ D n ( q ) adjoint q = 0 mod 2 q = 1 mod 2 q n q n + 2 q n + q n − SO ǫ n ( q ) q = 0 mod 2 q = 1 mod 2 q n q n + q n − HS n ( q ) q = 0 mod 2 q = 1 mod 2 q n q n + q nǫ E ( q ) adjoint, p = 2 q = 0 , − ǫ mod 3 q = ǫ mod 3 q q + 2 q E adjoint, p = 3 q = 0 mod 2 q = 1 mod 2 q q + q Table 1. Number of semisimple lasses for simple algebrai groups.the subgroup of Z( G sc ) of order r . If G r is de(cid:28)ned over F q with Frobenius map F ǫ , then put ǫ A rn ( q ) = G F ǫ r .Corollary 3.6. Let G be a simple algebrai group de(cid:28)ned over F q with orrespond-ing Frobenius map F . If G F is isomorphi to G F sc , then the number of semisimple lasses of G F is q n , where n is the semisimple rank of G . Otherwise, the numberof semisimple lasses of G F is given in Table 1. As usually, we denote by φ theEuler fun tion.Proof. Let G be a simple algebrai group de(cid:28)ned over F q with orresponding Frobe-nius F . Denote by ( G ∗ , F ∗ ) a pair dual to ( G , F ) . In table 2, we re all simplealgebrai groups in duality.Fix a linear hara ter ζ of Z ( G ∗ ) and denote by L ∗ ζ a uspidal Levi subgroup of L min (ker( ζ )) . Write G ∗ sc for a simple simply- onne ted group of the same versionas G ∗ and by π : G ∗ sc → G ∗ the universal over of G ∗ . The endomorphism F ∗ of OLIVIER BRUNAT G G ∗ A n G r G ( n +1) /r B n simply- onne ted C n of type adjointadjoint C n of type simply- onne ted D n +1 simply- onne ted adjoint SO n +2 SO n +2 D n simply- onne ted adjoint SO n SO n HS n HS n E simply- onne ted adjoint E simply- onne ted adjointTable 2. Groups in duality G ∗ is indu ed by a unique Frobenius map (also denoted by F ∗ ) of G ∗ sc . Now, put b L ∗ ζ = π − ( L ∗ ζ ) . Note that b L ∗ ζ is a Levi subgroup of G ∗ sc with the same semisimplerank as L ∗ ζ . Moreover, following [1, 2.10℄, we dedu e that b L ∗ ζ ∈ L min ( π − (ker( ζ ))) .Note that, sin e G ∗ is simple, one has ker( h b L ∗ ζ ) = π − (ker( ζ )) ; see [1, 2.9℄.Suppose now that Z ( G ∗ sc ) is y li of order N . Then Z ( G ∗ ) is y li of order N ′ = N/ | ker( π ) | . Sin e Im( ζ ) is a subgroup of C × of order o( ζ ) (we onsiderhere Irr( Z ( G ∗ )) as a group with produ t the tensor produ t of hara ters). it inparti ular follows that ker( ζ ) has order N ′ / o( ζ ) . But there is only one subgroup K of Z ( G ∗ ) of order N ′ / o( ζ ) and L ∗ ζ is then a standard Levi of L min ( K ) onlydepending on o( ζ ) . Furthermore, one has | π − ( K ) | = | K || ker( π ) | = N / o( ζ ) . Sin e Z ( G ∗ sc ) is y li , π − ( K ) is then the unique subgroup of order N/ o( ζ ) . Then b L ∗ ζ is a Levi subgroup of G ∗ sc satisfying | ker( h b L ∗ ζ ) | = N/ o( ζ ) .In [1, Table 2.17℄, Bonnafé expli itly omputes L min ( K ) for any subgroup K of Z ( G ∗ sc ) . In Table 3, we re all some information that we need. For more details, werefer to [1℄. For the notation in Table 3, we put µ n = { z ∈ F × p | z n = 1 } .Hen e, using Table 3 we then an (cid:28)nd the uspidal Levi subgroup (and itssemisimple rank) asso iated to every linear hara ter of Z ( G ∗ ) for G ∗ of type A n , B n , C n , E and E and D n +1 . For example, suppose G is of type A n . Thenusing the notation pre eding Corollary 3.6 , there is an integer r su h that G = G r .Moreover, one has G ∗ r = G r ′ with r ′ = ( n + 1) /r . Note that |Z ( G r ′ ) | = r . Let d be a divisor of r and let ζ be a linear hara ter of Z ( G r ′ ) of order d . Then b L ∗ ζ hassemisimple rank equal to n +1 d ( d − .Suppose G is of type D n and denote by π : G ∗ sc → G ∗ the universal overof G ∗ as above. The group Z ( G ∗ sc ) has order and exponent . Moreover, thethree non-trivial hara ters of Z ( G ∗ sc ) have distin t kernel. These kernels are thesubgroups of order of Z ( G ∗ sc ) denoted by c , c and c in Table 3. Note that ifOUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 9Type of G Z ( G ) K ss-rk ( L ) for L ∈ L min ( K ) Z ( L ) A n µ n +1 µ ( n +1) /d d | ( n + 1) p ∤ d n +1 d ( d − µ d B n p = 2 µ ⌊ n +12 ⌋ µ C n p = 2 µ µ D n +1 p = 2 µ µ n + 22 µ µ D n p = 2 µ × µ c c c n + 1 nn µ × µ µ µ µ E p = 3 µ µ E p = 2 µ µ Table 3. L min ( K ) for simple simply- onne ted groups ker( π ) = c then G ∗ = SO n and if ker( π ) ∈ { c , c } , then G ∗ = HS n . Let ζ bea non-trivial linear hara ter of Z ( G ∗ ) . Suppose (cid:28)rst that G ∗ = G ∗ sc . Then, the orresponding uspidal Levi L ∗ ζ is a uspidal standard Levi subgroup G ∗ sc su h that ζ and h L ζ have the same kernel. If G ∗ = SO n or G ∗ = HS n , then Z ( G ∗ ) hasorder and the semisimple rank of the uspidal Levi asso iated to the non-trivial hara ter of Z ( G ∗ ) equals the semisimple rank of any elements of L min (ker( π )) (inthe group G ∗ sc ).We now dis uss the onditions on q given in the se ond olumn of Table 1.Suppose that Z ( G ∗ ) is y li of order N . Then, using [7, Table 1.12.6, 1.15.2℄, weshow that the order of H ( F ǫ ∗ , Z ( G ∗ )) is the g d of N and q − ǫ . If Z ( G ∗ ) is not y li (i.e. G is of type D n ) and if p = 2 , then H ( F ǫ ∗ , Z ( G ∗ )) = Z ( G ∗ ) ; see [7,Table 1.12.6, 1.15.2℄.The result then follows from Theorem 3.5. (cid:3) G be a onne ted redu tive group de(cid:28)ned over F q (with Frobenius map F ) asabove and let ( G ∗ , F ∗ ) denote a dual pair of ( G , F ) . Write S (resp. T ) for a set ofrepresentatives of semisimple lasses of G ∗ F ∗ (resp. a set of representatives of F ∗ -stable semisimple lasses of G ∗ ). Moreover, we suppose that the elements of T are F ∗ -stable (whi h is possible be ause by Lang-Steinberg Theorem, we an hoose an F ∗ -stable representative in every F ∗ -stable geometri lass of G ∗ ). Put A G ∗ ( s ) =C G ∗ ( s ) / C G ∗ ( s ) ◦ . Re all that the lasses of G ∗ F ∗ with representative t ∈ G ∗ F ∗ onjugate to s in G ∗ are parametrized by the set of F ∗ - lasses of A G ∗ ( s ) . Moreover, A G ∗ ( s ) is abelian, implying | H ( F ∗ , A G ∗ ( s )) | = | A G ∗ ( s ) F ∗ | . Note that there isan inje tive morphism between A G ∗ ( s ) F ∗ and H ( F, Z ( G )) ∧ . Hen e | A G ∗ ( s ) F ∗ | divides | H ( F, Z ( G )) | and for every divisor d of | H ( F, Z ( G )) | , we put(7) T d = n s ∈ T | d = | A G ∗ ( s ) F ∗ | o . For s ∈ S and z ∈ H ( F, Z ( G )) , we set ρ s,z = D G ( χ s,z ) , where the hara ter χ s,z is the onstituent of the Gelfand-Graev hara ter Γ z de(cid:28)ned in Equation (6). Put Irr s ( G F ) = { ρ s,z | s ∈ S , z ∈ H ( F, Z ( G )) } . The irredu ible hara ters ρ s,z are the so- alled semisimple hara ters of G F .Proposition 4.1. With the above notation, we have | Irr s ( G F ) | = X d/ | H ( F, Z ( G )) | d |T d | . Proof. As explained in [6, p. 139℄, we embed G in a onne ted redu tive group with onne ted enter e G with the same derived subgroup and su h that G is normal in e G . We extend F to e G (denoted by the same symbol). The in lusion G ⊆ e G indu esa surje tive F ∗ -equivariant morphism i ∗ : e G ∗ → G ∗ . For s ∈ S , there is an F ∗ -stable semisimple e s of e G ∗ su h that i ∗ ( e s ) = s . Write ρ e s for the semisimple hara terof e G F orresponding to s (this hara ter is unique be ause H ( F, Z ( e G )) is trivial).Then by [6, 14.49℄, the hara ter ρ s, is a onstituent of Res e G F G F ( ρ e s ) . Moreover, theinertial group e G F ( s ) of ρ s, in e G F is su h that e G F / e G F ( s ) ≃ A G ∗ ( s ) F ∗ . Thusby Cli(cid:27)ord theory, sin e Res e G F G F ( ρ e s ) is multipli ity free (see [9℄), we dedu e that Res e G F G F ( ρ e s ) has | A G ∗ ( s ) F ∗ | onstituents. It follows that | Irr s ( G F ) | = X s ∈S | A G ∗ ( s ) F ∗ | = X t ∈T X s ∈S∩ [ t ] G ∗ | A G ∗ ( s ) F ∗ | = X t ∈T | A G ∗ ( t ) F ∗ | . The result follows. (cid:3)
Proposition 4.2. We keep the same notation as above and we suppose that p is agood prime for G . Suppose that H ( F, Z ( G )) has prime order ℓ . Let ζ be a nontrivial linear hara ter of H ( F, Z ( G )) . Write L for its asso iated uspidal Levisubgroup. Then we have | Irr s ( G F ) | = | Z( G ) ◦ F | (cid:16) q l + ( ℓ − q l − (ss-rk( L )) (cid:17) , where l denotes the semisimple rank of G . In parti ular, in Table 4, we give thenumber of semisimple hara ters of G F for simple groups G with Z( G )) F of primeorder. For the notation of Table 4, we put m = gcd( r, q − ǫ ) .OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 11 G F sc | Irr s ( G F sc ) | ǫ A rn ( q ) m prime q n + ( m − q n +1 m − B n ( q ) q = 1 mod 2 q n + 3 q ⌊ n/ ⌋ C n ( q ) q = 1 mod 2 q n + 3 q n − ǫ D n +1 ( q ) q = − ǫ mod 4 q n +1 + 3 q n − SO ǫ n ( q ) q = 1 mod 2 q n + 3 q n − HS n ( q ) q = 1 mod 2 q n + 3 q nǫ E ( q ) , p = 2 q = ǫ mod 3 q + 8 q E ( q ) , p = 3 q = 1 mod 2 q + 3 q Table 4. Number of semisimple hara ters.Proof. We denote by T and T ℓ the sets as de(cid:28)ned in Equation (7). We have |T | = |T | + |T ℓ | and |S| = |T | + ℓ |T ℓ | implying |T | = 1 ℓ − ℓ |T | − |S| ) and |T ℓ | = 1 ℓ − |S| − |T | ) . Furthermore, from [6, 14.42℄ we dedu e that |T | = | Z( G ) ◦ F | q l . Moreover, sin e ℓ isprime, all non trivial linear hara ters of H ( F, Z ( G )) are faithful on H ( F, Z ( G )) .Their orresponding hara ters of Z ( G ) then have the same kernel (equal to L ( Z ( G )) ).Thus, they are asso iated to a same uspidal Levi subgroup L , whi h is the standardLevi of L min ( L ( Z ( G ))) . Thanks to Theorem 3.5 we dedu e that |S| = | Z( G ) ◦ F | (cid:16) q l + ( ℓ − q l − (ss-rk( L )) (cid:17) . Now, using Proposition 4.1, we obtain | Irr s ( G F ) | = |T | + ℓ |T ℓ | = ( ℓ + 1) |S| − ℓ |T | = | Z( G ) ◦ F | (cid:0) ( ℓ + 1) q l + ( ℓ − q l − (ss-rk( L )) − ℓq l (cid:1) = | Z( G ) ◦ F | (cid:0) q l + ( ℓ − q l − (ss-rk( L )) (cid:1) . Now, Table 4 follows from Table 3. However, note that for G = SO n , we have todistinguish whether n is even or not. If n = 2 k + 1 , then the number of semisimple hara ters of SO ǫ k +2 ( q ) is q k +1 + 3 q k − = q n + 3 q n − . If n = 2 k , then the numberof semisimple hara ters of SO ǫ k ( q ) is q k + 3 q k − = q n + 3 q n − . (cid:3)
5. Chara ters of p ′ -order in Borel subgroups5.1. Formula for the number of p ′ - hara ters. In this se tion, we keep thesame notation as above. In parti ular, T denotes a maximal F -stable torus of G ontained in an F -stable Borel subgroup B of G . We onsider the group B = U ⋊ T , where U = B / U (see Ÿ3.1 for the notation). Note that B is F -stable and B F = U F ⋊ T F . Moreover, the set Irr p ′ ( B F ) is in bije tion with the set Irr( B F ) ;see [2, Lemma 4℄. As in the proof of Proposition 4.1, we onsider e G a onne ted2 OLIVIER BRUNATredu tive group with onne ted enter ontaining G and su h that they have thesame derived subgroup. We denote by e T the unique F -stable maximal torus of e G ontaining T . We denote by Ω and e Ω the sets of T F -orbits and e T F -orbits on Irr( U F ) , respe tively. As in Equation (1), we denote by O the set of F -orbits on ∆ . Moreover, for every ω ∈ O , we (cid:28)x a non-trivial hara ter φ ω of X Fω (for thenotation, see Equation (1)). For J ⊆ O , we set φ J = 1 J ⊗ Y ω ∈ J φ ω , where J = Q ω / ∈ J X Fω . Then by [4, 2.9, 8.1.2℄, the set { φ J | J ⊆ O} is a set ofrepresentatives of e Ω .Proposition 5.1. We keep the notation as above. For every J ⊆ O , we denoteby Ω J (resp. Ω J, ) the element of e Ω (resp. Ω ) ontaining φ J . Moreover, we set n J = | Ω J | / | Ω J, | . Then | Irr p ′ ( B F ) | = X J ⊆O n J | C T F ( φ J ) | . Proof. First remark that n J is an integer. Indeed, sin e T F ⊆ e T F , we dedu e that Ω J is a disjoint union of T F -orbits. In parti ular, there is k su h that(8) Ω J = k G i =1 Ω J,i , where Ω J,i ∈ Ω (the notation is hosen su h that φ J = φ J, ∈ Ω J, ). Moreover,for every ≤ i ≤ k , | Ω J,i | = | Ω J, | be ause Ω J,i and Ω J, are onjugate by anelement of e T F . Then | Ω J, | divides | Ω J | and n J = k . For ≤ i ≤ n J , (cid:28)x t i ∈ e T F su h that φ J,i = t i φ J, ∈ Ω J,i and denote by C T F ( φ J,i ) the stabilizer of φ J,i in T F .Then the inertial subgroup I J,i of φ J,i in B F is U F ⋊ C T F ( φ J,i ) . Moreover, sin e U F is abelian, we an extend φ J,i to I J,i setting e φ J,i ( ut ) = φ J,i ( u ) for u ∈ U F and t ∈ C T F ( φ J,i ) . Then, by Cli(cid:27)ord theory, the hara ters of B F su h that φ J,i is a onstituent of their restri tions to U F are exa tly the irredu ible hara ters Ind B F I J,i ( e φ J,i ⊗ ψ ) with ψ ∈ Irr(C T F ( φ J,i )) . There are | C T F ( φ J,i ) | su h hara ters.Hen e, we dedu e | Irr( B F ) | = X J ⊆O n J X i =1 | C T F ( φ J,i ) | . Furthermore, we have | C T F ( φ J,i ) | = | t i C T F ( φ J, ) | . The result follows. (cid:3) For J ⊆ O , we de(cid:28)ne(9) m ( J ) = G ω ∈ J ω. Note that m ( J ) ⊆ ∆ and F ( m ( J )) = m ( J ) .Lemma 5.2. We keep the notation as above. For J ⊆ O , we asso iate to φ J the F -stable standard Levi subgroup L m ( J ) where m ( J ) is the subset of ∆ de(cid:28)ned inRelation (9). Then we have n J = | H ( F, Z ( L m ( J ) )) | and | C T F ( φ J ) | = n J | Z( G ) ◦ F | Y ω ∈O\ J ( q | ω | − , OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 13where n J is the integer de(cid:28)ned in Proposition 5.1.Proof. Re all that Ω J (resp. Ω J, ) is the e T F -orbit (resp. T F -orbit) of φ J . ByEquation (8), one has | Ω J | = n J | Ω J, | . Moreover, as explained in the proof of [4, 8.1.2℄, we have | Ω J | = Q ω ∈ J ( q ω − . Itthen follows that | C T F ( φ J ) | = n J | T F | Q ω ∈ J ( q | ω | − . Furthermore, by [4, 2.9℄, we have | T F | = | Z( G ) ◦ F | Q ω ∈O ( q | ω | − . Hen e wededu e | C T F ( φ J ) | = n J | Z( G ) ◦ F | Y ω ∈O\ J ( q | ω | − . Let L m ( J ) be the standard F -stable Levi subgroup of G orresponding to the subsetof simple roots m ( J ) . Denote by B m ( J ) ⊆ B an F -stable Borel subgroup of L m ( J ) and by U m ( J ) the unipotent radi al of B m ( J ) . The set m ( J ) is the set of simpleroots of L m ( J ) asso iated to B m ( J ) . In parti ular, Equation (1) applied to the onne ted redu tive group L m ( J ) gives U F ,m ( J ) = Y ω ∈ J X Fω . We denote by φ ′ J the restri tion of φ J to U F ,m ( J ) . Then φ ′ J ∈ Irr( U F ,m ( J ) ) andthe map Irr( U F ,m ( J ) ) → Irr( U F ) , ϑ J ⊗ ϑ , is T F -equivariant. Moreover, notethat φ ′ J is a regular hara ter of U F ,m ( J ) . Hen e, using [6, 14.28℄, we dedu e that n J = | H ( F, Z ( L m ( J ) )) | as required. (cid:3) Corollary 5.3. With the above notation, one has | Irr p ′ ( B F ) | = | Z( G ) ◦ F | X J ⊆O |Z ( L m ( J ) ) F | Y ω ∈O\ J ( q | ω | − , where m ( J ) is the subset of ∆ asso iated to J as in Equation (9).Proof. It is a dire t onsequen e of Proposition 5.1 and Lemma 5.2 and the equality | H ( F, Z ( L m ( J ) )) | = |Z ( L m ( J ) ) F | . (cid:3) In the following, we will need the following result.Lemma 5.4. Fix I ∈ O and put I = O\ I . Then we have X I ⊆ J ⊆O Y ω / ∈ J ( q | ω | −
1) = q | m ( I ) | , where m is the map de(cid:28)ned in Equation (9).Proof. First remark that X I ⊆ J ⊆O Y ω / ∈ J ( q | ω | −
1) = X J ⊆ I Y ω ∈ J ( q | ω | − . Furthermore, for every (cid:28)nite set A and f : A → R , one has(10) Y a ∈ A ( f ( a ) + 1) = X J ⊆ A Y a ∈ J f ( a ) . A = I and f : I → R , ω q | ω | − and we dedu e X J ⊆ I Y ω ∈ J ( q | ω | −
1) = Y ω ∈ I q | ω | = q P ω ∈ I | ω | Moreover, Equation (9) implies | m ( I ) | = P ω ∈ I | ω | and the result follows. (cid:3) Remark 5.5. If the enter of G is onne ted, then the enter of every Levi subgroup L of G is onne ted (be ause the map h L is surje tive). In parti ular, Corollary 5.3and Lemma 5.4 (applied with I = ∅ ) give | Irr p ′ ( B F ) | = | Z( G ) F | q | m ( O ) | = | Z( G ) F | q | ∆ | , whi h is a well-known result; see [2, Remark 1℄.5.2. The ase of quasi-simple groups. In this se tion, we suppose that G isa quasi-simple algebrai group. We keep the notation as above. Re all that for I ⊆ ∆ , the map h L I : Z ( G ) → Z ( L I ) denotes the surje tive map indu ed bythe in lusion Z( G ) ⊆ Z( L I ) . Moreover, re all that for every subgroup K of Z ( G ) ,there is I ⊆ ∆ su h that K = ker( h L I ) (we use here the fa t that G is quasi-simple;see [1, 2.9℄). Then we denote by I K a subset of ∆ su h that K = ker( h L IK ) and I K is minimal (for the in lusion). In parti ular, L I K ∈ L min ( K ) .Proposition 5.6. With the above notation, if p is good for G , we have | Irr p ′ ( B F ) | = | Z( G ) ◦ F | X K ≤Z ( G ) F |Z ( G ) F | | K | q | I K | − X K ′ ∈ max( K ) q | I K ′ | , where max( K ) denotes the set of maximal proper subgroups of K .Proof. For a subgroup K of Z ( G ) , we de(cid:28)ne A K = { I ∈ ∆ | I K ⊆ I } and B K = { J ∈ O | ker( h L m ( J ) ) = K } . where m ( J ) is the subset of ∆ asso iated to J de(cid:28)ned in Equation (9). ThenCorollary 5.3 implies | Irr p ′ ( B F ) | = | Z( G ) ◦ F | X K ≤Z ( G ) F X J ∈ B K |Z ( L m ( J ) ) F | Y ω / ∈ J ( q | ω | − | Z( G ) ◦ F | X K ≤Z ( G ) F |Z ( L m ( J ) ) F | X J ∈ B K Y ω / ∈ J ( q | ω | − , be ause for J ∈ B K , the numbers |Z ( L m ( J ) ) F | are onstant. Furthermore, one has B K = { J ∈ O | ker( h L m ( J ) ) ⊆ K }\{ J ∈ O | ker( h L m ( J ) ) ( K } . Note that L I K is F -stable. Then I K is a union of some F -orbits lying in a subset e I K of O , su h that m ( e I K ) = I K . Sin e L I K ∈ L min ( K ) , it follows { J ∈ O | ker( h L m ( J ) ) ⊆ K } = { J ∈ O | e I K ⊆ J } . Moreover, one has { J ∈ O | ker( h L m ( J ) ) ( K } = G K ′ ∈ max( K ) { J ∈ O | ker( h L m ( J ) ) ⊆ K ′ } = G K ′ ∈ max( K ) { J ∈ O | e I K ′ ⊆ J } . OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 15Thus, if we put C K = { J ∈ O | e I K ⊆ J } , then it follows X J ∈ B K Y ω / ∈ J ( q | ω | −
1) = X J ∈ C K Y w / ∈ J ( q | w | − − X K ′ ∈ max( K ) X J ∈ C K ′ Y ω / ∈ J ( q | ω | − q | ∆ \ m ( e I K ) | − X K ′ ∈ max K q | ∆ \ m ( e I K ′ ) | . The last equality omes from Lemma 5.4. Moreover, we have h L m ( J ) ( Z ( G ) F ) = Z ( L m ( J ) ) F implying |Z ( L m ( J ) ) F | = |Z ( G ) F /K | . The result follows. (cid:3) Proposition 5.7. Let G be a onne ted redu tive group de(cid:28)ned over F q with or-responding Frobenius F . Suppose p is a good prime for G and Z ( G ) F has primeorder ℓ . Put r = | I | for L I in L min ( { } ) . Then we have | Irr p ′ ( B F ) | = | Z( G ) ◦ F | (cid:0) q l + ( ℓ − q l − r (cid:1) , where l is the semisimple rank of G .Proof. First remark that we do not suppose that G is quasi-simple. Indeed, the set L min ( { } ) is non-empty. If we denote by L I a standard Levi lying in L min ( { } ) ,then we have ker( h L I ) = { } . Hen e I = I { } (see the beginning of Ÿ5.2 for thenotation). Moreover, we always have I Z ( G ) F = ∅ . We an then apply the proof ofProposition 5.6. We obtain | Irr p ′ ( B F ) | = | Z( G ) ◦ F | (cid:16) |Z ( G ) F | q | ∆ |− r + q | ∆ | − q | ∆ |− r (cid:17) = | Z( G ) ◦ F | (cid:16) q | ∆ | + ( |Z ( G ) F | − q | ∆ |− r (cid:17) . Sin e | ∆ | is the semisimple rank of G , the result follows. (cid:3) Remark 5.8. For a group G as in Proposition 5.7, if ζ denotes a non-trivial hara ter of H ( F, Z ( G )) and L ζ its asso iated uspidal Levi of G , then L ζ is F -stable and ker( h L ζ ) is trivial. Then L ζ ∈ L min ( { } ) . In parti ular, the number r of Proposition 5.7 is equal to the semisimple rank of L ζ , implying | Irr p ′ ( B F ) | = | Z( G ) ◦ F | (cid:16) q l + ( ℓ − q l − (ss-rk( L ζ )) (cid:17) . Comparing with Proposition 4.2, we dedu e | Irr p ′ ( G F ) | = | Irr p ′ ( B F ) | . Hen e, this then proves that, if p is a good prime for G and H ( F, Z ( G ) has primeorder, then the M Kay onje ture holds for G in de(cid:28)ning hara teristi .Proposition 5.9. If G is a simple and simply- onne ted algebrai group of type D n , then | Irr p ′ ( B F ) | = q n + 3 q n − + 6 q n + 4 q n − . If G is a simple and simply- onne ted algebrai group of type D n +1 with H ( F, Z ( G )) of order , then | Irr p ′ ( B F ) | = q n +1 + 3 q n − + 12 q n − . G is simple and simply- onne ted group of type D n , then Z ( G ) F is a(cid:28)nite group of order with exponent . Denote by c , c and c its subgroups oforder . Moreover, using Table 3, we dedu e K | I K |{ } n − c nc nc n − Z ( G ) F n K | I K |{ } n − Z n − Z ( G ) F n + 1 Type D n Type D n +1 The result then follows from Proposition 5.6. (cid:3)
6. Restri tion of semisimple hara ters to the enterIn this se tion, we keep the notation as above. To simplify the notation, we set G = G F , Z = Z( G ) F and U = U F . For z ∈ H ( F, Z ( G )) and ν ∈ Irr( Z ) , we put Γ z,ν = Ind GZU ( ν ⊗ φ z ) , where φ z is the regular hara ter of U orresponding to z . Note that by Cli(cid:27)ordtheory, one has Ind
ZUU ( φ z ) = X ν ∈ Irr( Z ) ν ⊗ φ z . We then dedu e that Γ z = X ν ∈ Irr( Z ) Γ z,ν . Lemma 6.1. Denote by E z and E z,ν the set of onstituents of Γ z and Γ z,ν , respe -tively. Then E z,ν = { χ ∈ E z | h Res GZ ( χ ) , ν i Z = 0 } . Proof. We denote by R a set of representatives of the double osets ZU \ G/Z .Therefore, for ϕ ∈ Irr( ZU ) , Ma key's theorem implies Res GZ (Ind GZU ( ϕ )) = X r ∈ R Ind Z r ( ZU ) ∩ Z (cid:0) Res r ( ZU ) ∩ Z ( r ϕ ) (cid:1) = X r ∈ R Res Z ( r ϕ ) . (11)Fix now ν, ν ′ ∈ Irr( Z ) . Then Equation (11) applied with ϕ = ν ⊗ φ z implies h Γ z,ν , Ind GZ ( ν ′ ) i G = h Res GZ (Γ z,ν ) , ν ′ i Z = h X r ∈ R ν, ν ′ i Z = | R |h ν, ν ′ i Z = | R | δ ν,ν ′ . The result then follows. (cid:3)
OUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 17Remark 6.2. Note that if we denote by F z and F z,ν the set of onstituents of D G (Γ z ) and D G (Γ z,ν ) , respe tively, then F z,ν = { χ ∈ E z | h Res GZ ( χ, ν i Z = 0 } .Indeed, by [6, 12.8℄ and [10, 2.2℄, D G (Ind GZ ( ν )) = Ind GZ ( ν ) . In parti ular, D G indu es a bije tion between E z,ν and F z,ν .Lemma 6.3. With the above notation, for z, z ′ ∈ H ( F, Z ( G )) and ν, ν ′ ∈ Irr( Z ) ,one has h Γ z,ν , Γ z ′ ,ν i G = h Γ z,ν ′ , Γ z ′ ,ν ′ i G . Proof. We have to show that the s alar produ t h Γ z,ν , Γ z ′ ,ν i G does not depend on ν . First remark that it follows from Lemma 6.1 that h Γ z,ν , Γ z ′ ,ν i G = h Γ z,ν , Γ z ′ i G . Denote by R a set of representatives of the double osets U Z \ G/U . Then Ma key'stheorem implies h Γ z,ν , Γ z ′ i G = h Res GU (cid:16) Ind
GZU ( ν ⊗ φ z ) (cid:17) , φ z ′ i U = X r ∈ R h Ind U r ( UZ ) ∩ U (cid:0) Res r ( UZ ) ∩ U ( r ( ν ⊗ φ z ) (cid:1) , φ z ′ i U = X r ∈ R h Ind U r U ∩ U ( r φ z ) , φ z ′ i U . Note that the s alar produ t in the last equality does not depend on ν . This provesthe laim. (cid:3) Corollary 6.4. With the above notation, for z, z ′ ∈ H ( F, Z ( G )) and ν ∈ Irr( Z ) ,we have h Γ z,ν , Γ z ′ ,ν i G = 1 | Z | h Γ z , Γ z ′ i G . Proof. We have h Γ z , Γ z ′ i G = X ν, ν ′ ∈ Irr( Z ) h Γ z,ν , Γ z ′ ,ν ′ i G . If ν = ν ′ , we have h Γ z,ν , Γ z ′ ,ν ′ i G = 0 be ause by Lemma 6.1, the onstituents of Γ z,ν (resp. of Γ z ′ ,ν ′ ) are onstituents of Ind GZ ( ν ) (resp. Ind GZ ( ν ′ ) ) and the hara ters Ind GZ ( ν ) and Ind GZ ( ν ′ ) have no onstituents in ommon. Then h Γ z , Γ z ′ i G = X ν ∈ Irr( Z ) h Γ z,ν , Γ z ′ ,ν i G . The result is now a onsequen e of Lemma 6.3 (cid:3)
Proposition 6.5. With the above notation, if p is a good prime for G and the enter of G is onne ted, then for every linear hara ter ν of Z( G F ) , one has | Irr s ( G F | ν ) | = 1 | Z( G F ) | | Irr s ( G F ) | . Proof. Sin e the enter of G is onne ted, there is only one Gelfand-Graev hara ter Γ . Moreover, Remark 6.2 implies | Irr s ( G F | ν ) | = h Γ ,ν , Γ ,ν i G F . | Irr s ( G F ) | = h Γ , Γ i G F . The result follows from Lemma 6.4 (cid:3) Proposition 6.6. With the above notation, if p is a good prime for G and thegroup H ( F, Z ( G )) has prime order ℓ , then for every linear hara ter ν of Z( G F ) ,one has | Irr s ( G F | ν ) | = 1 | Z( G F ) | | Irr s ( G F ) | . Proof. We onsider e G a onne ted redu tive group with onne ted enter as in theproof of Proposition 4.1. Fix s a semisimple element of G ∗ F ∗ and e s a semisimpleelement of e G ∗ F ∗ su h that i ∗ ( e s ) = s . In the proof Proposition 4.1, we have seen that Res e G F G F ( ρ e s ) has | A G ∗ ( s ) F ∗ | onstituents. In fa t, the onstituents of Res e G F G F ( ρ e s ) arein bije tion with Irr( A G ∗ ( s ) F ∗ ) . We denote by ρ s,ϑ the onstituent orrespondingto ϑ ∈ Irr( A G ∗ ( s ) F ∗ ) . Moreover, this bije tion ould be hosen su h that thereis a surje tive morphism ω s : H ( F, Z ( G )) → Irr( A G ∗ ( s ) F ∗ ) satisfying ρ s,ϑ (for ϑ ∈ Irr( A G ∗ ( s ) F ∗ ) ) is a onstituent of D G (Γ z ) for z ∈ H ( F, Z ( G )) if and onlyif ω s ( z ) = ϑ . In parti ular, the hara ter ρ s,ϑ lies in | H ( F, Z ( G )) | / | A G ∗ ( s ) F ∗ | di(cid:27)erent duals of Gelfand-Graev hara ters of G F . Furthermore, H ( F, Z ( G )) hasprime order ℓ . It follows that a semisimple hara ter of G F is either a onstituentof only one D (Γ z ) or of all. We keep the notation of Remark 6.2 and put, for ν ∈ Irr(Z( G F )) F ν = \ z ∈ H ( F, Z ( G )) F z,ν . The above dis ussion implies that if z = z ′ , then(12) F z,ν ∩ F z ′ ,ν = F ν . Moreover, one has
Irr p ′ ( G F | ν ) = [ z ∈ H ( F, Z ( G )) F z,ν . Therefore, | Irr p ′ ( G F | ν ) | = | [ z ∈ H ( F, Z ( G )) F z,ν | = ℓ X k =1 ( − k +1 X I ⊆ H ( F, Z ( G )) , | I | = k | \ z ∈ I F z,ν | = X z | F z,ν | + | F ν | ℓ X k =2 ( − k +1 X I ⊆ H ( F, Z ( G )) , | I | = k X z | F z,ν | + | F ν | ℓ X k =2 ( − k +1 (cid:18) ℓk (cid:19) = X z | F z,ν | + | F ν | (1 − ℓ ) . Note that, sin e the hara ters Γ z,ν are multipli ity free, one has | F z,ν | = h Γ z,ν , Γ z,ν i G F and | F ν | = h Γ z,ν , Γ z ′ ,ν i G F where z and z ′ are two (cid:28)xed distin t elements ofOUNTING p ′ -CHARACTERS IN FINITE REDUCTIVE GROUPS 19 H ( F, Z ( G )) . Fix two su h elements z and z ′ . Then Corollary 6.4 implies | F z,ν | = 1 | Z( G F ) | h Γ z , Γ z i G F and | F ν | = 1 | Z( G F ) | h Γ z , Γ z ′ i G F . Denote by L the uspidal Levi subgroup asso iated to every non-trivial hara terof H ( F, Z ( G )) and by l the semisimple rank of G . Proposition 3.2 gives h Γ z , Γ z i = | Z ◦ | (cid:16) q l − ( ℓ − q l − (ss-rk( L )) (cid:17) and h Γ z , Γ z ′ i = | Z ◦ | (cid:16) q l − q l − (ss-rk( L )) (cid:17) , with Z ◦ = Z( G ) ◦ F . It follows | Irr s ( G F | ν ) | = 1 | Z( G F ) | | Z ◦ | (cid:16) q l − ( ℓ − q l − (ss-rk( L )) (cid:17) = 1 | Z( G F ) | | Irr s ( G F ) | . The last equality omes from Proposition 4.2. (cid:3)
Remark 6.7. As we remark in [3℄, the number | Irr p ′ ( B F | ν ) | does not depend on ν for all ν ∈ Z( G F ) and | Irr p ′ ( B F | ν ) | = 1 | Z( G F ) | | Irr p ′ ( B F ) | . Suppose now that p is a good prime for G and H ( F, Z ( G )) has prime order. Then,thanks to Remark 5.8 and Proposition 6.6, we dedu e | Irr p ′ ( B F | ν ) | = | Irr p ′ ( G F | ν ) | , for every ν ∈ Irr(Z( G F )) . This proves Theorem 1.1.A knowledgements. Part of this work was done during the programme (cid:16)Alge-brai Lie Theory(cid:17) in Cambridge. I gratefully a knowledge (cid:28)nan ial support by theIsaa Newton Institute.I wish to sin erely thank Jean Mi hel for pointing me in this dire tion and forvaluable and larifying dis ussions on the paper [5℄. I also wish to thank GunterMalle for his reading of the manus ript.Referen es[1℄ C. Bonnafé. Éléments unipotents réguliers des sous-groupes de Levi. Canad. J. Math.,56(2):246(cid:21)276, 2004.[2℄ O. Brunat. On the indu tive M Kay ondition in the de(cid:28)ning hara teristi . to appear inMath. Z.[3℄ O. Brunat and F. Himstedt. On equivariant bije tions of hara ters in (cid:28)nite redu tive groups.In preparation.[4℄ R.W. Carter. Finite groups of Lie type. Pure and Applied Mathemati s (New York). JohnWiley & Sons In ., New York, 1985. Conjuga y lasses and omplex hara ters, A Wiley-Inters ien e Publi ation.[5℄ F. Digne, G. I. Lehrer, and J. Mi hel. On Gel ′′