Derived categories and Deligne-Lusztig varieties II

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Cédric Bonnafé, Jean-François Dat, Raphaël Rouquier

To cite this version:

Cédric Bonnafé, Jean-François Dat, Raphaël Rouquier. Derived categories and Deligne-Lusztig va-rieties II. Annals of Mathematics, Princeton University, Department of Mathematics, 2017, 185 (2), pp.609-670. �10.4007/annals.2017.185.2.5�. �hal-01228904v3�

VARIETIES II

by

C

B

, J

-F

D

& R

R

Abstract. —This paper is a continuation and a completion of [BoRo1]. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group — this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decom-position that can be obtained via Deligne-Lusztig varieties. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic sub-groups.

We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of [BoRo1], the finer homo-topy equivalence was unknown. As a consequence, the equivalences preserve defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.

Contents

1. Introduction . . . 2

1.A. Jordan decomposition . . . 3

1.B. Generation of the derived category . . . 5

1.C. Independence of the Deligne-Lusztig induction of the parabolic in a given series . . . 5

1.D. Structure of the article . . . 6

2. Notations . . . 7

2.A. Modules . . . 7 The first author is partly supported by the ANR (Project No ANR-12-JS01-0003-01 ACORT). The second author thanks the Institut Universitaire de France and the project ANR-14-CE25-0002-01 PerCoLaTor for their support. The third author is partly supported by the NSF (grant DMS-1161999) and by a grant from the Simons Foundation (#376202, Raphaël Rouquier).

2.B. Finite groups . . . 8

2.C. Varieties . . . 8

2.D. Reductive groups . . . 9

2.E. Deligne-Lusztig varieties . . . 9

3. Generation . . . 10

3.A. Centralizers ofℓ-subgroups . . . 10

3.B. Perfect complexes and disconnected groups . . . 13

3.C. Generation of the derived category . . . 14

4. Rational series . . . 17

4.A. Rational series in connected groups . . . 17

4.B. Coroots of fixed points subgroups . . . 19

4.C. Centralizers and rational series . . . 21

4.D. Generation and series . . . 24

4.E. Decomposition map and Deligne-Lusztig induction . . 25

5. ComparingY_-varieties . . . 28

5.A. Dimension estimates and further . . . 28

5.B. Setting . . . 31

5.C. Preliminaries . . . 33

5.D. Comparison of complexes . . . 36

6. Independence with respect to the parabolic subgroup . . . 41

6.A. Isomorphisms . . . 41

6.B. Transitivity . . . 46

7. Jordan decomposition and quasi-isolated blocks . . . 47

7.A. Quasi-isolated setting . . . 47

7.B. Clifford theory . . . 49

7.C. Embedding in a group with connected center and Morita equivalence . . . 50

7.D. Splendid Rickard equivalence and local structure . . . . 52

Appendix A. Aboutℓ-permutation modules . . . 56

References . . . 59

1. Introduction

Let G be a connected reductive algebraic group over an algebraic closure of a

finite field, endowed with an endomorphism F, a power of which is a Frobenius

endomorphism. Letℓ be a prime number distinct from the defining characteristic

ofG and _K a finite extension of _Qℓ, large enough for the finite groups considered.

LetO be the ring of integers ofK overZℓandk the residue field. We will denote by

The main tool for the study of representations ofGF _{overΛis the Deligne-Lusztig}

induction. Let Lbe an _F-stable Levi subgroup of G contained in a parabolic

sub-group P with unipotent radical Vso that P_{= V ⋊ L}. Consider the Deligne-Lusztig

variety

Y_{P={g V ∈ G/V | g}−1_{F (g ) ∈ V · F (V)}.}

It has a left action ofGF _{and a right action ofL}F _{by multiplication. The}

correspond-ing complex ofℓ-adic cohomology induces a triangulated functor

RG L⊂P: D b_(ΛLF_{)→ D}b_(ΛGF_{), M 7→ R Γ} c(YP, Λ) ⊗LΛLF M and a morphism RLG⊂P= [R G L⊂P] : G0(ΛLF)→ G0(ΛGF).

This is the usual Harish-Chandra construction whenPis_F-stable.

1.A. Jordan decomposition. — LetG∗be a group Langlands dual toG, with

Frobe-nius F∗. Consider the setIrr(GF_{) of characters of irreducible representations ofG}F

overK. Deligne and Lusztig gave a decomposition ofIrr(GF_{)into rational series}

Irr(GF) =a

(s )

Irr(GF, (s ))

where (s ) runs over the set of G∗F∗-conjugacy classes of semi-simple elements of G∗F∗. The unipotent characters ofGF _{are those inIrr(G}F_{, 1).}

Let L be an _F-stable Levi subgroup of G with dual L∗ _{⊂ G}∗ containing _CG∗(s ).

Lusztig constructed a bijection

Irr(LF, (s ))−→ Irr(G∼ F, (s )), ψ 7→ ±RLG(ψ).

Ifs ∈ Z (L∗_{), then there is a bijection}

Irr(LF, (1))−→ Irr(L∼ F, (s )), ψ 7→ ηψ

where ηis the one-dimensional character ofLF _{corresponding to s, and we obtain}

a bijection

Irr(LF, (1))−→ Irr(G∼ F, (s )).

This provides a description of irreducible characters ofGF _{in the rational series(s )}

in terms of unipotent characters of an other group, whenCG∗(s )is a Levi subgroup

ofG∗.

Let us now consider the modular version of the theory described above. Lets be

a semi-simple element of G∗F∗

of order prime to ℓ. Consider`

tIrr(G

F_{, (t )), where}

(t )runs over conjugacy classes of semi-simple elements ofG∗F∗

whoseℓ′_{-part is(s ).}

of the corresponding block idempotents is an idempotent eGF s ∈ Z (O G F_{), and we} obtain a decomposition O GF-mod =M (s ) O GFe_sGF-mod where(s )runs overG∗F∗

-conjugacy classes of semi-simpleℓ′_{-elements ofG}∗F∗

. LetLbe an_F-stable Levi subgroup ofGwith dual L∗ containing_CG∗(s ). LetPbe

a parabolic subgroup ofGwith unipotent radicalVand Levi complementL. Broué

[Br2] conjectured that the (O GF_{, O L}F_{)-bimodule H}dim YP_(Y

P, O )eL F s induces a Morita equivalence betweenO GF_eGF s andO L F_eLF

s . This was proven by Broué [Br2] whenL

is a torus and in [BoRo1] in general.

Broué also conjectured that the truncated complex of cohomology GΓc(YV, O )e LF

s

(Rickard’s refinement ofRΓc(YV, O )e LF

s , well defined in the homotopy category [Ri1])

induces a splendid Rickard equivalence between O GF_eGF

s and O L F_eLF

s : it induces

not only an equivalence of derived categories, but even an equivalence of homo-topy categories, and it induces a similar equivalence for centralizers ofℓ-subgroups.

One of our main results here is a proof of that conjecture. In order to show that there is a homotopy equivalence, for connected groups, we show that the global functor induces local derived equivalences for centralizers of ℓ-subgroups. Since

such centralizers need not be connected, we need to extend the results of [BoRo1] to disconnected groups. So, part of this work involves working with disconnected groups.

We also extend the “Jordan decomposition equivalences” (Morita and splendid Rickard) to the “quasi-isolated case”: assume now only C◦

G∗(s ) ⊂ L

∗_{, and that L}∗

is minimal with respect to this property. We show that the right action of LF _on

Hdim YP(Y

P, O )e LF

s extends to an action ofN = NGF(L, eL F

s )commuting with the action

ofGF_{, and the resulting bimodule induces a Morita equivalence betweenO G}F_eGF

s

andO N eLF

s . Similarly, the complexGΓc(YV, O )e GF

s induces a splendid Rickard

equiv-alence betweenO GF_eGF

s andO N e

LF

s .

As a consequence, we deduce that the bijection between blocks ofO GF_eGF

s and

O N eLF

s preserves the local structure, and in particular, preserves defect groups.

Cabanes and Enguehard have proven this under some assumptions on ℓ [CaEn1,

Proposition 5.1], and Kessar and Malle in the setting of [BoRo1], when one of the blocks under consideration has abelian defect groups (modulo a centralℓ-subgroup)

[KeMa1, Theorem 1.3], an important step in their proof of half of Brauer’s height zero conjecture for all finite groups [KeMa1, Theorem 1.1] and the second half for quasi-simple groups [KeMa2, Main Theorem].

Let us summarize this.

Theorem 1.1. — AssumeC◦

G∗(s ) ⊂ L

The right action of LF _{on GΓ}

c(YP, O )eL

F

s extends to an action of N and the resulting

complex C induces a splendid Rickard equivalence between O GF_eGF

s and O N e

LF

s . The

bimoduleHdim YP(C )induces a Morita equivalence betweenO GF_eGF

s andO N e

LF

s .

The bijections between blocks ofO GF_eGF

s andO N e

LF

s induced by those equivalences

pre-serve the local structure.

Significant progress has been made recently on counting conjectures for finite groups, using the classification of finite simple groups, and [BoRo1] has proved very useful. We hope this theorem will lead to simplifications and new results.

The character-theoretic consequence of this theorem is that, for groups with dis-connected center, the Jordan decomposition shares many of the properties of that for the connected case (commutation with Deligne-Lusztig induction for example). In typeA, the Jordan decomposition of characters links all series to unipotent series

of smaller groups: even in that case, the good behaviour of those correspondences was known only whenq is large (Bonnafé [Bo3] forSL_n and Cabanes [Ca] forSU_n).

1.B. Generation of the derived category. — One of the two key steps in [BoRo1]

was the proof that the category of perfect complexes for O GF _{is generated by the}

complexesRΓc(YB), whereBruns over Borel subgroups ofGwith an F-stable

maxi-mal torus. We show here a more precise result of generation of the derived category ofO GF_{. LetE be the set{RΓ}

c(YB)⊗ L

O TF M }, whereTruns overF-stable maximal tori

ofG,Bover Borel subgroups ofGcontainingT, and_M over isomorphism classes of

O TF_-modules.

Theorem 1.2. — The setE generatesDb_{(O G}F_{)(as a thick subcategory) if and only if all}

elementary abelianℓ-subgroups ofGF _{are contained in tori.}

This, in turn, requires an extension of the results of Broué-Michel [BrMi] on the compatibility between Deligne-Lusztig series of characters and the Brauer mor-phism, to disconnected groups. We are able to achieve this by refining our result on the generation of the category of perfect complexes to a generation of the cat-egory of ℓ-permutation modules whose vertices are contained in tori (the crucial

case is that of connected groups). Such a result allows us to obtain a generating re-sult for the full derived category, under the assumption that all elementary abelian

ℓ-subgroups are contained in tori.

Note that the condition is automatically satisfied for GL_n and U_n (see Examples

3.17) and whenℓis very good forG.

1.C. Independence of the Deligne-Lusztig induction of the parabolic in a given

series. — It is known in most cases, and conjectured in general, that the mapRG

L⊂P

and [BoMi] whenq > 2andF is a Frobenius endomorphism overF_q). On the other

hand, the functorRG

L⊂Pdoes depend onP. Our main new geometrical result proves

the independence after truncating by a suitable series.

LetP₁andP₂be two parabolic subgroups admitting a common Levi complement L. Denote byV∗

i the unipotent radical of the parabolic subgroup ofG

∗_{corresponding}

toP_i.

Theorem 1.3. — Lets be a semi-simple element ofL∗F∗ _{of order prime toℓ. If}

CV∗ 1∩F ∗V ∗ 1(s ) ⊂ CV ∗ 2(s ) and CV ∗ 2∩F ∗V ∗ 2(s ) ⊂ CF ∗V ∗ 1(s )

then there is an isomorphism of functors between R_LG_⊂P 1: D b_(ΛLF e_sLF)−→ Db(ΛGFe_sGF) and RLG⊂P2[m] : D b_(ΛLF_eLF s )−→ D b_(ΛGF_eGF s ), wherem = dim(YG P₂)− dim(Y G P₁). For instance, ifCV∗ 1(s ) = CV ∗

2(s ), then the assumption of Theorem 1.3 is satisfied.

This is the key result to prove Theorem 1.1. This result shows that whenCG◦∗(s ) ⊂ L∗, the _{(O G}F_{, O L}F_{)-bimodule H}dim YP_(Y

P, O )eL

F

s is independent of P, a question left

open in [BoRo1]. We deduce that the bimodule is stable under the action of N = NGF(LF, eL

F

s ). Using an embedding in a group with connected center, we show that

the obstruction for extending the action ofLF to_N does vanish.

Remark.— Theorem 1.3 is used in [Dat] to construct equivalences of categories

be-tween tamely ramified blocks ofp-adic general linear groups. Roughly speaking,

the main idea of [Dat] is to “glue” the bimodules giving the Morita equivalences of [BoRo1] along a suitable building. The gluing process crucially uses the indepen-dence of the bimodules on the choice of parabolic subgroups.

1.D. Structure of the article. — We begin in §3 with the study of generation of

the category of perfect complexes, then we move to complexes of ℓ-permutation

modules and finally we derive our result on the derived category. A key tool, due to Rickard, is that the Brauer functor applied to the complex of cohomology of a variety is the complex of cohomology of the fixed point variety.

Section §4 is devoted to the study of rational series and their compatibility with local block theory. Broué and Michel proved a commutation formula between generalized decomposition maps and Deligne-Lusztig induction. We need to ex-tend the compatibility between Brauer and Deligne-Lusztig theory to disconnected

groups, and check that the local blocks obtained from a series satisfyingCG◦∗(s ) ⊂ L ∗

also satisfy a similar assumptionC_(C◦ ◦

G(Q ))∗(s ) ⊂ (L ∩ C

◦

G(Q ))

∗_.

From §5 onwards, the group G is assumed to be connected. Sections §5 and §6

are devoted to the study of the dependence of the Deligne-Lusztig induction with respect to the parabolic subgroup. The first section is devoted to the particular case of varieties associated with Borel subgroups (and generalizations involving sequences of elements). It is convenient there to work with a reference maximal torus. This is the crucial case, from which the general one is deduced in the latter section, where we go back to Levi subgroups that do not necessarily contain that fixed maximal torus.

The final section §7 is devoted to the Jordan decomposition. We start by provid-ing an extension of the action ofN on the cohomology bimodule by proving that the

cocycle obstruction would survive in a similar setting for a group with connected center, where the action does exist. The Rickard equivalence is obtained inductively, and that induction requires working with disconnected groups.

In an appendix, we provide some results on the homotopy category of complexes ofℓ-permutation modules for a general finite group.

We would like to thank the Referee for an extraordinarily thorough list of sug-gestions, which greatly improved our paper.

2. Notations

2.A. Modules. — Letℓbe a prime number,K a finite extension ofQℓlarge enough

for the finite groups considered,O its ring of integers overZℓandk its residue field.

We will denote byΛa ring that is either _K,_O or_k.

GivenC an additive category, we denote byCompb(C )the category of bounded complexes of objects ofC and byHob_{(C )its homotopy category.}

LetAbe aΛ-algebra, finitely generated and projective as aΛ-module. We denote

by Aopp _{the algebra opposite to A. We denote by A-mod the category of finitely}

generatedA-modules and byA-projits full subcategory of projective modules. We

denote byG0(A)the Grothendieck group of A-mod.

We putCompb(A) = Compb(A-mod),Db_{(A) = D}b_(A-mod)andHob_{(A) = Ho}b_(A-mod).

We denote byA-perf ⊂ Db_{(A)the thick full subcategory of perfect complexes}

(com-plexes quasi-isomorphic to objects ofCompb(A-proj)).

LetC ∈ Compb(A). There is a unique (up to a non-unique isomorphism) complex

non-zero direct summand that is homotopy equivalent to0. Note thatC ≃ Cred_{⊕ C}′

for someC′homotopy equivalent to zero.

We denote byEnd•_A(C )the totalHom-complex, with degreen termL

j −i =nHomA(Ci, Cj).

Let B be Λ-algebra, finitely generated and projective as a Λ-module. Let_C be a

bounded complex of (A ⊗ΛBopp)-modules, finitely generated and projective as left

A-modules and as right B-modules. We say that C induces a Rickard equivalence

between Aand B if the canonical mapB → End•_A(C )is an isomorphism in Ho(B ⊗Λ

Bopp_{)and the canonical mapA → End}•

Bopp(C )opp is an isomorphism inHo(A ⊗ΛAopp).

2.B. Finite groups. — Let G be a finite group. We denote by Gopp _{the opposite}

group toG. We put∆_{G = {(g , g}−1_{)|g ∈ G } ⊂ G × G}opp_{. Giveng ∈ G, we denote by|g |}

the order ofg.

Let H be a subgroup ofG and x ∈ G. We denote by x∗ the equivalence of

cate-gories

x∗: Λ(x−1H x )-mod ∼

−→ ΛH -mod

where x∗(M ) = M as aΛ-module and the action ofh ∈ H on x∗(M )is given by the

action of x−1_{h x on M. We also denote by x}

∗ the corresponding isomorphism of

Grothendieck groups

x∗: G0(Λ(x −1

H x )) −→ G∼ ₀(ΛH ). We assumeΛ_=O orΛ_{= k} in the remainder of §2.B.

Anℓ-permutationΛ_G-moduleis defined to be a direct summand of a finitely

gener-ated permutation module. We denote byΛ_{G -perm}the full subcategory ofΛ_{G -mod}

with objects theℓ-permutationΛ_G-modules.

LetQ be anℓ-subgroupQ ofG. We consider the Brauer functorBrQ : ΛG -perm →

k [NG(Q )/Q ]-perm. Given M ∈ ΛG -perm, we define BrQ(M ) as the image of MQ in

(k M )Q, where(k M )Q is the largest quotient ofk M = k ⊗ΛM on whichQacts trivially.

We denote bybrQ : (ΛG )Q → k CG(Q )the algebra morphism given bybrQ(

P

g ∈Gλgg ) =

P

g ∈CG(Q )λgg whereλg ∈ Λforg ∈ G. GivenM ∈ ΛG -permand e ∈ Z (ΛG ) an

idem-potent, we haveBrQ(M e ) = BrQ(M )brQ(e ).

LetH be a subgroup ofG, letb be an idempotent ofZ (ΛG )andc an idempotent

ofZ (ΛH ). LetC ∈ Compb(ΛG b ⊗ (ΛH c )opp_{). We say thatC is splendid if the (C}red₎i_’s

areℓ-permutation modules whose indecomposable direct summands have a vertex

contained in∆_H.

2.C. Varieties. — Let p be a prime number different from ℓ and F an algebraic

closure ofFp. By variety, we mean a quasi-projective algebraic variety overF.

Let X be a variety acted on by a finite group _G. There is an object _{GΓc(X, Λ)} of

Hob_{(ΛG -perm), well defined up to a unique isomorphism. It is a representative in}

étaleΛ-cohomology with compact support ofXconstructed as_{τ¶ 2 dim X}of the

Gode-ment resolution (cf [Rou1, §2], [DuRou, §1.2], and [Ri1]). We denote by RΓc(X, Λ)

the image ofGΓc(X, Λ)inDb(ΛG ).

Assume Λ _{= O} or _k and let_Q be an _ℓ-subgroup of _G. The inclusion XQ _{,→ X}

induces an isomorphism [Ri1, Theorem 4.2] GΓc(XQ, k )

∼

−→ BrQ(GΓc(X, Λ))in Hob(k NG(Q )-perm).

2.D. Reductive groups. — LetG be a (possibly disconnected) reductive algebraic

group endowed with an endomorphism F, a power Fδ _{of which is a Frobenius}

endomorphism defining a rational structure over a finite field Fq of characteristic

p. We refer to [DigMi2, DigMi3] for basic results on disconnected groups.

Recall that a torus of G is torus of G◦. Following the classical terminology (cf

for example [Sp, §6.2]), we define a Borel subgroup ofG to be a maximal connected

solvable subgroup of G. We define a parabolic subgroup of G to be a subgroup P

of G such that G_/P is complete. We define the unipotent radical V of a parabolic

subgroup P to be its unique maximal connected unipotent normal subgroup. A

Levi complementtoVinPis a subgroupLofPsuch thatP_{= V ⋊ L}.

Note that a closed subgroupPofGis a parabolic subgroup ofGif and only ifP◦_is

a parabolic subgroup ofG◦_{. LetPbe a parabolic subgroup ofG. We haveP}◦_{= P∩G}◦_.

The unipotent radicalVofPcoincides with that ofP◦_{. A Levi complement to Vin} Pis a subgroup of the form _NP(L◦), whereL_◦ is a Levi complement ofVinP_◦ (then L◦_{= L◦}andP_{= V⋊L}). Note that our definition of parabolic subgroup is more general

than that of “parabolic” subgroup of [DigMi2], which requiresP_{= NG(P}◦₎.

We denote by∇(G, F )the set of pairs(T, θ )whereTis an _F-stable maximal torus

ofGand_θ is an irreducible character ofTF_{. Note that hereTis a torus ofG}◦_.

Given an integer d, we denote by ∇_d′(G, F ) the set of pairs(T, θ ) ∈ ∇(G, F ) such

that the order ofθ is prime tod. We put∇Λ(G, F ) = ∇(G, F )ifΛ= K and∇Λ(G, F ) =

∇ℓ′(G, F )ifΛ=O ork (recall thatk is a field of characteristicℓ).

2.E. Deligne-Lusztig varieties. — GivenPa parabolic subgroup ofGwith

unipo-tent radicalVand_F-stable Levi complementL, we define the Deligne-Lusztig

vari-ety Y_{V= Y}G V= YP= Y G P={g V ∈ G/V | g −1_{F (g ) ∈ V · F (V)}.}

This is a smooth variety, as in the case of connected reductive groups. It has a left action by multiplication of GF _{and a right action by multiplication ofL}F _{(note that}

the left and right actions ofZ (G)F _{coincide). This provides a triangulated functor}

(2.1) RLG⊂P: D

b_(ΛLF_{) −→ D}b_(ΛGF₎

and a morphism RLG⊂P= [R G L⊂P] : G0(ΛLF)→ G0(ΛGF). We putXG P={g P ∈ G/P | g −1_{F (g ) ∈ P · F (P)} = Y}G P/L F_.

Remark 2.2. — SinceY_Pdepends only onV, it is endowed with an action of_NGF(P◦, L◦),

which is the group of rational points of the maximal Levi subgroup with connected componentL◦.

3. Generation

The aim of this section is to extend [BoRo1, Theorem A] to the case of discon-nected groups, and to deduce a generation theorem for the derived category.

In this section §3,Gis a (possibly disconnected) reductive algebraic group.

3.A. Centralizers ofℓ-subgroups. — LetPbe a parabolic subgroup ofGadmitting

an F-stable Levi complement L, and letV denote the unipotent radical of P. It is

easily checked [DigMi2, Proof of Proposition 2.3] that

(3.1) YG V= a g ∈GF_/G◦F g YGV◦= G F _× G◦F Y G◦ V .

It follows immediately from (3.1) that (3.2) R_LG_⊂P◦ IndL_LF◦F ≃ R G L◦_⊂P◦≃ Ind GF G◦F◦R G◦ L◦_⊂P◦.

IfG_{= P· G}◦, then the isomorphismG◦_{/V ×L}◦L≃ G/Vinduces an isomorphism

(3.3) RLG◦◦_⊂P◦◦ Res LF L◦F ≃ Res GF G◦F◦R G L⊂P.

Proposition 3.4. — LetQ be a finite solvablep′_{-group of automorphisms ofG that}

com-mute withF and normalize(P, L). (a) The groupGQ _{is reductive.}

(b) PQ _{is a parabolic subgroup ofG}Q _{whose unipotent radical isV}Q _{and admittingL}Q _as

anF-stable Levi complement. In particular,VQ _{is connected.}

(c) The natural mapGQ_/VQ_{→ (G/V)}Q _{is an isomorphism of(G}Q_{, N}

G(P◦, L◦)Q)-varieties.

(d) (V ·F_V)Q _{= V}Q_·F_(VQ_).

(e) The natural map YGQ

VQ → (Y

G V)

Q _{is an isomorphism of ((G}Q₎F_{, (N}

G(P◦, L◦)Q)F)-varieties.

IfQ is anℓ-group, it gives rise to an isomorphism BrQ(GΓc(Y

G V, k )) ∼ −→ GΓc(Y GQ VQ, k )in Hob_{(k ((G}Q₎F _{× (N} G(P◦, L◦)Q)F opp).

Proof. — Assume firstQ is cyclic, generated by an elementl.

(a) and (b) follow from [DigMi2, Proposition 1.3, Theorem 1.8, Proposition 1.11]. (c) Note that that both varieties are smooth (for(G/V)Q_{, this follows from the fact}

thatQ is ap′_{-group andG/Vis smooth). The injectivity of the map is clear.}

Let us prove the surjectivity. Letg V ∈ (G/V)Q_{. Then,g}−1_{l (g ) ∈ V. Denote byad(g )}

the automorphism x 7→ g x g−1 _{ofG. Sincead(g )}−1_{l ad(g ) is semisimple, it stabilizes}

a maximal torus of P◦ (see [St, Theorem 7.5]), hence it stabilizes the unique Levi

complement L′ of P◦ containing this maximal torus. Since all Levi complements

are conjugate under the action of V, there exists _{v ∈ V} such that _v−1L′_{v = L}◦. It

follows that(g v )−1l (g v ) ∈ Vand(g v )−1l (g v )normalizesL◦, hence_{(g v )}−1_{l (g v ) = 1}, so

g v ∈ GQ_{, as desired.}

The tangent space at V of_(G/V)Q _{is the Q-invariant part of the tangent space of} G_/VatV. That last tangent space is a quotient of the tangent space ofGat the origin.

It follows that the canonical map GQ _{→ (G/V)}Q _{induces a surjective map between}

tangent spaces at the origin. Consequently, the canonical map GQ_/VQ _{→ (G/V)}Q

induces a surjective map between tangent spaces at the origin. We deduce that the map is an isomorphism.

(d) The number of F-stable maximal tori of L is a power of _p (see [St,

Corol-lary 14.16]). SinceQ is ap′-group, it normalizes some F-stable maximal torus.

Us-ing now the root system with respect to this maximal torus, we deduce that there exists aQ-stable subgroupV′ofVsuch thatV_{= V}′_{· (V∩ F (V))}andV′_{∩ F (V) = 1}.

There-fore,V_{· F (V) = V}′_{· F (V)}and the result follows.

(e) follows immediately from (c) and (d).

We prove now the proposition by induction on|Q |. LetQ1be a normal subgroup

ofQ of index a prime number and let l ∈ Q, l 6∈Q1. LetQ2 be the subgroup of Q

generated byl. By induction, the proposition holds forQ replaced byQ1: we have a

reductive groupG_{1= G}Q1 _{and a parabolic subgroupP}

1= PQ1 with unipotent radical V_{1 = V}Q1 _{and an F-stable Levi complement L}

1 = LQ1. These are all stable under

Q2. The cyclic case of the proposition applied to the action ofQ2 on (G1, P1, V1, L1)

establishes the proposition for the action ofQ on(G, P, V, L).

Remark 3.5. — IfQ is a finite solvablep′-subgroup ofGF, then_{NG(Q )}is reductive.

If in additionQ normalizes(P, L), thenNP(Q )is a parabolic subgroup ofNG(Q )with

unipotent radicalVQ _{and Levi complementN}

L(Q ). The maps defined in (e) of

Propo-sition 3.4 are equivariant for the diagonal action ofNG(P◦, L◦,Q )F.

Lemma 3.6. — LetQ be a finite solvable p′-group of automorphisms of G_{that commute}

withF. We assumeQ stabilizes a maximal torus ofG_{and a Borel subgroup containing that}

maximal torus.

LetT_{Q be anF-stable maximal torus of}GQ _{contained in a Borel subgroup}B_{Q of}GQ_{. Then,}

CG◦(T_Q)is an F-stable maximal torus ofGthat is contained in aQ-stable Borel subgroupB

ofG_{such that(B}Q₎◦_{= B} Q.

Proof. — Note that the lemma holds for Q cyclic by [DigMi2, Theorem 1.8]. We

proceed by induction on |Q | as in the proof of Proposition 3.4, and we keep the notations of that proof. We know (Lemma forQ2) that TQ1 = CG◦₁(TQ)is an F-stable

maximal torus ofG◦

1. By induction, CG◦(T_Q) = NG◦(T_Q1)◦= CG◦(T_Q1)is anF-stable

max-imal torus ofG.

The existence of the Borel subgroup can be obtained as in [DigMi2, p.350]. LetT′

be a maximal torus ofGstable under_Q andB′be a Borel subgroup ofGcontaining T′and stable under_Q. By Proposition 3.4,(T′Q₎◦ _{is a maximal torus ofG}Q _and(B′Q₎◦

is a Borel subgroup containing it. So, there is x ∈ (GQ₎◦ _{such that T}

Q =x(T′Q)◦ and B_{Q =}x_(B′Q₎◦_{. LetB=}x_B′_{. This is aQ-stable Borel subgroup ofGcontainingC}

G◦(T_Q).

By Proposition 3.4,(BQ₎◦ _{is a Borel subgroup ofG}Q_{, hence(B}Q₎◦_{= B} Q.

To complete Proposition 3.4, note the following result.

Lemma 3.7. — LetP be anℓ-subgroup of GF _{× N}

GF(P, L)opp such that(YG_V)P 6= ∅. ThenP

is(GF _{× 1)-conjugate to a subgroup of∆N}

GF(P, L).

Proof. — ReplacingLby_{NG(P, L)}, we can assume that_{NG(P, L) = L}.

LetQ ⊂ LF _{(respectivelyR ⊂ G}F_{) denote the image of P through the second}

(re-spectively first) projection and lety V ∈ (YG V)

P_.

Ifg ∈ R, then there existsl ∈ Q such that(g , l ) ∈ P. Therefore, g y l V = y V, hence y−1_{g y V = l}−1_{V. This implies that y}−1_{R y ⊂ Q V. We denote byη : R → Q the}

composi-tionR −→ y∼ −1_{R y ,→ Q V ։ Q. SinceR (respectivelyQ) acts freely onG/Vas they are}

ℓ-groups, the previous computation shows thatηis an isomorphism, and that P = {(g , η(g )) | g ∈ R }.

Now, there exists a positive integer m such that Fm_{(P) = P and y}−1_{R y ⊂ P}Fm

. So y−1R y acts by left translation on PFm_/LFm_{. Since y}−1_{R y is a finite ℓ-group and}

|PFm

/LFm

| = |VFm

|is a power ofp, it follows thaty−1R y has a fixed point inPFm_/LFm.

Consequently, there exists v ∈ V such that y−1_{l y v L = v L for all l ∈ R. In other}

words, (y v )−1_{R (y v ) ⊂ L. This means that, by replacing y by y v if necessary, we}

Now, y−1F (y ) ∈ V · F (V)but, sinceF (y l y−1) = y l y−1 for alll ∈ Q, we deduce that y−1F (y ) ∈ CG(Q ). So

y−1F (y ) ∈ (V · F (V)) ∩ CG(Q ) = CV(Q ) · F (CV(Q )) ⊂ CG◦(Q )

(see Proposition 3.4(b) and (d)). So, by Lang’s Theorem, there exists x ∈ CG◦(Q )such

that y−1F (y ) = x−1F (x ). This implies thath = y x−1∈ GF_{, and}

P = {(hl h−1, l ) | l ∈ Q }, as expected.

Corollary 3.8. — The indecomposable summands of theO (GF_×N

GF(P, L)opp)-moduleGΓ_c(YG_V, O )red

have a vertex contained in∆_{NGF(P, L).}

Proof. — LetQbe anℓ-subgroup ofGF_×N

GF(P, L)oppthat is not(GF×1)-conjugate to a

subgroup of∆_NGF(P, L). We haveBr_Q(GΓ_c(YG_V, O )) ≃ GΓ_c((YG_V)Q, k ) ≃ 0inHob(k NGF_×(LF₎opp(Q ))

by Lemma 3.7. The result follows now from Lemma A.2.

3.B. Perfect complexes and disconnected groups. — GivenM a simpleΛGF-module,

we denote byY (M )the set of pairs(T, B)such thatTis an_F-stable maximal torus of GandBis a (connected) Borel subgroup ofGcontainingTsuch that_RHom•

ΛGF(R Γc(YB, Λ), M ) 6=

0. We then setd (M ) = min(T,B)∈Y (M )dim(YB). The following two theorems are proved

in [BoRo1, Theorem A] wheneverGis connected.

Theorem 3.9. — Let M be a simple ΛGF_{-module. Then Y (M ) 6= ∅. Moreover, given}

(T, B) ∈ Y (M )such thatd (M ) = dim(YB), we have

HomDb_(ΛGF₎(R Γ_c(YB, Λ), M [−i ]) = 0 for alli 6= d (M ). Proof. — By (3.2), we have HomDb_(ΛGF₎(R Γ_c(Y G B, Λ), M [−i ]) = HomDb(ΛG◦F)(R Γc(Y G◦ B, Λ), Res GF G◦FM [−i ]).

Since M is simple and G◦F _{Ã G}F_{, it follows that Res}GF

G◦F M is semisimple. Since the

theorem holds inG◦F (see [BoRo1, Proof of Theorem A]), we know that_{Y (M )}is not

empty. The second statement follows from the fact that, if two simpleΛG◦F-modules M1andM2occur in the semisimple moduleRes

GF

G◦FM, then they are conjugate under GF_{, and sod (M}

Theorem 3.10. — The triangulated categoryΛGF_{-perfis generated by the complexesR Γ}

c(YB, Λ),

whereT_{runs over the set ofF-stable maximal tori of}G_andB_{runs over the set of Borel}

sub-groups ofG_containingT_.

3.C. Generation of the derived category. — In this subsection §3.C, we assume

Λ_=O or_k. We refer to Appendix A for the needed facts about_ℓ-permutation

mod-ules.

Let Q be an ℓ-subgroup of GF _{and let M be an indecomposable ℓ-permutation}

Λ_[GF _{× Q}opp_]-module with vertex ∆_Q. We denote by _{Y [M ]} the set of pairs _{(T, B)}

satisfying the following conditions:

– Tis an_F-stable maximal torus ofGcontained in a Borel subgroupBofGsuch

thatQ normalizes(T, B)

– M is a direct summand of a term of the complex ResGF×NGF(B,T)opp

GF_×Qopp GΓc(YB, Λ)

red . We setd [M ] = min(T,B)∈Y [M ]dim(Y

CG◦(Q )

C◦

B(Q )).

Lemma 3.11. — If Q normalizes a pair (T ⊂ B) where T _{is an F-stable maximal torus}

and B_{a Borel subgroup of}G_{, thenY [M ] 6= ∅. Moreover, given(T, B) ∈ Y [M ] such that}

d [M ] = dim(YC◦

B(Q )), the degreei term of the complex Res

GF_×N

GF(B,T)opp

GF_×Qopp GΓc(YB, Λ)

red

has no direct summand isomorphic toM ifi 6= d [M ].

Proof. — Note that NGF_×Qopp(∆Q ) = (CG(Q )F × 1)∆Q, and we identify CG(Q )F with

NGF_×Qopp(∆Q )/∆Q via the first projection. LetV = Br∆_Q(M ), an indecomposable

pro-jectivek CG(Q )F-module. LetL be the simple quotient ofV.

Now, let B_Q be a Borel subgroup of _{CG(Q )} admitting an _F-stable maximal torus T_Q. By Lemma 3.6,_CG(TQ)◦ _{is anF-stable maximal torus ofGand it is contained in}

a Borel subgroupBofGsuch thatB_{Q = C}◦

B(Q ). We setD = ResCG(Q ) F_×TF opp Q CG(Q )F×1 GΓc(Y CG(Q ) B_Q , k )
red

. By Proposition 3.4(e), we have Br∆_Q(GΓ_c(Y G B, Λ)) ≃ GΓc (Y G B) ∆_Q , k
≃ GΓc(Y CG(Q ) B_Q , k ) ≃ D

inHob(k CG(Q )F). It follows from Lemma A.2 thatM is a direct summand of thei-th

term of ResG F_×TF opp Q GF_×Qopp GΓ_c(Y G B, Λ)
red

if and only ifV is a direct summand ofDi_{. So the}

result follows from Theorem 3.9. Note thatd [M ] = d [V ] = d (L ) = dim YCG(Q )

B_Q .

Recall that given a Borel subgroupBofG with an_F-stable maximal torus T, the

Let A be the thick subcategory of Hob(ΛGF_{) generated by the complexes of the}

form

GΓc(YB, Λ) ⊗ΛQL ,

where

– Truns over_F-stable maximal tori ofG

– Bruns over Borel subgroups ofGcontainingT

– Q is anℓ-subgroup ofNGF(T, B)

– andL is aΛ_Q-module, free of rank₁overΛ.

Let B be the full subcategory of ΛGF_{-mod consisting of modules whose}

inde-composable direct summands have a one-dimensional source and a vertexQ which

normalizes a pair (T⊂ B)where Tis an _F-stable maximal torus andBa Borel

sub-group ofG.

Theorem 3.12. — We haveA = Hob_(B).

Proof. — GivenN an indecomposableΛGF_{-module with a one-dimensional source}

L and a vertexQ which normalizes a pair(T⊂ B), where Tis an _F-stable maximal

torus andBa Borel subgroup, we set_{d [N ]}to be the minimum of the numbers_{d [M ]},

whereM runs over the set of indecomposableℓ-permutationΛ_(GF _{× Q}opp₎-modules

with vertex∆_Q and such that_N is a direct summand of_{M ⊗ΛQ L}.

Note that ifM is an indecomposableℓ-permutationΛ_(GF_×Qopp₎-module with

ver-tex properly contained in∆_Q, then the indecomposable direct summands of_{M ⊗ΛQ} L have vertices of size< |Q | and a one-dimensional source. Since theΛ_(GF _{× Q}opp₎

-moduleΛ_G is a direct sum of indecomposable modules with vertices contained in ∆_Q, we deduce that there is an indecomposable_ℓ-permutationΛ_(GF_×Qopp_)-module

M with vertex∆_Q and such that_N is a direct summand of_{M ⊗ΛQ L}.

We now proceed by induction on the pair(|Q |, d [N ])(ordered lexicographically) to show thatN ∈ A. FixM an indecomposableℓ-permutationΛ_(GF _×Qopp_)-module

M with vertex∆_Q and such that _N is a direct summand of _{M ⊗ΛQL}, with_{d [N ] =} d [M ]. Let(T, B) ∈ Y [M ]be such thatdim(YB) = d [M ]and letD = Res

GF_×N GF(T,B)opp GF_×Qopp GΓ_c(Y G B, Λ)
red . If i 6= d [M ], then Lemma 3.11 and Corollary 3.8 show that the indecomposable

direct summandsM′ofDi _{have vertices of size< |Q |, or have vertex∆Q and satisfy}

d [M′]< d [M ]. Therefore, the indecomposable direct summandsN′ofDi_⊗

ΛQL have

vertices of size< |Q |or have vertexQ and satisfyd [N′]< d [N ]. We deduce from the

induction hypothesis thatDi_⊗

Λ_Q L ∈ A fori 6= d [N ]. SinceN is a direct summand

Corollary 3.13. — Assume that every elementary abelian ℓ-subgroup of GF _normalizes

a pair(T⊂ B) whereT_{is an F-stable maximal torus and}B_{a Borel subgroup of} G_{. Then}

Db(ΛGF_{) is generated, as a triangulated category closed under direct summands, by the}

complexesRΓc(YB, Λ) ⊗Λ_QL, whereTruns over the set ofF-stable maximal tori ofG,Bruns

over the set of Borel subgroups of G _containing T_{, Q runs over the set of ℓ-subgroups of}

NGF(T, B)andL runs over the set of (isomorphism classes) ofΛQ-modules which are free of

rank1overΛ_.

Proof. — Since the categoryDb_(ΛGF_{)is generated, as a triangulated category closed}

under taking direct summands, by indecomposable modules with elementary abelian vertices and one-dimensional source [Rou3, Corollary 2.3], the statement follows from Theorem 3.12.

Remark 3.14. — It is easy to show conversely that if Db_(ΛGF_{)is generated by the}

complexes RΓc(YB, Λ) ⊗Λ_Q L as in Corollary 3.13, then Db(ΛGF) is generated by

in-decomposable modules with a one-dimensional source and an elementary abelian vertex that normalizes a pair(T⊂ B)whereTis an_F-stable maximal torus andBa

Borel subgroup.

In particular, the generation assumption for Λ _{= k} implies that all elementary

abelianℓ-subgroups ofGF _{are contained in maximal tori.}

The particular caseGF _{= GL}

n(Fq)(for arbitraryn) is enough to ensure thatDb(H )

is generated by indecomposable modules with elementary abelian vertices and one-dimensional source, for any finite group H — this fact is a straightforward

conse-quence of Serre’s product of Bockstein’s Theorem, but we know of no other proof. It would be interesting to find a direct proof of that result forGL_n(Fq).

Recall that an element ofG0(ΛGF)is uniform if it is in the image of

P

TR G

T(G0(ΛTF)),

whereTruns over the set of_F-stable maximal tori ofG.

One can actually describe exactly which complexes are “uniform”.

Corollary 3.15. — Let T be the full triangulated subcategory ofDb_(ΛGF_{) generated by}

the complexesRΓc(YB, Λ) ⊗ΛNGF(T,B)M whereTruns over the set of F-stable maximal tori of

G_,B_{runs over the set of Borel subgroups of}G_containingT_{andM runs over the set of}

(iso-morphism classes) of finitely generatedΛ_{NGF(T, B)-modules. Assume that every elementary}

abelian ℓ-subgroup of GF _{normalizes a pair (T⊂ B)whereTis an F-stable maximal torus}

andB_{a Borel subgroup of}G_.

An objectC ofDb_(ΛGF_{)is inT if and only if[C ] ∈ G}

0(ΛGF)is uniform.

Proof. — The statement follows from Corollary 3.13 and from Thomason’s classifi-cation of full triangulated dense subcategories [Tho, Theorem 2.1].

Remark 3.16. — Note that Corollary 3.15 holds also forΛ_{= K}: in the proof,

Corol-lary 3.13 is replaced by Theorem 3.10.

Examples 3.17. — (1) IfG_{= GLn(F)}orSL_n(F), then all abelian subgroups consisting

of semisimple elements are contained in maximal tori. This just amounts to the classical result in linear algebra which says that a family of commuting semisimple elements always admits a basis of common eigenvectors.

(2) Assume Gis connected. Let_π(G)denote the set of prime numbers which are

bad for G or divide _|(Z(G∗_)/Z(G∗₎◦₎F∗_{|. If ℓ 6∈ πand if t is an ℓ-element of G}F_{, then}

CG(t )is a Levi subgroup ofGandπ(CG(t )) ⊂ π(G)[CaEn2, Proposition 13.12(iii)]. An

induction argument shows the following fact.

(3.18) Ifℓ 6∈ π, then all abelianℓ-subgroups ofGF _{are contained in maximal tori.}

So Corollary 3.13 can be applied ifℓ 6∈ π. This generalizes (1).

Counter-example 3.19. — Assume here, and only here, that ℓ = 2 (so that p 6= 2)

and thatG_{= PGL2(F)}. Let _t (respectively _t′_{) denote the class of the matrix}1 0

0 −1 ‹

(respectively 0 1 1 0 ‹

) in G. Then _{〈t , t}′_〉 is an elementary abelian ₂-subgroup of G

which is not contained in any maximal torus of G(indeed, since G has rank ₁, all

finite subgroups of maximal tori ofGare cyclic).

4. Rational series

4.A. Rational series in connected groups. — We assume in this subsection §4.A

thatGis connected.

Let d be a positive integer divisible by δ and such that (w F )d_{(t ) = t}qd /δ

for all

t ∈ Tandw ∈ NG(T). Letζbe a generator ofF_q×d /δ. Recall [DigMi1, Proposition 13.7]

that the map

N : Y (T) −→ TF

λ 7−→ NFd_/F(λ(ζ)) = λ(ζ)F(λ(ζ)) · · ·F d −1

(λ(ζ))

is surjective and it induces an isomorphismY (T)/(F − 1)(Y (T))−→ T∼ F_{. The morphism}

Y (Y) × X (T) → K×, (λ, µ) 7→ ζ〈µ,λ+F (λ)+···+Fd −1(λ)〉

factors throughN × 1and induces a morphismTF _{× X (T) → K}×_{. The corresponding}

morphism X (T) → Hom(TF_{, K}×_{) = Irr(T}F_{)is surjective and induces an isomorphism}

Let (G∗, T∗, F∗) be a triple dual to (G, T, F ) [DeLu, Definition 5.21]. The isomor-phisms X (T)/(F − 1)(X (T))−→ Irr(T∼ F_{)and X (T)/(F − 1)(X (T)) = Y (T}∗_)/(F∗_{− 1)(Y (T}∗_))−→∼ T∗F induce an isomorphism_Irr(TF)−→ T∼ ∗F∗

.

Let(T, θ ) ∈ ∇(G, F )and letΦ(respectivelyΦ∨_{) denote the root (respectively coroot)}

system ofGrelative toT.

We setθY _{=θ ◦ N : Y (T) → K}×_and

Φ∨_{(θ ) = Φ}∨_{∩ Ker(θ}Y_).

Note thatΦ∨_{(θ )is closed and symmetric, hence it defines a root system. We}

de-note byWG◦(T, θ )its Weyl group. It is a subgroup of the Weyl groupNG(T)/Tand it

is contained in the stabilizerWG(T, θ )ofθY.

This can be translated as follows in the dual group [DigMi1, Proposition 2.3]. Let

s ∈ T∗F∗

be the element corresponding to θ. Identifying the coroot systemΦ∨ with

the root system ofG∗_{, we obtain that}

Φ∨_{(θ ) = {α}∨_{∈ Φ}∨ _{| α}∨_{(s ) = 1}}

is the root system ofCG∗(s ). IfV∗ is a unipotent subgroup of G∗ normalized byT∗,

then CV∗(s ) is generated by the one-parameter subgroups of G∗ normalized by T∗,

contained inV∗_{, and corresponding to elements ofΦ}∨_{(θ ).}

The group W◦

G(T, θ ) is identified with the Weyl group W

◦_(T∗_{, s )ofC}◦

G∗(s ) relative

toT∗while_{WG(T, θ )}is identified with the Weyl group_{W (T}∗_{, s )}of_CG∗(s ).

Recall that(T₁, θ1) and(T2, θ2) are in the same geometric series if there exists x ∈ G

such that (T₂, θY 2 ) =

x_(T

1, θ1Y) and x

−1_{F (x )T}

1 ∈ WG(T1, θ1). The pairs are in the same

rational series if in addition the element s2 ∈ T∗F ∗ 1 corresponding to x−1 θ2 is G∗F ∗ -conjugate tos1. We have now a direct description of rational series.

Proposition 4.1. — The pairs (T₁, θ1) and (T2, θ2) are in the same rational series if and

only if there existsx ∈ Gsuch that(T₂, θY 2 ) =

x_(T

1, θ1Y)andx

−1_{F (x )T}

1∈ WG◦(T1, θ1).

Proof. — Note that givenx ∈ Gsuch that x_T

1isF-stable, then x−1F (x ) ∈ NG₁(T1).

LetT∗

i be anF

∗_{-stable maximal torus ofG}∗ _{and let s} i ∈ T∗F

∗

i be such that the G ∗F∗

-orbit of (T∗_i, si) corresponds to the GF-orbit of (Ti, θi). Then the statement of the

proposition is equivalent to the following: (∗) s1 and s2 are G∗F

∗

-conjugate if and only if there exists x ∈ G∗ _{such that (T}∗ 2, s2) = x_(T∗ 1, s1)andx−1F∗(x )T∗1∈ W ◦_(T∗ 1, s1). So let us prove(∗).

First, ifs1ands2areG∗F ∗

-conjugate, then there existsx ∈ G∗F∗

such thats2= x s1x−1.

Then T∗

1 and x −1_T∗

2x are two maximal tori ofC ◦

that y T∗₁y−1= x−1T∗ 2x. Then(T ∗ 2, s2) = x y_(T∗ 1, s1)and (x y )−1F∗(x y ) = y−1F∗(y ) ∈ CG◦∗(s1), as desired.

Conversely, assume that there existsx ∈ G∗_{such that(T}∗ 2, s2) = x_(T∗ 1, s1)andx −1_F∗_{(x )T}∗ 1∈ W◦_(T∗

1, s1). By Lang’s Theorem applied to the connected group CG◦∗(s1), there exists

y ∈ C◦

G∗(s1) such that x−1F∗(x ) = y−1F∗(y ). Then x y−1 ∈ G∗F ∗

and s2 = x y−1s1y x−1.

The proof of(∗)is complete.

We can now translate the properties of regularity and super-regularity defined in [BoRo1, §11.4]. LetPbe a parabolic subgroup ofGand letLbe a Levi subgroup

ofP. We assume thatLis_F-stable. Let_{X ⊂ ∇(L, F )}be a rational series.

Proposition 4.2. — The rational seriesX is(G, L)-regular (respectively(G, L)-super-regular)

if and only ifW◦

G(T, θ ) ⊂ L(respectivelyWG(T, θ ) ⊂ L) for some (or any) pair(T, θ ) ∈ X.

Proof. — This follows immediately from [BoRo1, Lemma 11.6].

4.B. Coroots of fixed points subgroups. — We consider now again a non-necessarily

connected reductive groupG.

We fix an element g ∈ G which stabilizes a pair (T, B) where B is a Borel

sub-group ofGandTis a maximal torus ofB. Such an element is called quasi-semisimple

in [DigMi2] and [DigMi3]. For instance, any semisimple element of G is

quasi-semisimple. Recall from [DigMi2, Theorem 1.8] that CG(g )◦ is a reductive group,

thatCB(g )◦= B∩ CG(g )◦ is a Borel subgroup ofCG(g ) and thatCT(g )◦= T∩ CG(g )◦ is a

maximal torus ofCB(g ). We shall be interested in determining the coroot system of

the fixed points subgroupCG(g )◦.

LetΦ(respectivelyΦ∨) be the root (respectively coroot) system ofG◦_{relative toT.}

Let Φ_{(g )} (respectively Φ∨_{(g )) denote the root (respectively coroot) system of C}

G(g )◦

relative to CT(g )◦. If Ωis a g-orbit in Φ, we denote by cΩ ∈ F× the scalar by which

g|Ω| acts on the one-parameter unipotent subgroup associated withα(through any

identification of this one-parameter subgroup with the additive group F). We de-note by (Φ/g )a _{the set of g-orbits Ω in Φ such that there exist α, β ∈ Ω such that}

α + β ∈ Φ. We denote by(Φ/g )b _{the set of other orbits. We set}

Φ_{[g ] = {Ω ∈ (Φ/g )}a _{| cΩ= 1}and_{p 6=2} ∪ {Ω ∈ (Φ/g )}b _{| cΩ= 1}.}

Finally, ifΩ_{∈ (Φ/g )}a (respectivelyΩ_{∈ (Φ/g )}b), letΩ∨_{= 2}P

α∈Ωα

∨ _{(respectively Ω}∨₌

P

α∈Ωα

∨_{). Note thatΩ}∨ _{isg-invariant, so it belongs toY (T)}g _{= Y (C}