aa r X i v : . [ m a t h . G R ] F e b FORMULAE FOR TWO-VARIABLE GREEN FUNCTIONS
FRANC¸ OIS DIGNE AND JEAN MICHEL
Abstract.
Based on results of Digne-Michel-Lehrer (2003) we give two formu-lae for two-variable Green functions attached to Lusztig induction in a finitereductive group. We present applications to explicit computation of theseGreen functions, to conjectures of Malle and Rotilio, and to scalar productsbetween Lusztig inductions of Gelfand-Graev characters.
Let G be a connected reductive group with Frobenius root F ; that is, somepower F δ is a Frobenius endomorphism attached to an F q δ -structure on G , where q δ is a power of a prime p . Let L be an F -stable Levi subgroup of a (non-necessarily F -stable) parabolic subgroup P of G . Let U be the unipotent radical of P and let X U = { g U ∈ G / U | g − F g ∈ U · F U } be the variety used to define the Lusztiginduction and restriction functors R GL and ∗ R GL . For u ∈ G F , v ∈ L F unipotentelements, the two-variable Green function is defined as Q GL ( u, v ) = Trace(( u, v ) | X i ( − i H ic ( X U )) . In this paper, using the results of [5], we give two different formulae for two-variable Green functions, and some consequences of these, including proving someconjectures of [13].The two-variables Green functions occur in the character formulae for Lusztiginduction and restriction. In particular, for unipotent elements these formulae read
Lemma 1. • If u is a unipotent element of G F , and ψ a class function on L F , we have R GL ( ψ )( u ) = | L F |h ψ, Q GL ( u, − ) i L F . • If v is a unipotent element of L F , and χ a class function on G F , we have ∗ R GL ( χ )( v ) = | L F |h χ, Q GL ( − , v − ) i G F . Two formulae for two-variable Green functions
For an element u in a group G we denote by u G the G -conjugacy class of u . Proposition 2. If p is almost good for G and u is regular, Q GL ( u, − ) vanishesoutside a unique regular unipotent class of L F . For v in that class, we have Q GL ( u, v ) = | v L F | − .Proof (Rotilio). Let γ G u be the normalized characteristic function of the G F -conjugacyclass of u ; that is, the function equal to 0 outside the class of u and to | C G ( u ) F | on that class. For v ′ ∈ L F unipotent, Lemma 1 gives ∗ R GL ( γ G u )( v ′ ) = | L F |h γ G u , Q GL ( − , v ′− ) i G F = | L F | Q GL ( u, v ′− ) . Now, by [1, Theorem 15.2], there exists v ∈ L F such that the left-hand side isequal to γ L v ( v ′ ). In [1, Theorem 15.2] q is assumed large enough so that the Mackeyformula holds and the results of [10] hold. The Mackey formula is known to hold if q > p almost good without condition on q by [15]. The proposition follows. (cid:3) As in [5], we consider irreducible G -equivariant local systems on unipotentclasses. These local systems are partitioned into “blocks” parametrised by cus-pidal pairs formed by a Levi subgroup and a cuspidal local system supported on aunipotent class of that Levi subgroup. Let us call Q L , I wF the function defined in [5,3.1(iii)] relative to an F -stable block I of unipotently supported local systems on L and to wF ∈ W L ( L I ) F where ( L I , ι I ) is the cuspidal datum of I (for a Levi L of a reductive group G , we set W G ( L ) = N G ( L ) / L ). Proposition 3.
Assume p almost good for G . For u a unipotent element of G F and v a unipotent element of L F , we have Q GL ( u, v ) = | L F | − X I X w ∈ W L ( L I ) | Z ( L I ) wF || W L ( L I ) | Q G , I G wF ( u ) Q L , I wF ( v ) , where I runs over the F -stable blocks of L and where I G is the block of G withsame cuspidal data as I . The part of the above sum for I the principal block is the same formula as [7,Corollaire 4.4]. Proof.
Lemma 1 applied with ψ = Q L , I wF gives, if we write R G F L F instead of R GL tokeep track of the Frobenius, h Q L , I wF , Q GL ( u, − ) i L F = | L F | − R G F L F ( Q L , I wF )( u )Now we have by [5, Proposition 3.2] Q L , I wF = R L F L wF I ˜ X ι I ,wF where ˜ X ι I ,wF is q c ι I times the characteristic function of ( ι I , wF ), a class function on L wF I . Here, as in[5, above Remark 2.1], for an irreducible G -equivariant local system ι , we denoteby C ι the unipotent G -conjugacy class which is the support of ι , and if ( L I , ι I ) isthe cuspidal datum of ι we set c ι = (codim C ι − dim Z ( L I )). In [5, Proposition3.2] it is assumed that q is large but by [15], p almost good is enough.By the transitivity of Lusztig induction we get h Q L , I wF , Q GL ( u, − ) i L F = | L F | − R G F L wF I ( ˜ X ι I ,wF )( u ) = | L F | − Q G , I G wF ( u ) , where I G is the block of G with same cuspidal data as I . Using the orthogonalityof the Green functions Q L , I wF , see [5, Corollary 3.5], and the fact they form a basisof unipotently supported class functions on L F , indexed by the W L ( L I )-conjugacyclasses of W L ( L I ) F , we get the proposition. (cid:3) Proposition 3 gives a convenient formula to compute automatically two-variableGreen functions. Table 1 gives an example, computed with the package
Chevie (see [14]).We denote by Y ι the characteristic function of the F -stable local system ι , andby A ( u ) the group of components of the centralizer of a unipotent element u . ORMULAE FOR TWO-VARIABLE GREEN FUNCTIONS 3
Proposition 4.
Assume p almost good for G and let R ι,γ be the polynomials whichappear in [5, Lemma 6.9] . Then Q GL ( u, v ) = | v L F | − | A ( v ) | − X I X ι ∈I G F ,γ ∈I F Y ι ( u ) Y γ ( v ) R ι,γ q c ι − c γ , where c ι = (codim C ι − dim Z ( L I )) .Proof. For a block I of L and ι ∈ I F , let ˜ Q ι be the function of [5, (4.1)]. Then by[5, (4.4)] applied respectively in G and L we have Q G , I G wF ( u ) = X ι ∈I G F ˜ Q ι ( wF ) ˜ Y ι ( u ) and Q L , I wF ( v ) = X κ ∈I F ˜ Q κ ( wF ) ˜ Y κ ( v )where ˜ Y ι = q c ι Y ι . Thus, using the notation Z L I as in [5, 3.3] to denote the function wF
7→ | Z ( L I ) wF | on W G ( L I ) F , the term relative to a block I in the formula ofProposition 3 can be written | L F | − hZ L I X κ ∈I F ˜ Q κ ˜ Y κ ( v ) , X ι ∈I G F ˜ Y ι ( u ) Res W G ( L I ) FW L ( L I ) F ˜ Q ι i W L ( L I ) F . Applying now [5, Lemma 6.9] this is equal to | L F | − hZ L I X κ ∈I F ˜ Q κ ˜ Y κ ( v ) , X ι ∈I G F ,γ ∈I F ˜ Y ι ( u ) R ι,γ ˜ Q γ i W L ( L I ) F , We use now [5, Corollary 5.2] which says that, h ˜ Q γ , Z L I ˜ Q κ i W L ( L I ) F = 0 unless C γ = C κ and in this last case is equal to | A ( v ) | − X a ∈ A ( v ) | C L ( v a ) F | q − c γ Y γ ( v a ) Y κ ( v a )Thus the previous sum becomes | L F | − X ι ∈I G F ,γ ∈I F ˜ Y ι ( u ) R ι,γ | A ( v ) | − X a ∈ A ( v ) | C L ( v a ) F | q − c γ Y γ ( v a ) X κ ∈I F Y κ ( v a ) ˜ Y κ ( v ) . But by [5, (4.5)] we have P κ Y κ ( v a ) ˜ Y κ ( v ) = ( q c κ | A ( v ) F | if v a = v κ runsover all local systems. Thus, summing over all the blocks, we get the formula inthe statement. (cid:3) Corollary 5.
Under the assumption of Proposition 4, for any unipotent elements u ∈ G F and v ∈ L F we have: (i) Q GL ( u, v ) vanishes unless v G ⊆ u G ⊆ Ind GL ( v L ) , where Ind GL ( v L ) is theinduced class in the sense of [12] . (ii) | v L F || A ( v ) | Q GL ( u, v ) is an integer and is a polynomial in q with integralcoefficients.Proof. For (i), we use [5, Lemma 6.9(i)] which states that R ι,γ = 0 unless C γ ⊆ C ι ⊆ Ind GL ( C γ ). Since ˜ Y κ ( v ) vanishes unless C κ ∋ v , the only non-zero terms inthe formula of proposition 4 have C γ ∋ v , whence the result since ˜ Y ι ( u ) vanishesunless C ι ∋ u .For (ii), we start with Lemma 6. q c ι − c γ R ι,γ is a polynomial in q with integral coefficients. FRANC¸ OIS DIGNE AND JEAN MICHEL
Proof.
The defining equation of the matrix ˜ R = { q c ι − c γ R ι,γ } ι,γ reads (see the proofof [5, Lemma 6.9(i)]): ˜ R = P G C G IC − L P − L where C G is the diagonal matrix with diagonal coefficients q c ι for ι ∈ I G , and C L is the similar matrix for L and I , where P G is the matrix with coefficients { P ι,ι ′ } ι,ι ′ ∈I G where these polynomials are those defined in [11, 6.5], and P L is thesimilar matrix for L and I , and finally I is the matrix with coefficients I ι,γ = h Ind W G ( L I ) FW L ( L I ) F ˜ ϕ γ , ˜ ϕ ι i W G ( L I ) F where ˜ ϕ γ is the character of W L ( L I ) F which corresponds by the generalised Springercorrespondence to γ (and similarly for ˜ ϕ ι ). Since P L and P G are unitriangular ma-trices with coefficients integral polynomials in q , thus P − L also, it suffices to provethat C G IC − L has coefficients polynomial in q , or equivalently thatif h Ind W G ( L I ) FW L ( L I ) F ˜ ϕ γ , ˜ ϕ ι i W G ( L I ) F = 0, then c ι − c γ ≥ . We now use [5, Proposition 2.3(ii)] which says that the non-vanishing above implies C γ ⊆ C ι ⊆ Ind GL ( C γ ). We now use that, according to the definitions, c ι − c γ =dim B G u − dim B L v where B G u is the variety of Borel subgroups of G containing anelement u of the support of ι , and where B L v is the variety of Borel subgroups of L containing an element v of the support of γ . Now the lemma follows from thefact that by [12, Theorem 1.3 (b)] we have dim B G u = dim B L v if u is an element ofInd GL ( C γ ), and that dim B G u is greater for u ∈ Ind GL ( C γ ) − Ind GL ( C γ ). (cid:3) Now (ii) results from the lemma: since the ˜ Y have values algebraic integers, byProposition 4 the expression in (ii) is a polynomial in q with coefficients algebraicintegers. But, since | L F | Q GL ( u, v ) is a Lefschetz number (see for example [8, 8.1.3]),the expression in (ii) is a rational number; since this is true for an infinite number ofintegral values of q the expression in (ii) is a polynomial with integral coefficients. (cid:3) Scalar products of induced Gelfand-Graev characters
The pretext for this section is as follows: in [2, Remark 3.10] is pointed theproblem of computing h R GL Γ ι , R GL Γ ι i G F when ( G , F ) is simply connected of type E , when L is of type A × A , and when ι corresponds to a faithful characterof Z ( L ) /Z ( L ), and checking that the value is the same as given by the Mackeyformula. We show now various ways to do this computation, assuming p good (thusnot solving the problem of loc. cit. where we need q = 2).From now on, we assume p good for G . Let Z = Z ( G ), and let Γ z be the Gelfand-Graev character parameterized by z ∈ H ( F, Z ), see for instance [3, Definition 2.7].Let u z be a representative of the regular unipotent class parametrized by z . As in[11, 7.5 (a)] for ι an F -stable local system on the regular unipotent class we defineΓ ι = c P z ∈ H ( F,Z ) Y ι ( u z )Γ z where c = | Z/Z || H ( F,Z ) | .Note that the cardinality | C G ( u z ) F | is independent of z ; actually it is equal to | Z ( G ) F | q rkss G (see [1, 15.5]). Thus we will denote this cardinality | C G F ( u ) | where u ∈ G F is any regular unipotent element. There exists a character ζ of H ( F, Z )and a root of unity b ι (see [4, above 1.5]) such that Y ι ( u z ) = b ι ζ ( z ). With thesenotations, we have ORMULAE FOR TWO-VARIABLE GREEN FUNCTIONS 5
Proposition 7.
We have Γ ι = η G σ − ζ c | C G F ( u ) | D Y ι where η G and σ ζ are definedas in [4, 2.5] ,Proof. This proposition could be deduced from [6, Theorem 2.8] using [4, Theorem2.7]. We give here a more elementary proof.With the notations of [3, (3.5’)] we have D Γ z = P z ′ ∈ H ( F,Z ) c z,z ′ γ z ′ . By [4,lemma 2.3] we have c z,z ′ = c zz ′− , and P z ∈ H ( F,Z ) ζ ( z ) c z, = η G σ − ζ . It followsthat c − b − ι D Γ ι = X z ∈ H ( F,Z ) ζ ( z ) D Γ z = X z,z ′ ∈ H ( F,Z ) ζ ( z ) c z,z ′ γ z ′ = X z,z ′ ∈ H ( F,Z ) c zz ′− , ζ ( z ) γ z ′ = X z ′ ∈ H ( F,Z ) ζ ( z ′ ) γ z ′ X z ′′ ∈ H ( F,Z ) c z ′′ , ζ ( z ′′ )= η G σ − ζ X z ′ ∈ H ( F,Z ) ζ ( z ′ ) γ z ′ = η G σ − ζ b − ι | C G F ( u ) |Y ι (cid:3) Proposition 8. If ι is a local system supported on the regular unipotent class of L and I denotes its block, we have h R GL Γ L ι , R GL Γ L ι i G F = | Z ( L ) Z ( L ) | X w ∈ W L ( L I ) | Z ( L I ) wF || W G ( L I ) || W L ( L I ) | | ( wF ) W G ( L I ) ∩ W L ( L I ) || ( wF ) W G ( L I ) | . Note that in a given block I there is at most one local system supported by theregular unipotent class (see [4, Corollary 1.10]). Proof.
When ι is supported by the regular unipotent class we have ˜ Q ι = 1, see thebegining of section 7, bottom of page 130 in [5]. Using this in the last formula ofthe proof of [5, Proposition 6.1], we get that Γ L ι is up to a root of unity equal to | A ( C ι )) || W L ( L I ) | − P w ∈ W L ( L I ) | Z ( L I ) wF | Q L , I wF . Since R GL Q L , I wF = Q G , I G wF , we get h R GL Γ ι , R GL Γ ι i G F = | A ( C ι )) | | W L ( L I ) | − X w,w ′ ∈ W L ( L I ) | Z ( L I ) wF || Z ( L I ) w ′ F |h Q G , I G wF , Q G , I G w ′ F i G F . By [5, 3.5] the last scalar product is zero unless wF and w ′ F are conjugate in W G ( L I ), and is equal to | C W G ( L I ) ( wF ) | / | Z ( L I ) wF | otherwise. We get h R GL Γ ι , R GL Γ ι i G F = | A ( C ι )) | | W L ( L I ) | − X w ∈ W L ( L I ) | Z ( L I ) wF || C W G ( L I ) ( wF ) || ( wF ) W G ( L I ) ∩ W L ( L I ) | , which gives the formula of the proposition since A ( C ι ) = Z ( L ) /Z ( L ) . (cid:3) Corollary 9.
Let ι and ι ′ be local systems supported on the regular unipotent classof G , and I , I ′ be their respective blocks: then FRANC¸ OIS DIGNE AND JEAN MICHEL (i) h Γ G ι , Γ G ι ′ i G F = ( if ι = ι ′ , | Z ( G ) Z ( G ) | | Z ( G ) F | q dim Z ( L I ) − dim Z ( G ) if ι = ι ′ . (ii) hY ι , Y ι ′ i G F = ( if ι = ι ′ ,q − rkss G | Z ( G ) F | − if ι = ι ′ . Proof.
The functions Q G , I G wF and Q G , I ′ G w ′ F are orthogonal to each other when I G = I ′ G (see [9, V, 24.3.6] where the orthogonality is stated for the functions X ι ). Since thereis a unique ι in a given block supported on the regular unipotent class, we get theorthogonality in (i). In the case ι = ι ′ in (i), the specialization L = G in Proposition8 is h Γ G ι , Γ G ι ′ i G F = | Z ( G ) Z ( G ) | P w ∈ W G ( L I ) | Z ( L I ) wF || W G ( L I ) | . By [5, Corollary 5.2], wherewe use that ˜ Q ι = 1 when ι has regular support, we have P w ∈ W G ( L I ) | Z ( L I ) wF || W G ( L I ) | = q − c ι | C G ( u ) F | . Whence h Γ G ι , Γ G ι ′ i G F = | Z ( G ) Z ( G ) | q − rk G +dim Z ( L I ) | C G ( u ) F | . Using | C G ( u ) F | = q rkss G | Z ( G ) F | , we get (i).For (ii), we apply Proposition 7 in (i), using that D is an isometry and that σ ζ σ ζ = q rkss L I by [4, proposition 2.5]. (cid:3) A particular case of Proposition 8 is
Corollary 10. If ( L , ι ) is a cuspidal pair, that is L = L I , then h R GL Γ L ι , R GL Γ L ι i G F = | Z ( L ) Z ( L ) | | W G ( L ) || Z ( L ) F | . We remark that this coincides with the value predicted by the Mackey formula h R GL Γ L ι , R GL Γ L ι i G F = X x ∈ L F \S ( L , L ) / L F h ∗ R LL ∩ x L (Γ L ι ) , ∗ R x LL ∩ x L ( x Γ L ι ) i ( L ∩ x L ) F Indeed, since the block I which contains the local system ζ is reduced to theunique cuspidal local system ( C, ζ ) where C is the regular class of L , all terms inthe Mackey formula where L ∩ x L = L vanish. Thus the Mackey formula reducesto h R GL Γ L ι , R GL Γ L ι i G F = X x ∈ W G ( L ) h Γ L ι , x Γ L ι i L F and any x in W L ( L ) acts trivially on H ( F, Z ( L )) since, the map h L being sur-jective, any element of H ( F, Z ( L )) is represented by an element of H ( F, Z ( G ));thus all the terms in the sum are equal, and we get the same result as Corollary 10by applying Corollary 10 in the case G = L .Another method for computing h R GL D Γ i , R GL D Γ i i G F would be to use Lemma1 and the values of the two-variable Green functions. We give these values in thefollowing table in the particular case of E for the F -stable standard Levi subroupof type A × A . Note that the table shows that the values of | v L F | Q GL ( u, v ) are notin general polynomials with integral coefficients but may have denominators equalto | A ( v ) | . Table 1.
Values of | v L F | Q GL ( u, v ) for G = E ( q ) simply connected and L = A ( q )( q − , for q ≡ − ORMULAE FOR TWO-VARIABLE GREEN FUNCTIONS 7 v \ u E E ζ ) E ζ ) E ( a ) E ( a ) ( ζ ) E ( a ) ( ζ ) D E ( a )111 ,
111 0 0 0 0 0 0 0 021 ,
21 0 0 0 0 0 0 1 4 q + 13 , q + 1) / / / q Φ / q + 2 q − q/ , ( ζ ) / q + 1) / / q Φ / q − q Φ / , ( ζ ) / / q + 1) / q Φ / q − q Φ / v \ u E ( a ) ( − ζ ) E ( a ) ( ζ ) E ( a ) ( − E ( a ) ( ζ ) E ( a ) ( − ζ ) ,
111 0 0 0 0 021 ,
21 2 q + 1 4 q + 1 2 q + 1 4 q + 1 2 q + 13 , q Φ ( q − q Φ / q + 2) q ( q − q Φ / q Φ , ( ζ ) (3 q + 2) q ( q − q Φ / q Φ (7 q + 2 q − q/ q Φ , ( ζ ) q Φ (7 q + 2 q − q/ q Φ ( q − q Φ / q + 2) q v \ u A A ζ ) A ζ ) D ( a )111 ,
111 0 0 0 021 ,
21 ( − q − ( − q − ( − q − q + 13 , q Φ Φ / − q − q + 1) q Φ / − q − q + 1) q Φ / q Φ Φ / , ( ζ ) ( − q − q + 1) q Φ / q Φ Φ / − q − q + 1) q Φ / q Φ Φ / , ( ζ ) ( − q − q + 1) q Φ / − q − q + 1) q Φ / q Φ Φ / q Φ Φ / v \ u A + A D A D ( a ) (111) ,
111 0 1 0 4 q + 121 ,
21 Φ Φ q Φ Φ (3 q + q + q + 1)Φ (8 q + 2 q + 4 q − q Φ , q + 1) q Φ / q Φ Φ Φ / q Φ (4 q + 1) q Φ / , ( ζ ) (2 q + 1) q Φ / q Φ Φ Φ / q Φ (4 q + 1) q Φ / , ( ζ ) (2 q + 1) q Φ / q Φ Φ Φ / q Φ (4 q + 1) q Φ / v \ u D ( a ) (21) D ( a ) A + A A ,
111 2 q + 1 Φ ( − q − (3 q + 2 q + 1)Φ ,
21 (8 q + 6 q + 2 q + 2) q (2 q + 1) q Φ Φ ( − q − q − q + 1) q Φ (3 q − q + 2 q − q Φ Φ , q + 1) q Φ Φ / q Φ Φ / − q − q Φ / , ( ζ ) (2 q + 1) q Φ Φ / q Φ Φ / − q − q Φ / , ( ζ ) (2 q + 1) q Φ Φ / q Φ Φ / − q − q Φ / v \ u A + A A + A ζ ) A + A ζ ) A ,
111 Φ Φ Φ Φ Φ Φ Φ Φ Φ ,
21 (2 q + 2 q + 4 q + 1) q Φ (2 q + 2 q + 4 q + 1) q Φ (2 q + 2 q + 4 q + 1) q Φ q Φ Φ Φ , q Φ Φ q Φ Φ Φ , ( ζ ) q Φ Φ , ( ζ ) q Φ Φ v \ u A ζ ) A ζ ) A +2 A A + A ,
111 Φ Φ Φ Φ Φ Φ (2 q + q + q + q + 1)Φ Φ Φ Φ ,
21 3 q Φ Φ Φ q Φ Φ Φ ( q + 2 q + 1) q Φ (3 q + 2 q + 1) q Φ Φ , , ( ζ ) q Φ Φ Φ , ( ζ ) q Φ Φ Φ v \ u A A ,
111 (3 q + q + q + 1)Φ Φ q + 3 q + 4 q + 4 q + 5 q + 4 q + 2 q + 2 q + 2 q + 121 ,
21 (4 q + q + 1) q Φ Φ (2 q + 1) q Φ Φ Φ Φ Φ , , ( ζ ) , ( ζ ) FRANC¸ OIS DIGNE AND JEAN MICHEL v \ u A A ,
111 ( − q − q − q − q − q − q − q − q − (2 q + q + q + q + 1)Φ Φ Φ ,
21 ( − q − q Φ Φ q Φ Φ Φ Φ , , ( ζ ) , ( ζ ) v \ u A ,
111 (2 q + q + q + q + 2 q + q + q + q + q + 1)Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ ,
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Laboratoire Ami´enois de Math´ematique Fondamentale et Appliqu´ee, CNRSUMR 7352, Universit´e de Picardie-Jules Verne, 80039 Amiens Cedex France.
Email address : [email protected] URL : (J. Michel) Institut Math´ematique de Jussieu – Paris rive gauche, CNRS UMR 7586,Universit´e de Paris, Bˆatiment Sophie Germain, 75013, Paris France.
Email address : [email protected] URL : webusers.imj-prg.fr/ ∼∼