aa r X i v : . [ m a t h . R T ] J a n Harish-Chandra Cuspidal Pairs
Jay Taylor
Abstract.
The irreducible characters of a finite reductive group are partitionedinto Harish-Chandra series that are labelled by cuspidal pairs. In this note, wedescribe how one can algorithmically calculate those cuspidal pairs using resultsof Lusztig. Introduction
As is well known, the set of (ordinary) irreducible characters Irr ( G F ) of a finitereductive group G F is partitioned into Harish-Chandra seriesIrr ( G F ) = G [ L , δ ] ∈ Cusp ( G F ) / G F E ( G F , [ L , δ ]) .Here Cusp ( G F ) / G F are the orbits, under the natural conjugation action, of cuspidal pairs consisting of: L a ( G , F ) -split Levi subgroup of G , meaning L is the Levi complement ofan F -stable parabolic subgroup of G , and δ ∈ Irr ( L F ) is a cuspidal character.This approach to studying Irr ( G F ) has played a crucial role in many recent workssolving open problems in the representation theory of finite groups [KM13; MS16; SF19;Hol19]. The utility here often comes from the fact that cuspidal characters posses uniquefeatures that are not enjoyed by all irreducible characters.To make this effective one needs to explicitly understand the set Cusp ( G F ) / G F . Inprinciple this is known by Lusztig’s extensive results on the classification of Irr ( G F ) [Lus84; Lus88]. However, in practice, determining exactly which pairs arise requiressome work. As we explain in Section 3, the work here involves answering the followingquestion: Assume H G is an F -stable closed connected subgroup of G containing amaximal torus of G . Then what is the smallest ( G , F ) -split Levi subgroup containing H ?In Section 2 we give an answer to this question in the abstract setting of root systems.Following work of Borel–Tits [BT65] and Bonnaf´e [Bon06] we give a characterisationof split Levi subsystems in a root system which generalises the usual characterisationfound in [Bou02], see 2.3 and Corollary 2.11. The connection to the above question isspelled out in Proposition 3.7.In [BT65, §6] the authors work with a reductive group G defined over a field k . If K/k is a Galois extension of fields then one can define an action of a finite quotient Γ ofthe Galois group Gal ( K/k ) on the roots of G . One can ask a similar question to the oneabove in this more general setting, and our statements in Section 2 are phrased so thatthey apply here as well.Implementing this in Michel’s development version of CHEVIE [Mic15] is easy andwe have made this available at [Tay20]. Thus, we get an algorithmic solution to theabove question.
In [BM92] Brou´e–Malle introduced the notion of a d -split Levi subgroup for aninteger d > . This generalises the notion of a ( G , F ) -split Levi subgroup, with the ( G , F ) -split Levi subgroups being the -split Levi subgroups of Brou´e–Malle. We also generaliseour characterisation of split Levi subsystems to this setting, see Corollary 2.19. Thus weobtain an algorithmic solution to the above question with “ ( G , F ) -split” replaced by “ d -split”.There is already a lot of functionality within CHEVIE for working with d -Harish-Chandra theory. For instance, one can construct all d -split Levi subgroups using thecommand SplitLevis and one can find all d -cuspidal unipotent characters using thecommand CuspidalUnipotentCharacters . Our observations here are concerned withbringing this information for unipotent characters to bear on all cuspidal characters.The d -split Levi subgroups supporting a d -cuspidal unipotent character are classi-fied by Brou´e–Malle–Michel [BMM93]. Reducing to the irreducible case, with storeddata for the exceptional types computed using the above general CHEVIE functions, wehave implemented this classification in [Tay20]. With this it is reasonably straightfor-ward to recreate part of the data contained in the tables in [KM13; Hol19]. In thesesources one also finds information on the relative Weyl groups, but we shall discuss thecomputational aspects of this elsewhere.
In Section 3 we recall results of Lusztig that give a parameterisation of the set ofcuspidal pairs Cusp ( G F ) / G F in terms of the Jordan decomposition of characters. Thisreduces us to a question about unipotent characters on the connected centraliser of asemisimple element. As such, we can easily reduce to the setting where our groupis adjoint. In Section 4 we apply the methods of the previous sections when G is anadjoint simple group of classical type and the semisimple element is isolated. When G is of exceptional type one finds the answer in the tables of [KM13], or this can berecomputed using [Tay20]. We cannot formally reduce to the isolated case but this givesa good flavour of the computations involved. If a group G acts on a set X then we will denote by X/G the resultingset of orbits. Typically the G -orbit of x ∈ X will be denoted by [ x ] ∈ X/G and we willsometimes write G x for the stabiliser of x . Acknowledgements
Calculations that culminated in our observations here were originally carried outwhile the author visited the TU Kaiserslautern. He gratefully acknowledges the supportreceived from grant SFB TRR-195 for this visit as well as the support of the Bell SystemFellowship while participating at the IAS. We thank Gunter Malle and Jean Michel fortheir comments on an earlier version of this work. Split Parabolic Subgroups of Reflection Groups
Let ( X , q X ) be a pair of finite rank free Z -modules equipped with a perfect pairing h − , − i : X × q X → Z and let K be a field of characteristic . We let X K = K ⊗ Z X and q X K = K ⊗ Z q X be corresponding K -vector spaces and we denote again by h − , − i : X K × q X K → K the form obtained from extension by linearity. Recall we have a contravariant bijection q : End K ( X K ) → End K ( q X K ) such that h φx , y i = h x , q φy i for all x ∈ X K and y ∈ q X K . Forany pair of non-zero vectors ( α , γ ) ∈ X × q X we have an endomorphism s α , γ ∈ End Z ( X ) defined by s α , γ ( x ) = x − h x , γ i α . Subsets of Root Systems
We assume Φ ⊆ X is a non-empty finite subset and q : Φ → q X is an injectionsuch that h α , q α i = and s α , q α ( Φ ) ⊆ Φ for all α ∈ Φ . We have Φ is a root system, in thesense of [Bou02, VI, §1.1, Def. 1], in the subspace of X R that it spans. We will assume, inaddition, that Φ is reduced. We let W = h s α , q α | α ∈ Φ i GL ( X R ) be the correspondingreflection group, which is finite.If Σ ⊆ X R is any subset then we denote by X R , Σ ⊆ X R the R -subspace spanned by Σ . We take the terms closed , symmetric , and parabolic , subset Ψ ⊆ Φ to be defined asin [Bou02, VI, §1.7, Def. 4], and say that Ψ ⊆ Φ is a closed subsystem if it is closed andsymmetric. Such a subsystem Ψ ⊆ Φ is naturally a root system in X R , Ψ . As W is finite we may choose a W -invariant positive definite symmetric bilinearform (− | −) : X R × X R → R . If λ ∈ X R then the set Ψ λ := { α ∈ Φ | ( α | λ ) > } isa parabolic set. Moreover, a subset Ψ ⊆ Φ is parabolic if and only if Ψ = Ψ λ for some λ ∈ X R , see [DM20, Prop. 3.3.8]. Given a parabolic set Ψ ⊆ Φ we have the symmetricpart Ψ s := Ψ ∩ (− Ψ ) is a closed subsystem, which we call a Levi subsystem .If E ⊆ X R is a subspace then we have a closed subsystem Φ E := Φ ∩ E ⊆ Φ and by[Bou02, VI, §1.7, Prop. 24] this is a Levi subsystem. More precisely, a subset Ψ ⊆ Φ isa Levi subsystem if and only if Ψ = Φ ∩ X R , Ψ . Thus, given a subset Σ ⊆ Φ we have Φ ∩ X R , Σ is the unique minimal Levi subsystem containing Σ . We recall the followingterminology to be used later. Definition 2.4.
We say a subset Σ ⊆ Φ is Φ -isolated if Φ = Φ ∩ X R , Σ . In other words, Σ is not contained in any proper Levi subsystem of Φ . Automorphisms
Suppose now that Γ N GL ( X R ) ( W ) is finite. As WΓ GL ( X R ) is also finitewe may assume our form (− | −) is WΓ -invariant. An element g ∈ Γ must preservethe reflections in W , hence permutes the lines spanned by the roots. More precisely, asour root system is reduced we have a well-defined permutation ρ g : Φ → Φ such that gα = c g , α ρ g ( α ) for some real number c g , α > 0 . We consider Φ to be a Γ -set via thispermutation action. Definition 2.6.
A subset Σ ⊆ Φ is said to be a Γ -split Levi subsystem if Σ = Ψ s for some Γ -stable parabolic subset Ψ ⊆ Φ . If φ ∈ N GL ( X R ) ( W ) has finite order then we say Σ ⊆ Φ is a φ -split Levi subsystem if it is a h φ i -split Levi subsystem. Our goal now is to prove Corollary 2.11. If E ⊆ X R is a Γ -stable subspace thenlet E Γ be the Γ -fixed subspace of E . If θ : X R → X Γ R is the orthogonal projection onto X Γ R then for v ∈ X R we have θv = | Γ | X g ∈ Γ gv . (2.8)Hence ( θx | y ) = ( θx | θy ) = ( x | θy ) for all x , y ∈ X R . If E ⊆ X R is a subset then we let Φ E = Φ E ⊥ = Φ ∩ E ⊥ , where E ⊥ = { x ∈ X R | ( x | e ) = for all e ∈ E } is the perpendicularspace. If λ ∈ X R then the symmetric part of the parabolic set Ψ λ is Φ λ := Φ { λ } . As in[Bon06, Prop. 2.2] we get the following characterisation of Γ -split Levi subsystems. Proposition 2.9.
Suppose Σ ⊆ Φ is a subset. Then the following are equivalent: (a) Σ is a Γ -split Levi subsystem, (b) Σ = Φ E for some subspace E ⊆ X Γ R , (c) Σ = Φ λ for some λ ∈ X Γ R . Proof. (c) ⇒ (b). Clear.(b) ⇒ (a). For α ∈ Φ denote by E α ⊆ E the hyperplane { e ∈ E | ( e | α ) = } . Bydefinition α ∈ Φ E if and only if E α = E . Now let F = S α ∈ Φ \ Φ E E α ⊆ E , which isempty if Φ = Φ E . As Φ \ Φ E is finite and R is infinite it is well known that E = F sothere exists a vector λ ∈ E \ F . Taking Ψ = Ψ λ we see that Φ E = Φ λ = Ψ s and clearly ρ g ( Ψ λ ) = Ψ gλ = Ψ λ for any g ∈ Γ .(a) ⇒ (c). As Σ is a Levi subsystem we have by 2.3 that Σ = Φ λ for some λ ∈ X R . Itsuffices to show that Σ = Φ λ = Φ θλ . Now, if α ∈ Φ then by (2.8) we have ( α | θλ ) = | Γ | X g ∈ Γ ( α | gλ ) = | Γ | X g ∈ Γ c g , α ( ρ g ( α ) | λ ) .If α ∈ Φ λ then as Φ λ is Γ -stable we have ( ρ g ( α ) | λ ) > for all g ∈ Γ so ( α | θλ ) > ,which means Φ λ ⊆ Φ θλ . Similarly, if α ∈ Φ \ Φ λ then as Φ \ Φ λ is Γ -stable we have ( ρ g ( α ) | λ ) < 0 for all g ∈ Γ so ( α | θλ ) < 0 , which means Φ \ Φ λ ⊆ Φ \ Φ θλ . (cid:4) Lemma 2.10. If E ⊆ X Γ R is a subspace then Φ θ − ( E ) is a Γ -split Levi subsystem of Φ . Proof.
By the equivalence of (a) and (b) in Proposition 2.9 it suffices to show that θ − ( E ) = θ ( E ⊥ ) ⊥ as then Φ θ − ( E ) = Φ θ ( E ⊥ ) . We have θ − ( E ) ⊆ θ ( E ⊥ ) ⊥ because if v ∈ θ − ( E ) and x ∈ E ⊥ then ( v | θx ) = ( θv | x ) = . Conversely, by assumption we have ( X Γ R ) ⊥ ⊆ E ⊥ which implies that θ ( E ⊥ ) = E ⊥ ∩ X Γ R and so θ ( E ⊥ ) ⊥ = ( E ⊥ ) ⊥ + ( X Γ R ) ⊥ = E ⊕ ( X Γ R ) ⊥ . Hence θ ( θ ( E ⊥ ) ⊥ ) = E which gives θ − ( E ) = θ ( E ⊥ ) ⊥ . (cid:4) Corollary 2.11.
For any subset Σ ⊆ Φ we have Φ ∩ θ − ( θ ( X R , Σ )) is the smallest Γ -splitLevi subsystem containing Σ . In particular, Σ is a Γ -split Levi subsystem if and only if Σ = Φ ∩ θ − ( θ ( X R , Σ )) . Proof.
By Lemma 2.10 we certainly have Φ ∩ θ − ( θ ( X R , Σ )) is a Γ -split Levi subsys-tem containing Σ . However, if Ψ ⊇ Σ is a Γ -split Levi subsystem then X R , Σ ⊆ X R , Ψ which implies that θ − ( θ ( X R , Σ )) ⊆ θ − ( θ ( X R , Ψ )) . So Φ ∩ θ − ( θ ( X R , Σ )) ⊆ Ψ = Φ ∩ θ − ( θ ( X R , Ψ )) . (cid:4) In CHEVIE [Mic15] one finds the command
EigenspaceProjector which canbe used to calculate the projector onto the φ -fixed subspace of X R for any finite orderelement φ ∈ N GL ( X R ) ( W ) . With this it is easy to find those roots α ∈ Φ satisfying θα ∈ θ ( X R , Σ ) = X R , θ ( Σ ) , see the function SplitLeviCover in [Tay20] for more details.
A Generalisation
We now consider a generalisation of these ideas following Brou´e–Malle [BM92].For this we assume φ ∈ GL ( X ) is an automorphism of finite order n > 0 , which weconsider as an element of GL ( X K ) in the natural way. For an integer d > 0 we denote by µ d C × the subgroup of d th roots of unity and by µ ∗ d ⊆ µ d the subset of elements oforder d , i.e., the primitive d th roots of unity.Assume K ⊆ C is a subfield and E ⊆ X K is a φ -invariant K -subspace. If ζ ∈ K × thenwe denote by E ( φ , ζ ) ⊆ E the ζ -eigenspace of φ acting on E . Moreover, for an integer d > 0 we let E ( φ , d ) = M ζ ∈ µ ∗ d E ( φ , ζ ) .Of course, E ( φ , d ) = { } if d ∤ n and E = L d | n E ( φ , d ) if µ n ⊆ K . From now on K = Q ( µ n ) ⊆ C is the n th cyclotomic field and G = Gal ( K / Q ) .We have a natural semilinear action of G on X K = K ⊗ Q X Q , in the sense of [Bou03,V, §10.4], where σ ∈ G acts via σ ⊗ Q Id on X K . This action commutes with the action of φ so it is clear that σX K ( φ , ζ ) = X K ( φ , σ ( ζ )) for any σ ∈ G . Hence the space X K ( φ , d ) is G -stable so there is a canonical Q -subspace X Q ( φ , d ) := X K ( φ , d ) G , the subspace of G -invariants, satisfying X K ( φ , d ) = K ⊗ Q X Q ( φ , d ) , see [Bou03, V, §10.4, Cor.]. Fix an indeterminate q . For an integer d > 0 we denote by Φ d ( q ) the d th cy-clotomic polynomial Q ζ ∈ µ ∗ d ( q − ζ ) ∈ Z [ q ] . We have X Q = L d | n X Q ( φ , d ) and the char-acteristic polynomial of φ on X Q ( φ , d ) is given by Φ d ( q ) a d , where a d = dim K ( X K ( φ , ζ )) with ζ ∈ µ ∗ d . Here we use that G acts transitively on µ ∗ d , see [Bou03, V, §11.5, Thm. 2]. If E ⊆ X Q ( φ , d ) is a φ -invariant Q -subspace then the characteristic polynomial of φ on E is given by Φ d ( q ) b for some b a d .We assume (− | −) is a non-degenerate W h φ i -invariant positive definite symmetricbilinear form X Q . We then consider (− | −) as a bilinear form on X K by scalar extension.As before, E ⊥ ⊆ X K is the perpendicular space of E ⊆ X K and Φ E = Φ ∩ E ⊥ . After[BM92] we make the following definition. Definition 2.15. If d > 0 is an integer then a subset Σ ⊆ Φ is a ( φ , d ) -split Levi subsystem if Σ = Φ E for some φ -invariant Q -subspace E ⊆ X Q ( φ , d ) . Lemma 2.16.
Fix a root of unity ζ ∈ µ ∗ d . For any subset Σ ⊆ Φ the following are equivalent: (a) Σ is ( φ , d ) -split, (b) Σ = Φ E = Φ ∩ E ⊥ for some subspace E ⊆ X K ( φ , ζ ) . Proof.
Let E ⊆ X Q ( φ , d ) be a φ -invariant Q -subspace and set E K = K ⊗ Q E and ˜ E = E K ( φ , ζ ) . It suffices to show that Φ E = Φ ˜ E . As above, we have G acts on E K and E K = L σ ∈ G / G ζ σ ˜ E . However, if α ∈ Φ and e ∈ ˜ E then σ ( α | e ) = ( α | σe ) . This showsthat Φ ˜ E = Φ E K = Φ E . (cid:4) For ζ ∈ K × we denote by θ ζ : X K → X K ( φ , ζ ) the orthogonal projection ontothe ζ -eigenspace of φ . The exact same proofs used to obtain Lemma 2.10 and Corollary 2.11yield the following, which give a characterisation of ( φ , d ) -split Levi subsystems. Lemma 2.18. If ζ ∈ µ ∗ d is a primitive d th root of unity and E ⊆ X K ( φ , ζ ) is a K -subspace then Φ ∩ θ − ( E ) is a ( φ , d ) -split Levi subsystem of Φ . Corollary 2.19.
Let ζ ∈ µ ∗ d be a primitive d th root of unity. Then for any subset Σ ⊆ Φ wehave Φ ∩ θ − ( θ ζ ( X K , Σ )) is the smallest ( φ , d ) -split Levi subsystem containing Σ . In particular, Σ is a ( φ , d ) -split Levi subsystem if and only if Σ = Φ ∩ θ − ( θ ζ ( X K , Σ )) . The eigenspace projector θ ζ can also be obtained through the CHEVIE com-mand
EigenspaceProjector and so the function
SplitLeviCover provided in [Tay20]will also deal with this more general setting. Now suppose φ ∈ N GL ( X R ) ( W ) has finiteorder n > 0 . Jean Michel has informed us that in this situation one simply takes (b),with K = Q ( µ n ) , of Lemma 2.16 as the definition for ( φ , d ) -split. With this definitionCorollary 2.19 also applies in this situation. Jordan Decomposition and Cuspidal Pairs
From now on G is a connected reductive algebraic group defined over an alge-braic closure F = F p of the finite field of prime order p > 0 . We let F : G → G be aFrobenius root as in [DM20]. The set of unipotent characters of the finite group G F willbe denoted by UCh ( G , F ) .We fix an F -stable maximal torus T of G and a tuple ( G ⋆ , T ⋆ , F ⋆ ) dual to ( G , T , F ) .The roots of G are denoted by Φ ⊆ X = X ( T ) = Hom ( T , F × ) and the Weyl group by W = N G ( T ) / T . We let : N G ( T ) → W be the natural quotient map. For an element g ∈ G we let ι g : G → G denote the inner automorphism given by ι g ( x ) = gxg − andwe let gF := ι g ◦ F . Consider the subset G ss ⊆ G of semisimple elements. The set T( G , F ) = { ( s , n ) ∈ G ss × G | n F ( s ) = s } is a G -set with the action given by g · ( s , n ) = ( g s , gnF ( g ) − ) . (3.3)Note the stabiliser of ( s , n ) under this action is the finite group C G ( s ) nF . By the Lang–Steinberg Theorem we have a bijective map G F ss / G F → T( G , F ) / G given by [ s ] [ s , ] ,where G F ss / G F is the set of semisimple conjugacy classes of G F .For s ∈ T we let C W ( s ) := N C G ( s ) ( T ) / T and C ◦ W ( s ) = N C ◦ G ( s ) ( T ) / T , whereC ◦ G ( s ) is the connected component of the centraliser C G ( s ) . Let A W ( s , F ) be the set ofcosets a ∈ C ◦ W ( s ) \ W such that w F ( s ) = s for some (any) w ∈ a . Then T G ( T , F ) denotesthe set of pairs ( s , a ) with s ∈ T and a ∈ A W ( s , F ) . There is a natural action of W on T G ( T , F ) , defined exactly as in (3.3). We remark that we have the following well-knownparameterisation of the semisimple classes of G F . Lemma 3.4.
We have a well-defined bijection T G ( T , F ) /W → T( G , F ) / G defined by sending [ s , C ◦ W ( s ) n ] [ s , n ] , where n ∈ N G ( T ) . Remark 3.5.
If we define T( G , T , F ) = { ( s , n ) ∈ T × N | n F ( s ) = s } ⊆ T( G , F ) then, byLemma 3.4, each orbit T( G , F ) / G has a representative in T( G , T , F ) . Let S G be an F -stable torus and let X R ( S ) = R ⊗ Z X ( S ) . We have an inducedmap F ∗ : X ( S ) → X ( S ) and on X R ( S ) this factors uniquely as qτ with q ∈ R >0 a positivereal number and τ ∈ GL ( X R ( S )) of finite order. We say S is F -split if τ is the identity.Any F -stable subtorus of G contains a unique maximal F -split subtorus. We denote byZ ◦ F ( G ) this subtorus of Z ◦ ( G ) . The following result is well known. We give a few detailsjust to make the connection with the material in Section 2. Proposition 3.7 (Borel–Tits).
Suppose H G is an F -stable closed connected maximal ranksubgroup of G . Then there is a unique minimal ( G , F ) -split Levi subgroup containing H . More-over, this subgroup is C G ( Z ◦ F ( H )) . In particular, H is a ( G , F ) -split Levi subgroup if and only if H = C G ( Z ◦ F ( H )) . Proof.
By assumption we have g T H for some g ∈ G with g − F ( g ) = n ∈ N G ( T ) .Hence, we can assume that H > T if we replace F by nF . Let qφ be the factorisation of ( nF ) ∗ on X R ( T ) so that q ∈ R >0 and φ ∈ GL ( X R ( T )) has finite order.As in [DM20, §3.3] we have a map Σ G Σ which gives a bijection between thelattice of quasi-closed subsets of Φ and the closed connected subgroups of G containing T . If Γ = h φ i then this maps the Γ -invariant parabolic subsets of Φ onto the F -stableparabolic subgroups of G containing T , see [DM20, Prop. 3.4.5], and thus the Γ -splitLevi subsystems of Φ onto the F -stable Levi subgroups of G containing T . The firststatement now follows from Corollary 2.11. The second statement follows from [GM20,Lem. 3.1.3] and the last statement is obvious. (cid:4) Jordan Parameters
We use a language for Jordan decompositions similar to that used by Cabanes–Sp ¨ath [CS17, §8]. If ( s , n ) , ( s , n ) ∈ T( G , F ) are such that ( s , n ) = ( g s , gn F ( g ) − ) ,for some g ∈ G , then we have ι g ◦ n F = n F ◦ ι g . We define Jor ◦ ( G , F ) to be the set ofpairs (( s , n ) , ψ ) with ( s , n ) ∈ T( G , F ) and ψ ∈ UCh ( C ◦ G ( s ) , nF ) . In other words, we haveJor ◦ ( G , F ) = G ( s , n ) ∈ T( G , F ) UCh ( C ◦ G ( s ) , nF ) .Moreover, G acts on Jor ◦ ( G , F ) via g · (s , ψ ) = ( g · s , ψ ◦ ι − ) .Now let G ad = G / Z ( G ) , which for these purposes we consider to be the adjointgroup of G . We denote again by F the Frobenius root of G ad induced by that of G . If g Z ( G ) ∈ G F ad then for any x ∈ G F we have g x ∈ G F and this gives a well-defined actionof G F ad on G F , hence also of G F ad on Irr ( G F ) and Cusp ( G F ) . Let us recall that in [Lus84; Lus88] Lusztig has shown the existence of a
Jordandecomposition J : Irr ( G F ) → Jor ◦ ( G ⋆ , F ⋆ ) / G ⋆ which is a certain surjective map whosefibres are the orbits Irr ( G F ) / G F ad . In what follows we will need to assume some compat-ibility properties between Jordan decompositions. We will write J • to denote a familyof Jordan decompositions J L : Irr ( L F ) → Jor ◦ ( L ⋆ , F ⋆ ) / L ⋆ , one for each F -stable Levi sub-group L G . Implicitly this involves the choice of an F ⋆ -stable Levi subgroup L ⋆ G ⋆ corresponding to L under duality, see [DM20, §11.4].We say the family J • is G F -invariant if for each F -stable Levi subgroup L G andelement g ∈ G F the following hold:• ( g L ) ⋆ = L ⋆ • J L ( χ ) = J g L ( χ ◦ ι − ) for all χ ∈ Irr ( L F ) .Requiring these properties is equivalent to choosing one map J L for each G F -orbit ofLevi subgroups. Remark 3.10.
Note this assumption implies that if L , L G are two F -stable Levisubgroups such that L ⋆ = g L ⋆ for some g ∈ G ⋆ F ⋆ then L ⋆ = L ⋆ and L is G F -conjugateto L . Cuspidal Characters
We start by reducing our problem concerning cuspidal pairs to a question inthe dual group using the Jordan decomposition. For this, let us define the set J ◦ cusp ( G , F ) of tuples (( s , n ) , ( L , ψ )) such that:• ( s , n ) ∈ T( L , F ) ,• L G is the smallest ( G , nF ) -split Levi subgroup containing C ◦ L ( s ) ,• ψ ∈ UCh ( C ◦ L ( s ) , nF ) is a cuspidal character.The second condition is equivalently stated as Z ◦ nF ( L ) = Z ◦ nF ( C ◦ L ( s )) , see Proposition 3.7.Of course, there is an action of G on J ◦ cusp ( G , F ) given by g · (s , ( L , ψ )) = ( g · s , ( g L , ψ ◦ ι − )) . The following is a consequence of Lusztig’s characterisation of cuspidal charactersgiven in [Lus78, 2.18] and [Lus77, 7.8]. Theorem 3.12 (Lusztig).
Assume J • is a G F -invariant family of Jordan decompositions as in3.9. Then we have a well-defined bijection Cusp ( G F ) / G F ad → J ◦ cusp ( G ⋆ , F ⋆ ) / G ⋆ [ L , δ ] [s , ( L ⋆ , ψ )] , (3.13) where J L ( δ ) = [s , ψ ] . Proof.
That this map makes sense and is surjective follows from [GM20, Thm. 3.2.22].Now assume ( L , δ ) , ( L , δ ) ∈ Cusp ( G F ) satisfy ( L , δ ) = ( g L , g δ ) for some g ∈ G F .If J L ( δ ) = [s , ψ ] and J L ( δ ) = [s , ψ ] then by our assumption that J • is G F -invariantwe get that J L ( δ ) = J L ( δ ) so (s , ψ ) = x · (s , ψ ) for some x ∈ L ⋆ = L ⋆ . Thus themap is well defined.Now suppose J L ( δ ) = [( s , ) , ψ ] and J L ( δ ) = [( s , ) , ψ ] and the correspondingpairs (( s , ) , ( L ⋆ , ψ )) = x · (( s , ) , ( L ⋆ , ψ )) are in the same G ⋆ -orbit for some x ∈ G ⋆ .Note x ∈ G ⋆ F ⋆ so L ⋆ = L ⋆ and L = g L for some g ∈ G F . Now we have J L ( δ ) =[( s , ) , ψ ] = [( s , ) , ψ ] = J L ( δ ) = J L ( g δ ) . It follows that lg δ = δ for someelement l Z ( L ) ∈ ( L / Z ( L )) F .By [Bon06, Prop. 4.2] we have Z ( L ) = Z ◦ ( L ) Z ( G ) so by the Lang–Steinberg The-orem there exists x ∈ Z ◦ ( L ) such that ( xl ) − F ( xl ) ∈ Z ( G ) . If z = xl then ( L , δ ) =( zg L , zg δ ) with zgZ ( G ) ∈ G F ad so the map is injective. (cid:4) After Theorem 3.12 we are reduced to understanding the set J ◦ cusp ( G , F ) / G .For this we define Jor ◦ cusp ( G , F ) to be the set of all tuples (( s , n ) , ( M , ψ )) with ( s , n ) ∈ T( G , F ) and ( M , ψ ) ∈ Cusp ( C ◦ G ( s ) nF ) a cuspidal pair with ψ ∈ UCh ( M , nF ) a unipotentcharacter. Clearly we have an action of G on Jor ◦ cusp ( G , F ) by setting g · (s , ( M , ψ )) =( g · s , ( g M , ψ ◦ ι − )) .The set Jor ◦ cusp ( G , F ) / G parameterises the Harish-Chandra series of unipotent char-acters and is completely understood via Lusztig’s explicit description of all cuspidalunipotent characters, which one finds in [Lus78, Part 3] or [Car93, §13.7]. The followingrelates Jor ◦ cusp ( G , F ) / G with J ◦ cusp ( G , F ) / G . Proposition 3.15.
We have a well-defined bijection J ◦ cusp ( G , F ) / G → Jor ◦ cusp ( G , F ) / G [( s , n ) , ( L , ψ )] [( s , n ) , ( C ◦ L ( s ) , ψ )] . Proof.
That this is well defined is obvious. Now assume we have pairs (( s , ) , ( L , ψ )) and (( s , ) , ( L , ψ )) such that (( s , ) , ( C ◦ L ( s ) , ψ )) = g · (( s , ) , ( C ◦ L ( s ) , ψ )) forsome g ∈ G . Then g ∈ G F and Z ◦ F ( L ) = Z ◦ F ( C ◦ L ( s )) = g Z ◦ F ( C ◦ L ( s )) = g Z ◦ F ( L ) .Hence, by Proposition 3.7 we have L = C G ( Z ◦ F ( L )) = g C G ( Z ◦ F ( L )) = g L . Thus themap is injective and it is surjective because [( s , n ) , ( M , ψ )] ∈ Jor ◦ cusp ( G , F ) / G is in theimage of [( s , n ) , ( C G ( Z ◦ nF ( M )) , ψ )] , which is in J ◦ cusp ( G , F ) / G by Proposition 3.7. (cid:4) From this we see that to understand the set J ◦ cusp ( G , F ) / G it suffices to computethe inverse of the map in Proposition 3.15. In other words, given an nF -split Levi sub-group M C ◦ G ( s ) we need to calculate the smallest ( G , nF ) -split Levi subgroup L G containing M . This we can do with the approach of Section 2. We also remark that itis straightforward to reduce the classification of J ◦ cusp ( G , F ) / G to the case where G is anadjoint simple group. We refer the reader to [DM20, §11.5] for more details.0Now assume that F is a Frobenius endomorphism. Then the above can be eas-ily generalised to the d -cuspidal setting as follows. For an integer d > we defineCusp d ( G F ) , J ◦ d ,cusp ( G , F ) , and Jor ◦ d ,cusp ( G , F ) , in the same way except replacing “cus-pidal” by “ d -cuspidal” and “ ( G , F ) -split” by “ d -split”, see [GM20, §3.5]. The map inTheorem 3.12 then gives an injection Cusp d ( G F ) / G F ad → J ◦ d ,cusp ( G , F ) / G and the mapin Proposition 3.15 gives a bijection J ◦ d ,cusp ( G , F ) / G → Jor ◦ d ,cusp ( G , F ) / G . Again, in thiscase we can invert the map in Proposition 3.15 using the results of Section 2. Cuspidal Pairs in Isolated Lusztig Series
Isolated Elements
In this section we assume that G is an adjoint simple group and that F is aFrobenius endomorphism. Let Z ( p ) be the localisation of Z at the prime ideal ( p ) . If ι : Z ( p ) / Z → K × is a fixed isomorphism then we denote by ˜ ι : Q → K × the homomor-phism obtained as the composition Q → Q / Z → Z ( p ) / Z ι → K × .Here the first map is the natural projection map and the second map is obtained byquotienting out the p -torsion subgroup of Q / Z .If q V = Q ⊗ Z q X , with q X = Hom ( F × , T ) , then we have a unique surjective homomor-phism of abelian groups ˜ ι : q V → T satisfying ˜ ι ( k ⊗ γ ) = γ ( ˜ ι ( k )) . We fix a simple andpositive system of roots ∆ ⊆ Φ + ⊆ Φ and denote by q Ω = ( q ̟ α ) α ∈ ∆ the basis of q V dualto ∆ ⊆ V with respect to h − , − i : X × q X → Z (extended linearly). As we assume G isadjoint we have X = Z ∆ and q X = Z q Ω .Recall that s ∈ T is said to be G -isolated if C ◦ G ( s ) is not contained in any proper Levisubgroup of G . If Φ ( s ) = { α ∈ Φ | α ( s ) = } then s is G -isolated if and only if Φ ( s ) is Φ -isolated, see Definition 2.4. Denote by α = P α ∈ ∆ n α α ∈ Φ the highest root and let e ∆ = ∆ ∪ { − α } be the extended set of simple roots. By convention we set q ̟ − α = and n − α = . If e ∆ p ′ is the set of α ∈ e ∆ such that gcd ( n α , p ) = then the following is shownin [Bon05, Thm. 5.1]. Proposition 4.2.
Every G -isolated element in T is W -conjugate to some h α := ˜ ι ( q ̟ α /n α ) with α ∈ e ∆ p ′ . Moreover, the root system Φ ( h α ) of the centraliser C ◦ G ( h α ) with respect to T has a simple system given by ∆ ( h α ) := e ∆ \ { α } . To understand the W -orbits of pairs ( s , a ) ∈ T G ( T , F ) , with s ∈ T a G -isolatedelement, it suffices to understand the orbits of C W ( h α ) acting on the set of cosets A W ( h α , F ) . However, after [Bon05, Thm. 5.1] this is easy once we find an element w ∈ W satisfying w F ( h α ) = h α . As we will now explain this is straightforward except for a fewcases in E .Let V = Q ⊗ Z X . As F is a Frobenius endomorphism the induced map F ∗ on X factorsas qφ with φ ∈ GL ( X ) of finite order and q ∈ Z >0 a positive integer. Note φ∆ = ∆ andwe may view φ as an element of GL ( V ) . If α ∈ e ∆ p ′ let w α ∈ C ◦ W ( h α ) be the unique1element satisfying w α ∆ ( h α ) = − ∆ ( h α ) so that w := w − α is the longest element of W with respect to ∆ . Lemma 4.4. If α ∈ e ∆ p ′ then either F ( h α ) = h α or w F ( h α ) = h α unless n α = , in whichcase G is of type E , and q ≡ , ( mod ) . Proof.
A straightforward calculation shows that F ( h α ) = h qφ − α and w h α = h − − α where ε = − w ∈ GL ( X ) satisfies ε∆ = ∆ . From the classification of irreducible rootsystems we see that n α . Note n α is the order of h α so if n α = thencertainly F ( h α ) = h α . Assume n α > 2 then one easily checks that α is fixed under allautomorphisms of the Dynkin diagram. Hence, if q ≡ ( mod n α ) then F ( h α ) = h α and if q ≡ − ( mod n α ) then w F ( h α ) = h α . As gcd ( q , n α ) = one of these conditionsmust hold unless n α = . Finally suppose n α = . Then either α is fixed by φ or G is oftype E and φ = ε . In either case the statement holds as ε has order . (cid:4) In what follows we determine the tuples [( s , n ) , ( L , ψ )] ∈ J ◦ cusp ( G , F ) / G when s is G -isolated and G is of classical type, see 4.14 for groups of exceptional type. Thisinformation is given in Table 4.1, although we omit the information regarding ψ . Westart with a pair ( h α , a ) ∈ T G ( T , F ) , with α ∈ ˜ ∆ p ′ , and choose an element n ∈ N G ( T ) such that n = w ∈ a is the unique element of the coset satisfying wφ ∆ ( h α ) = ∆ ( h α ) . Remark 4.6. If w F ( h α ) = h α then w = w α w is the unique element of the cosetC ◦ W ( h α ) w satisfying wφ ∆ ( h α ) . Using Lusztig’s classification of cuspidal unipotent characters [Lus78] it is easyto list all the cuspidal pairs ( M , ψ ) ∈ Cusp ( C ◦ G ( h α ) , nF ) with T M C ◦ G ( h α ) astandard Levi subgroup, determined by a subset of ∆ ( h α ) , and ψ ∈ UCh ( C ◦ G ( h α ) , nF ) .We then calculate the smallest ( G , nF ) -split Levi subgroup L containing M , so that M = C ◦ L ( s ) , using the approach of Section 2.The root system of M with respect to T will be denoted by Σ ⊆ Φ ( h α ) . We let θ : V → V wφ be the projection onto the wφ -fixed space of V . Our aim is to calculatethe smallest wφ -split Levi subsystem Ψ = θ − ( θ ( V Σ )) containing Σ ; here V Σ ⊆ V isthe subspace spanned by Σ . We will specify Ψ by specifying the set of positive roots Ψ + = Ψ ∩ Φ + . Classical Types
The case of type A n is trivial, as the only isolated element is the identity, sowe assume that G has type B n ( n > ), C n ( n > ), or D n ( n > ). We take the rootsystem and simple roots ∆ = ( α , . . . , α n ) to be labelled as in [Bou02]. We will assumethat ( e , . . . , e n ) is a basis of the Q -vector space V so that the roots are expressed as in[Bou02] with respect to this basis.Let E = { e , . . . , e n , − e n , . . . , − e } ⊆ V . The natural action of W on V preserves theset E so we have a permutation representation W → S E into the symmetric group on E .This representation is faithful and we will identify W with its image. For i < j n S E : p i , j = p j , i = ( i , j )(− i , − j ) and u i , j = u j , i =( i , − j )(− i , j ) . Moreover, for i n we define c i = ( i , − i ) . We now consider a few of the more difficult cases from Table 4.1. We identifythe case by listing the type of C ◦ G ( s ) nF . In all the special cases we consider we have φ is the identity. Before proceeding we make a few comments on the conventions usedin Table 4.1. Here we use the following notation for the sets of square and triangularintegers:• (cid:3) m = { ( + m ) | k ∈ Z > } \ { } so that (cid:3) = { , , , . . . } and (cid:3) = { , , , . . . } ,• △ = { k ( k + ) /2 | k ∈ Z > } = { , , , , , , . . . } .These integers occur in the classification of cuspidal unipotent characters for groups ofclassical type, see [Lus77, Thm. 8.2].Suppose the root system Φ ( s ) of C ◦ G ( s ) has two indecomposable components X a · Y n − a . In this case a component of C ◦ L ( s ) with rank d a is taken to be contained inthe component X a and a component with rank m − d n − a or m n is taken tobe contained in the component Y n − a . Finally, let us note that each row of the tablecorresponds to exactly one orbit of pairs ( s , a ) ∈ T G ( T , F ) except the rows marked by( ⋆ ) which correspond to two such orbits. C n2 ( q ) . We take w = u , n u , n − · · · u n/2 , n/2 + . The fixed point space V w has as basis ( v i | i n2 ) with v i = e i − e n + − i . Moreover, the subspace θ ( V Σ ) ⊆ V w has as basis ( v , . . . , v m2 ) . If J = { , . . . , m2 , n + − m2 , . . . , n } then Ψ + consistsof { i , e i ± e j | i , j ∈ J and i < j } , giving a component of type C m , and { e i + e n + − i | m2 < i n2 } giving the n − m2 components of type A . D a ( q ) · B n − a ( q ) . Recall that a n . We take w = c . The fixedpoint space V w has as basis ( e , e , . . . , e n ) . There are two cases to consider. If M hastype B m then θ ( V Σ ) has as basis ( e n − m + , . . . , e n ) . If J = { , n − m + , . . . , n } then Ψ + is given by { e i , e i ± e j | i , j ∈ J and i < j } which is of type B m + .If M has type D d · B m − d then θ ( V Σ ) has as basis ( e , . . . , e d , e n − m + d + , . . . , e n ) . If J = { , . . . , d , n − m + d + , . . . , n } then Ψ + is given by { e i , e i ± e j | i , j ∈ J and i < j } which is of type B m . D a ( q ) · D n − a ( q ) . Recall that a n2 . We take w = c c n . Thefixed point space V w has as basis ( e , e , . . . , e n − ) . First, if M = T is a torus then Ψ + = { e − e n , e + e n } , which has type D = A A .If M has type D m then θ ( V Σ ) has as basis ( e n − m + , . . . , e n − ) . If J = { , n − m + , n − m + , . . . , n } then Ψ + is given by { e i ± e j | i , j ∈ J and i < j } which is of type D m + . The case where M is of type D d or D d · D m − d is similar to previous cases. D n2 ( q ) . Here there are two orbits of pairs ( s , a ) such that C ◦ G ( s ) nF hastype D n2 ( q ) . We take w to be one of the following two elements: u , n u , n − · · · u n/2 , n/2 + p , n u , n − · · · u n/2 , n/2 + V w has as basis ( v i | i n2 ) with v i = e i − e n + − i for i > 1 and v = e − e n for the first choice of w and v = e + e n for the second choice of w . The Exceptional Types If G is of exceptional type then the analogous data to that found in Table 4.1,amongst other things, can be extracted from the tables of [KM13]. This data can alsobe easily recomputed using [Tay20] and commands provided in CHEVIE . Specifically, in[Tay20] we have implemented the command
JordanCuspidalLevis which can be usedto obtain the relevant Levi subgroups. We give an example in the case of E , see [KM13,Table 7, no. 19–24] gap> GF := CoxeterCoset(CoxeterGroup("E",8), ());;gap> s := SemisimpleElement(Group(GF), [0,0,0,0,0,1/3,0,0]);;gap> CG := Centralizer(Group(GF), s).group;;gap> w := LongestCoxeterElement(CG)*LongestCoxeterElement(G);;gap> CGF := CoxeterSubCoset(GF, InclusionGens(CG), w);2E6<1,5,3,4,2,96>x2A2gap> JordanCuspidalLevis(CGF);[ [ E8, 2E6<1,5,3,4,2,96>x2A2 ],[ E7<6,2,5,4,3,1,74>.(q-1), 2E6<1,5,3,4,2,96>.(q-1)(q+1) ],[ E7<1,2,3,4,13,7,8>.(q-1), 2A5<1,3,4,2,96>x2A2<7,8>.(q-1) ],[ D6<2,13,4,3,1,74>.(q-1)^2, 2A5<1,3,4,2,96>.(q-1)^2(q+1) ],[ D4<26,8,7,27>.(q-1)^4, 2A2<7,8>.(q-1)^4(q+1)^2 ],[ A1<26>xA1<27>xA1<74>.(q-1)^5, (q-1)^5(q+1)^3 ] ] Table 4.1: Isolated Elements in Classical Types. G F C ◦ G ( s ) nF p ? L nF C ◦ L ( s ) nF Conditions B n ( q ) B n ( q ) − B m ( q ) · Φ n − m1 B m ( q ) · Φ n − m1 m ∈ △ D a ( q ) · B n − a ( q ) ( a n ) = B m ( q ) · Φ n − m1 D d ( q ) · B m − d ( q ) · Φ n − m1 d ∈ (cid:3) , m − d ∈ △ D a ( q ) · B n − a ( q ) ( a n ) = B m + ( q ) · Φ n − m − B m ( q ) · Φ n − m − · Φ m ∈ △ B m ( q ) · Φ n − m1 2 D d ( q ) · B m − d ( q ) · Φ n − m1 d ∈ (cid:3) , m − d ∈ △ C n ( q ) C n ( q ) − C m ( q ) · Φ n − m1 C m ( q ) · Φ n − m1 m ∈ △ C a ( q ) · C n − a ( q ) ( a n/2 ) = C m ( q ) · Φ n − m1 C d ( q ) · C m − d ( q ) · Φ n − m1 d , m − d ∈ △ C n2 ( q ) ( n even ) = A ( q ) n − m2 · C m ( q ) · Φ n − m2 C m2 ( q ) · Φ n − m2 · Φ n − m2 m ∈ △ D n ( q ) D n ( q ) − D m ( q ) · Φ n − m1 D m ( q ) · Φ n − m1 m ∈ (cid:3) D a ( q ) · D n − a ( q ) ( a n/2 ) = D m ( q ) · Φ n − m1 D d ( q ) · D m − d ( q ) · Φ n − m1 d , m − d ∈ (cid:3) D a ( q ) · D n − a ( q ) ( a n/2 ) = D ( q ) · Φ n − Φ n − · Φ D d + ( q ) · Φ n − d −
11 2 D d ( q ) · Φ n − d − · Φ d ∈ (cid:3) D m + ( q ) · Φ n − m −
11 2 D m ( q ) · Φ n − m − · Φ m ∈ (cid:3) D m ( q ) · Φ n − m1 2 D d ( q ) · D m − d ( q ) · Φ n − m1 d , m − d ∈ (cid:3) ( ⋆ ) D n2 ( q ) ( n even ) = A ( q ) n − m2 · D m ( q ) · Φ n − m2 D m2 ( q ) · Φ n − m2 · Φ n − m2 m ∈ (cid:3) D n ( q ) D n ( q ) − Φ n − · Φ Φ n − · Φ D m ( q ) · Φ n − m1 2 D m ( q ) · Φ n − m1 m ∈ (cid:3) D a ( q ) · D n − a ( q ) ( a n − ) = n − · Φ Φ n − · Φ D m ( q ) · Φ n − m1 D d ( q ) · D m − d ( q ) · Φ n − m1 d ∈ (cid:3) , m − d ∈ (cid:3) D a ( q ) · D n − a ( q ) ( a n − ) = n − · Φ Φ n − · Φ D m + ( q ) · Φ n − m − D m ( q ) · Φ n − m − · Φ m ∈ (cid:3) D m ( q ) · Φ n − m1 2 D d ( q ) · D m − d ( q ) · Φ n − m1 d ∈ (cid:3) , m − d ∈ (cid:3) ( ⋆ ) D n2 ( q ) ( n even ) = A ( q ) n2 − · D ( q ) · Φ n2 − Φ n2 − · Φ n2 − · Φ A ( q ) n − m2 · D m ( q ) · Φ n − m2 D m2 ( q ) · Φ n − m2 m ∈ (cid:3) References [Bon05] C. Bonnaf´e,
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