On the unipotent support of character sheaves
aa r X i v : . [ m a t h . R T ] J a n ON THE UNIPOTENT SUPPORT OF CHARACTER SHEAVES
MEINOLF GECK AND DAVID H´EZARD
Abstract.
Let G be a connected reductive group over F q , where q is largeenough and the center of G is connected. We are concerned with Lusztig’s the-ory of character sheaves , a geometric version of the classical character theory ofthe finite group G ( F q ). We show that under a certain technical condition, therestriction of a character sheaf to its unipotent support (as defined by Lusztig)is either zero or an irreducible local system. As an application, the general-ized Gelfand-Graev characters are shown to form a Z -basis of the Z -module ofunipotently supported virtual characters of G ( F q ) (Kawanaka’s conjecture). Dedicated to Professors Ken-ichi Shinoda and Toshiaki Shoji on their 60thbirthday Introduction
Let G be a connected reductive algebraic group over F p , an algebraic closure ofthe finite field with p elements where p is a prime. Let q be a power of p and assumethat G is defined over the finite field F q ⊆ F p , with corresponding Frobenius map F : G → G . Then it is an important problem to determine and to understand thevalues of the irreducible characters (in the sense of Frobenius) of the finite group G F . For this purpose, Lusztig [12] has developed the theory of character sheaves ;see [15] for a general overview. This theory produces some geometric objects over G (provided by intersection cohomology with coefficients in Q ℓ , where ℓ = p is aprime) from which the irreducible characters of G F can be deduced for any q . Inthis way, the rather complicated patterns involved in the values of the irreduciblecharacters of G F are seen to be governed by geometric principles.In this paper, we discuss an example of this interrelation between geometricprinciples and properties of character values. On the geometric side, we will beconcerned with the restriction of a character sheaf A to the unipotent variety of G . Under some restriction on p , Lusztig [14] has associated to A a well-definedunipotent class O A of G , called its unipotent support . We will be interested inthe restriction of A to O A . Under a certain technical condition (formulated in [4],following a suggestion of Lusztig) the restriction of A to O A is either zero or anirreducible G -equivariant local system on O A (up to shift); see Section 3. Theverification of that technical condition can be reduced to a purely combinatorialproblem, involving the induction of characters of Weyl groups, the Springer corre-spondence and the data on families of characters in Chapter 4 of Lusztig’s book[10]. The details of the somewhat lengthy case-by-case verification are worked outin the second author’s thesis [6]; the main ingredients will be explained in Section 2. Mathematics Subject Classification.
Primary 20C15; Secondary 20G40.
On the character-theoretic side, we will consider the generalized Gelfand–Graevrepresentations (GGGR’s for short) introduced by Kawanaka [7], [8]. In Section 4,assuming that p, q are large and the center of G is connected, we deduce thatKawanaka’s conjecture [9] holds, that is, the characters of the various GGGR’s of G F form a Z -basis of the Z -module of unipotently supported virtual characters of G F . As a further application, in Proposition 4.6, we obtain a new characterisationof GGGR’s in terms of vanishing properties of their character values.2. The Springer correspondence, families and induction
In this section, we deal with the combinatorial basis for the discussion of theunipotent support of character sheaves. We keep the basic assumptions of the intro-duction: G is a connected reductive algebraic group over F p ; we assume throughoutthat p is a good prime for G and that the center of G is connected. Let B ⊆ G bea Borel subgroup and T ⊆ B a maximal torus. Let W = N G ( T ) /T be the Weylgroup of G , with set of generators S determined by the choice of T ⊆ B .Let Irr( W ) be the set of irreducible characters of W (over an algebraically closedfield of characteristic 0). The Springer correspondence associates with each E ∈ Irr( W ) a pair ( u, ψ ) where u ∈ G is unipotent (up to G -conjugacy) and ψ is anirreducible character of the group of components A G ( u ) = C G ( u ) / C G ( u ) ◦ ; see [10, § E ↔ ( u, ψ ).Now we can define three invariants a E , b E and d E for E ∈ Irr( W ). b E is the smallest i > E appears with non-zero multiplicity in the i th symmetric power of the reflection representation of W ; see [10, (4.1.2)]. a E is the largest i > u i divides the generic degree D E ( u ) ∈ Q [ u ]defined in terms of the generic Iwahori–Hecke algebra over Q [ u / , u − / ];see [10, (4.1.1)]. d E is dim B u where B u is the variety of Borel subgroups containing a unipotent u ∈ G such that E ↔ ( u, ψ ) for some ψ ∈ Irr(A G ( u )); see [10, § Lemma 2.1.
We have a E d E b E for all E ∈ Irr( W ) .Proof. See [14, Cor. 10.9] for the first inequality and [18, § a E b E was first observed by Lusztig; see [10, 4.1.3]. (cid:3) Recall that Irr( W ) is partitioned into families and that each family contains aunique special E ∈ Irr( W ), that is, a character such that a E = b E ; see [10, 4.1.4].Furthermore, in [10, Chap. 4], Lusztig associates with any family F ⊆
Irr( W ) afinite group G F , case-by-case for each type of finite Weyl group. (The groups G F form a crucial ingredient in the statement of the Main Theorem 4.23 of [10].) If G is simple modulo its center, then G F ∼ = S , S , S or ( Z / Z ) e for some e > G ∗ be the Langlands dual of G , with Borel subgroup B ∗ and maximaltorus T ∗ ⊆ G ∗ . Let W ∗ = N G ∗ ( T ∗ ) /T ∗ be the Weyl group of G ∗ , with generatingset S ∗ determined by T ∗ ⊆ B ∗ . We can naturally identify W and W ∗ . Note that a E and b E are independent of whether we regard E as a representation of W or of W ∗ . However, it does make a difference as far as d E is concerned.Let s ∈ G ∗ be semisimple and W s be the Weyl group of C G ∗ ( s ). (Note thatC G ∗ ( s ) is a connected reductive group since the center of G is connected.) Replacing s by a conjugate, we may assume that s ∈ T ∗ . Then W s is a subgroup of W ∗ and, nipotent support of character sheaves 3 hence, may be identified with a subgroup of W . So we can consider the inductionof characters from W s to W . Proposition 2.2.
Let s ∈ G ∗ be semisimple and F ⊆
Irr( W s ) be a family. If E is the special character in F , then we have Ind WW s ( E ) = E ′ + a combination of ˜ E ∈ Irr( W ) with b ˜ E > d ˜ E > b E , where E ′ ∈ Irr( W ) is such that b E ′ = d E ′ = b E ; furthermore, E ′ ↔ ( u, underthe Springer correspondence, where stands for the trivial character.Proof. See [14, §
10] and [10, § (cid:3) We are now looking for a condition which guarantees that all ˜ E = E ′ occurring inthe decomposition of Ind WW s ( E ) have d ˜ E > b E . Following a suggestion of Lusztig,such a condition has been formulated in [4, 4.4]. In order to state it, we introducethe following notation.Let S G be the set of all pairs ( s, F ) where s ∈ G ∗ is semisimple (up to G ∗ -conjugacy) and F ⊆
Irr( W s ) is a family. Following [10, § G : S G → { unipotent classes of G } , as follows. Let ( s, F ) ∈ S G and E ∈ F be special. Then consider the inductionInd WW s ( E ) and let E ′ be as in Proposition 2.2. Now define O = Φ G ( s, F ) to be theunipotent class containing u where E ′ ↔ ( u,
1) under the Springer correspondence.
Proposition 2.3 (H´ezard [6]) . Assume that s ∈ G ∗ is semisimple and isolated, thatis, C G ∗ ( s ) is not contained in a Levi complement of any proper parabolic subgroupof G ∗ . Let F ⊆
Irr( W s ) be a family and assume that ( ∗ ) |G s, F | = | A G ( u ) | where u ∈ O = Φ G ( s, F ) . Then the following sharper version of Proposition 2.2 holds: If E is the specialcharacter in F , then we have Ind WW s ( E ) = E ′ + a combination of ˜ E ∈ Irr( W ) with d ˜ E > b E . Proof.
In the setting of Proposition 2.2, let us writeInd WW s ( E ) = E ′ + E ′′ + a combination of ˜ E ∈ Irr( W ) with d ˜ E > b E where E ′′ is the sum of all ˜ E ∈ Irr( W ) such that d ˜ E = b E , ˜ E = E ′ and ˜ E appears in Ind WW s ( E ). Thus, we must show that E ′′ = 0 if ( ∗ ) holds. By standardarguments, this can be reduced to the case where G is simple modulo its center.The reflection subgroups of W which can possibly arise as W s for some semisim-ple element s ∈ G ∗ are classified by a standard algorithm; see [2].Now, if G is of exceptional type, E ′′ can be computed in all cases using explicittables for the Springer correspondence [18] and induce/restrict matrices for thecharacters of Weyl groups; see [6, § E ′′ can be foundfor each type of G . By inspection of these tables, one checks that if ( ∗ ) holds, then E ′′ = 0.If G is of classical type, the induction of characters of Weyl groups and theSpringer correspondence can be described in purely combinatorial terms, involvingmanipulations with various kinds of symbols ([11, § ∗ ) can alsobe formulated in purely combinatorial terms. Using this information, it is then Geck and H´ezard possible to check that, if ( ∗ ) holds, then E ′′ = 0. For the details of this verification,see [6, Chap. 3].We remark that, for G of type B n , Lusztig [13, 4.10] has shown that E ′′ = 0even without assuming that ( ∗ ) holds. (cid:3) Finally, the following result settles the question of when condition ( ∗ ) is actuallysatisfied. Proposition 2.4 (Lusztig [10, 13.3, 13.4] ; see also H´ezard [6]) . Let O be a unipo-tent class. Then |G s, F | | A G ( u ) | for all ( s, F ) ∈ S G such that u ∈ O = Φ G ( s, F ) . Furthermore, there exists some ( s, F ) where s is isolated and we have equality. If O is F -stable (where F is a Frobenius map on G ), then such a pair ( s, F ) can bechosen to be F -stable, too.Proof. Again, this can be reduced to the case where G is simple modulo its center,where the assertion is checked case-by-case along the lines of the proof of Proposi-tion 2.3. The existence of suitable semisimple elements s ∈ G ∗ with centralisers ofthe required type is checked using the tables in [1], [2] (for G of exceptional type) orusing explicit computations with suitable matrix representations (for G of classicaltype). Again, see [6] for more details. (cid:3) It would be interesting to find proofs of Propositions 2.3 and 2.4 which do notrely on a case-by-case argument.3.
Unipotent support
Recall that G is assumed to have a connected center and that we are workingover a field of good characteristic. Now let ˆ G be the set of character sheaves on G (up to isomorphism) over Q ℓ where ℓ is a prime, ℓ = p . By Lusztig [12, § G = a ( s, F ) ∈S G ˆ G s, F where ˆ G s, F − ←→ M ( G F ) . Here, as in Section 2, G F is the finite group associated to a family F ⊆
Irr( W s ) asin [10, Chap. 4]. Furthermore, for any finite group Γ, the set M (Γ) consists of allpairs ( x, σ ) (up to conjugacy) where x ∈ Γ and σ ∈ Irr(C Γ ( x )).Also recall that we have a natural map Φ G : S G → { unipotent classes of G } ,defined as in [10, § p is large enough, so that themain results of Lusztig [14] hold. (Here, “large enough” means that we can operatewith the Lie algebra of G as if we were in characteristic 0, e.g., we can use exp todefine a morphism from the nilpotent variety in the Lie algebra to the unipotentvariety of G .) Theorem 3.1 (Lusztig [14, Theorem 10.7]) . Let ( s, F ) ∈ S G and O = Φ G ( s, F ) bethe associated unipotent class. Then the following hold. (a) There exists some A ∈ ˆ G s, F and an element g ∈ G with Jordan decom-position g = g s g u = g s g u (where g s is semisimple and g u ∈ O ) such that A | { g } = 0 . Note added January 2008: A new recent preprint by Lusztig [16] provides a detailed proof ofthe statements in [10, 13.3, 13.4]. nipotent support of character sheaves 5 (b)
For any A ∈ ˆ G s, F , any unipotent class O ′ = O with dim O ′ > dim O , andany g ′ ∈ G with unipotent part in O ′ , we have A | { g ′ } = 0 . Consequently, the class O is called the unipotent support for the character sheavesin ˆ G s, F . Note that it may actually happen that A | O = 0 for A ∈ ˆ G s, F .Given a unipotent class O , we denote by I O the set of irreducible G -equivariant Q ℓ -local systems on O (up to isomorphism). Theorem 3.2 (Geck [4, Theorem 4.5]; see also the remarks in Lusztig [13, 1.6]) . Let s ∈ G ∗ be semisimple and F ⊆
Irr( W s ) be a family. Let O = Φ G ( s, F ) bethe associated unipotent class and assume that condition ( ∗ ) in Proposition 2.3 issatisfied. Then, for any A ∈ ˆ G s, F , the restriction A | O is either zero or an irreducible G -equivariant local system (up to shift). Furthermore, the map A A | O defines abijection from the set of all A ∈ ˆ G s, F with A | O = 0 onto I O . (Note: In [4, Theorem 4.5], the conclusion of Proposition 2.3, i.e., the validity ofthe sharper version of Proposition 2.2, was added as an additional hypothesis; thiscan now be omitted.)Now let q be a power of p and assume that G is defined over F q ⊆ F p , withcorresponding Frobenius map F : G → G . We translate the above results to classfunctions on the finite group G F .If A is a character sheaf on G then its inverse image F ∗ A under F is again acharacter sheaf. There are only finitely many A such that F ∗ A is isomorphic to A ; such a character sheaf will be called F -stable. Let ˆ G F be the set of F -stablecharacter sheaves. For any A ∈ ˆ G F we choose an isomorphism φ : F ∗ A ∼ → A andwe form the characteristic function χ A,φ . This is a class function G F → Q ℓ whosevalue at g is the alternating sum of traces of φ on the stalks at g of the cohomologysheaves of A . Now φ is unique up to scalar hence χ A,φ is unique up to scalar.Lusztig [12, §
25] has shown that { χ A,φ | A ∈ ˆ G F } is a basis of the vector space of class functions G F → Q ℓ . Let O be an F -stable unipotent class of G . We denote by I F O the set of all E ∈ I O such that E is isomorphic to its inverse image F ∗ E under F . For any such E ,we can define a class function Y E : G F → Q ℓ as in [12, (24.2.2)–(24.2.4)]. We have Y E ( g ) = 0 for g
6∈ O F and Y E ( g ) = Trace( ψ, E g ) for g ∈ O F , where ψ : F ∗ E ∼ → E is asuitably chosen isomorphism. On the level of characteristic functions, Theorem 3.2translates to the following statement (see [13, § § Corollary 3.3.
Let ( s, F ) ∈ S G be F -stable and O = Φ G ( s, F ) be the associatedunipotent class (which is F -stable). Assume that condition ( ∗ ) in Proposition 2.3holds. Then, for any F -stable A ∈ ˆ G s, F , we have either χ A,φ ( g ) = 0 for all g ∈ O F or φ can be normalized such that χ A,φ ( g ) = Y E ( g ) for all g ∈ O F where E = A | O . Now let us consider the irreducible characters of G F . Lusztig [10] has shownthat we have a natural partitionIrr( G F ) = a ( s, F ) ∈S FG Irr s, F ( G F ) . Furthermore, each piece Irr s, F ( G F ) in this partition is parametrized by a “twisted”version of the set M ( G F ); see [10, Chap. 4]. Lusztig [12] gave a precise conjecture Geck and H´ezard about the expression of the characteristic functions of F -stable character sheavesas linear combinations of the irreducible characters of G F . Since we are assumingthat G has a connected center (and p is large), this conjecture is known to hold byShoji [17]. In particular, the following statement holds: Proposition 3.4 (Shoji [17]) . Let ( s, F ) ∈ S FG and A ∈ ˆ G s, F be F -stable. Then χ A,φ is a linear combination of the irreducible characters in
Irr s, F ( G F ) . We can now deduce the following result, whose statement only involves the valuesof the irreducible characters of G F , but whose proof relies in an essential way onthe above results on character sheaves. Corollary 3.5.
Let O be an F -stable unipotent class and u , . . . , u d be represen-tatives for the G F -conjugacy classes contained in O . Let ( s, F ) ∈ S G be F -stablesuch that O = Φ G ( s, F ) and condition ( ∗ ) in Proposition 2.3 holds. Then thereexist ρ , . . . , ρ d ∈ Irr s, F ( G F ) such that the matrix (cid:0) ρ i ( u j ) (cid:1) i,j d has a non-zerodeterminant.Proof. By the proof of [12, 24.2.7], there are precisely d irreducible G -equivariantlocal systems E , . . . , E d on O (up to isomorphism) which are isomorphic to theirinverse image under F ; furthermore, the matrix ( Y E i ( u j )) i,j d is non-singular.By Theorem 3.2, we can find A , . . . , A d ∈ ˆ G s, F such that A i | O = E i for all i . Since each E i is isomorphic to its inverse image under F , the same is truefor A i as well. (Indeed, since ( s, F ) is F -stable, we have F ∗ A i ∈ ˆ G s, F for all i ;furthermore, F ∗ A i | O ∼ = F ∗ E i ∼ = E i . So we must have F ∗ A i ∼ = A i by Theorem 3.2.)By Corollary 3.3, we have χ A i ,φ i = Y E i for all i (where φ i is normalized suitably).It follows that the matrix (cid:0) χ A i ,φ i ( u j ) (cid:1) i,j d has a non-zero determinant.By Proposition 3.4, every χ A i ,φ i can be expressed as a linear combination of thecharacters in Irr s, F ( G F ). Hence there must exist ρ , . . . , ρ d ∈ Irr s, F ( G F ) such thatthe matrix (cid:0) ρ i ( u j ) (cid:1) i,j d has a non-zero determinant. (cid:3) Kawanaka’s conjecture
Kawanaka [8] has shown that, assuming we are in good characteristic, one canassociate with every unipotent element u ∈ G F a so-called generalized Gelfand–Graev representation Γ u (GGGR for short). They are obtained by inducing certainirreducible representations from unipotent radicals of parabolic subgroups of G F .At the extreme cases when u is trivial or a regular unipotent element we obtainthe regular representation of G F or an ordinary Gelfand–Graev representation, re-spectively. Subsequently, assuming that p, q are large, Lusztig [14] gave a geometricinterpretation of GGGR’s in the framework of the theory of character sheaves. Conjecture 4.1 (Kawanaka [7, (3.3.1)]) . The characters of the various
GGGR ’sof G F form a Z -basis of the Z -module of unipotently supported virtual charactersof G F . By Kawanaka [9, Theorem 2.4.3], the conjecture holds if the center of G isconnected and G is of type A n or of exceptional type. In this section, assumingthat p, q are large enough, we will show that it also holds for G of classical type.Given a unipotent element u ∈ G F , denote by γ u the character of the GGGRΓ u . The usual hermitian scalar product for class functions on G F will be denoted nipotent support of character sheaves 7 by h , i . The following (easy) result provides an effective method for verifying thatthe above conjecture holds. Lemma 4.2.
Let u , . . . , u n be representatives for the conjugacy classes of unipo-tent elements in G F . Assume that there exist virtual characters ρ , . . . , ρ n of G F such that the matrix of scalar products ( h ρ i , γ u j i ) i,j n is invertible over Z . ThenConjecture 4.1 holds.Proof. Since the above matrix of scalar products is invertible, γ u , . . . , γ u n are lin-early independent class functions on G F . Consequently, they form a basis of the Q ℓ -vectorspace of unipotently supported class functions on G F . In particular, givenany unipotently supported virtual character χ of G F , we can write χ = P ni =1 a j γ j where a j ∈ Q ℓ , and it remains to show that a j ∈ Z for all j .To see this, consider the scalar products of χ with the virtual characters ρ i . Weobtain P j a j h ρ i , γ j i = h ρ i , χ i ∈ Z for all i = 1 , . . . , n . Since the matrix of scalarproducts ( h ρ i , γ j i ) is invertible over Z , we can invert these equations and concludethat a j ∈ Z for all j , as desired. (cid:3) Let D G be the Alvis–Curtis–Kawanaka duality operation on the character ringof G F . For any ρ ∈ Irr( G F ), there is a sign ε ρ = {± } such that ρ ∗ := ε ρ D G ( ρ ) ∈ Irr( G F ) . The following result will be crucial for dealing with groups of classical type. Weassume from now on that the center of G is connected and that p, q are large, sothat the results in Section 3 can be applied. Proposition 4.3.
Let O be an F -stable unipotent class and u , . . . , u d be represen-tatives for the G F -conjugacy classes contained in O . Let ( s, F ) ∈ S G be F -stablesuch that O = Φ G ( s, F ) and condition ( ∗ ) in Proposition 2.3 holds.Assume that G F is abelian. Then there exist ρ , . . . , ρ d ∈ Irr s, F ( G F ) such that h ρ ∗ i , γ u j i = δ ij for i, j d .Proof. The following argument is inspired by the proof of [3, Proposition 5.6]. By[14, Theorem 11.2] and the discussion in [5, Remark 3.8], we have d X i =1 [A G ( u i ) : A G ( u i ) F ] h ρ ∗ , γ u i i = | A G ( u ) | n ρ for any ρ ∈ Irr s, F ( G F ) , where n ρ > E ∈ Irr( W s )be the special character in F . Then ρ (1) = ± n − ρ q a E N where N is an integer, N ≡ q ;note also that n ρ is divisible by bad primes only.Now, Lusztig [10, 4.26.3] actually gives a precise formula for the integer n ρ , interms of a certain Fourier coefficient. In the case where G F is abelian, this Fouriercoefficient evaluates to |G F | − . Thus, we have n ρ = |G F | − . So, since ( ∗ ) is assumedto hold, we obtain d X i =1 [A G ( u i ) : A G ( u i ) F ] h ρ ∗ , γ u i i = 1 for any ρ ∈ Irr s, F ( G F ) . Now note that each term [A G ( u i ) : A G ( u i ) F ] is a positive integer and each term h ρ ∗ , γ u i i is a non-negative integer. It follows that, given ρ ∈ Irr s, F ( G F ), there Geck and H´ezard exists a unique i ∈ { , . . . , d } such that h ρ ∗ , γ u i i = 1 and h ρ ∗ , γ i ′ i = 0 for i ′ ∈{ , . . . , d } \ { i } . Thus, we have a partition Irr s, F ( G F ) = I ∐ I ∐ · · · ∐ I d such that h ρ ∗ , γ u i i = (cid:26) ρ ∈ I i , ρ ∈ I j where j = i. Assume, if possible, that I r = ∅ for some r ∈ { , . . . , d } . This means that h ρ, D G ( γ u r ) i = h D G ( ρ ) , γ u r i = 0 for all ρ ∈ Irr s, F ( G F ). Thus, by the definitionof the scalar product, we have0 = 1 | G F | X g ∈ G F ρ ( g ) D G ( γ u r )( g ) for all ρ ∈ Irr s, F ( G F ) . Let g ∈ G F and assume that the corresponding term in the above sum is non-zero.First of all, since D G ( γ u r ) is unipotently supported, g must be unipotent. Let O ′ be the conjugacy class of g . By [14, 6.13(i) and 8.6], we have D G ( γ u r )( g ) = 0 unless O is contained in the closure of O ′ . Furthermore, by [14, Theorem 11.2], we have ρ ( g ) = 0 unless O ′ = O or dim O ′ < dim O . Hence, to evaluate the above sum, weonly need to let g run over all elements in O F . Thus, we have0 = d X j =1 | C G F ( u j ) | ρ ( u j ) D G ( γ u r ( u j )) for all ρ ∈ Irr s, F ( G F ) . In particular, this holds for the characters ρ , . . . , ρ d in Corollary 3.5. The invert-ibility of the matrix of values in Corollary 3.5 then implies that D G ( γ u r )( u j ) = 0for 1 j d . Thus, the restriction of D G ( γ u r ) to O F is zero. Now, the relationsin [4, (2.4a)] (which are formally deduced from the main results in [14]) imply that h D G ( γ u r ) , Y E i equals Y E ( u r ) times a non-zero scalar, for any E ∈ I F O . Hence, wehave Y E ( u r ) = 0 for any E ∈ I F O . However, this contradicts the fact that the matrixof values (cid:0) Y E ( u j ) (cid:1) is invertible (see the remarks at the beginning of the proof ofCorollary 3.5). This contradiction shows that we have I i = ∅ for all i . Now choose ρ i ∈ I i for 1 i d . Then we have h ρ ∗ i , γ u j i = δ ij for 1 i, j d , as desired. (cid:3) Remark 4.4.
In the setting of Proposition 4.3, let us drop the assumption that G F is abelian and assume instead that G F is isomorphic to S , S or S . (Thesecases occur when G is simple modulo its center and of exceptional type.) Then, bythe Main Theorem 4.23 of [10], we have a bijection Irr s, F ( G F ) ↔ M ( G F ).Let u , . . . , u d be representatives for the G F -conjugacy classes contained in O F .Since condition ( ∗ ) in Proposition 2.3 is assumed to hold, we can identify M ( G F )with the set of all pairs ( u i , σ ) where 1 i d and σ ∈ Irr(A G ( u i ) F ). Thus, viathe above-mentioned bijection, we have a parametrizationIrr s, F ( G F ) = { ρ ( u i ,σ ) | i d, σ ∈ Irr(A G ( u i ) F ) } . On the other hand, Kawanaka [8], [9] obtained explicit formulas for the values ofthe characters of the GGGR’s (for G of exceptional type). Using these formulas,one can check that h ρ ∗ u i ,σ , γ u j i = (cid:26) σ (1) if i = j, . Thus, setting ρ i := ρ ( u i , for 1 i d (where 1 stands for the trivial character),we see that the conclusion of Proposition 4.3 holds in these cases as well. nipotent support of character sheaves 9 Theorem 4.5.
Recall our standing assumption that p, q are large enough and thecenter of G is connected. Then Kawanaka’s Conjecture 4.1 holds.Proof. By standard reduction arguments, we can assume without loss of generalitythat G is simple modulo its center. If G is of type A n or of exceptional type,the assertion has been proved by Kawanaka [9, Theorem 2.4.3], using his explicitformulas for the character values of GGGR’s. The following argument covers thesecases as well.Let O , . . . , O N be the F -stable unipotent classes of G , where the numbering ischosen such that dim O · · · dim O N . By Proposition 2.4, for each i , we canfind an F -stable pair ( s i , F i ) ∈ S G such that O i = Φ G ( s i , F i ) and condition ( ∗ ) inProposition 2.3 holds.For each i , let u i, , . . . , u i,d i be a set of representatives for the G F -conjugacyclasses contained in O Fi . Let ρ i, , . . . , ρ i,d i be irreducible characters as in Proposi-tion 4.3 (if G is of classical type) or as in Remark 4.4 (if G is of exceptional type).We claim that h ρ ∗ i ,j , γ u i ,j i = 0 if i < i . This is seen as follows. We have h ρ ∗ i ,j , γ u i ,j i = ±h ρ i ,j , D G ( γ u i ,j ) i . By thedefinition of the scalar product, we have h ρ i ,j , D G ( γ u i ,j ) i = 1 | G F | X g ∈ G F ρ i ,j ( g ) D G ( γ u i ,j )( g ) . We now argue as in the proof of Proposition 4.3 to evaluate this sum. First of all, it’senough to let g run over all unipotent elements of G F . Now let g ∈ G F be unipotentand assume, if possible, that the corresponding term in the above sum is non-zero.The fact that ρ i ,j ( g ) = 0 implies that the class of g either equals O i or hasdimension < dim O i . Furthermore, the fact that D G ( γ u i ,j )( g ) = 0 implies that O i is contained in the closure of the class of g . Since we numbered the unipotentclasses according to increasing dimension, we conclude that dim O i = dim O i ;furthermore, g ∈ O i and O i is contained in the closure of the class of g , whichfinally shows that O i = O i , a contradiction. Thus, our assumption was wrong,and the above scalar product is zero.Together with the relations in Proposition 4.3 (or Remark 4.4), we now see thatthe matrix of all scalar products h ρ ∗ i ,j , γ u i ,j i i ,i N, j d i , j d i is a block triangular matrix where each diagonal block is an identity matrix. Hencethat matrix of scalar products is invertible over Z and so Kawanaka’s conjectureholds by Lemma 4.2. (cid:3) Proposition 4.6 (Characterisation of GGGR’s) . Recall that p, q are large enoughand the center of G is connected. Let O be an F -stable unipotent class in G and χ be a character of G F . Then χ = γ u for some u ∈ O F if and only if the followingthree conditions are satisfied: (a) If χ ( g ) = 0 for some g ∈ G F , then the conjugacy class of g is contained inthe closure of O . (b) If D G ( χ )( g ) = 0 for some g ∈ G F , then O is contained in the closure ofthe conjugacy class of g . (c) We have χ (1) = | G F | q − dim O / . Proof. If χ = γ u for some u ∈ O F , then (a) and (c) are easily seen to hold by theconstruction of Γ u ; see Kawanaka [8]. Condition (b) is obtained as a consequence of[14, 6.13(i) and 8.6]. To prove the converse, by standard reduction arguments, wecan assume without loss of generality that G is simple modulo its center. Assumenow that (a), (b) and (c) hold for χ . Since χ is unipotently supported, we can write χ as an integral linear combination of the characters of the various GGGR’s of G F ;see Theorem 4.5.Now, given any F -stable unipotent class O ′ , the characters γ u , where u ∈ O ′ F ,satisfy (a) with respect to O ′ . Hence, all characters γ u , where u is contained in theclosure of O , satisfy (a). One easily deduces that any class function satisfying (a) is alinear combination of various γ u where u is contained in the closure of O . Similarly, any class function satisfying (b) is a linear combination of various D G ( γ u ) where O is contained in the closure of the class of u . Hence, a class function satisfying both(a) and (b) will be a linear combination of various γ u such that u ∈ O F .Let u , . . . , u d be representatives for the G F -conjugacy classes in O F . Then theabove discussion shows that we can write χ = P dj =1 a j γ u j where a j ∈ Z for all i .Now consider the characters ρ i in Proposition 4.3 (for G of classical type) orin Remark 4.4 (for G of exceptional type). Taking scalar products of χ with ρ ∗ i ,we find that a i > i and so χ is a positive sum of characters of variousGGGR’s associated with O F . All these GGGR’s have dimension | G F | q − dim O / .Hence χ (1) is a positive integer multiple of | G F | q − dim O / . Condition (c) now forcesthat χ = γ u for some u ∈ O F , as required. (cid:3) References [1] D. I. Deriziotis: The centralizers of semisimple elements of the Chevalley groups E and E ,Tokyo J. Math. (1983), 191–216.[2] D. I. Deriziotis: Conjugacy classes and centralizers of semisimple elements in finite groupsof Lie type, Vorlesungen aus dem Fachbereich Mathematik der Universit¨at Essen, Heft 11(1984).[3] M. Geck: Basic sets of Brauer characters of finite groups of Lie type, III, Manuscripta Math. (1994), 195–216.[4] M. Geck: Character sheaves and generalized Gelfand–Graev characters, Proc. London Math.Soc. (1999), 139–166.[5] M. Geck and G. Malle: On the existence of a unipotent support for the irreducible charactersof finite groups of Lie type, Trans. Amer. Math. Soc. 352 (2000), 429–456.[6] D. H´ezard: Sur le support unipotent des faisceaux-caract`eres, Ph. D. 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Lusztig: On the character values of finite Chevalley groups at unipotents elements,Journal of Algebra, (1986), 146–194. nipotent support of character sheaves 11 [14] G. Lusztig: A unipotent support for irreducible representations, Advances in Math. 94,139–179 (1992).[15] G. Lusztig: Character sheaves and generalizations, in “The unity of mathematics” (ed.P. Etingof et al.), Progress in Math. , Birkhauser, Boston, 2006, 443–455.[16] G. Lusztig: Unipotent classes and special Weyl group representations, preprint (November2007), available at arXiv:0711.4287 .[17] T. Shoji: Character sheaves and almost characters of reductive groups, Advances in Math. (1995), 244–313; II, 314–354.[18] N. Spaltenstein: On the generalized Springer correspondence for exceptional groups, Ad-vanced Studies in Pure Math. Kinokuniya, Tokyo, and North-Holland, Amsterdam, 1985,317–338.
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