$\theta$-semisimple twisted conjugacy classes of type D in $\operatorname{PSL}_n(q)$

0.28 MB·English

by  Giovanna Carnovale

θ-SEMISIMPLE CLASSES OF TYPE D IN PSL (q)
n
5 GIOVANNACARNOVALE1,AGUST´INGARC´IAIGLESIAS2
1
0
2 ABSTRACT. Letpbeanoddprime,m ∈ Nandsetq = pm,G = PSLn(q).
Letθ beastandardgraphautomorphism ofG, dbeadiagonal automorphism
l
u and Frq be the Frobenius endomorphism of PSLn(Fq). We show that every
J (d◦θ)-conjugacy class of a (d◦θ,p)-regular element in G isrepresented in
6 someFrq-stablemaximaltorusofPSLn(Fq)andthatmostofthemareoftype
1 D.Wewriteoutthepossibleexceptionsandshowthat,inparticular,ifn≥5is
eitheroddoramultipleof4andq > 7,thenallsuchclassesareoftypeD.We

] developgeneralargumentstodealwithtwistedclassesinfinitegroups.
A
Q
.
h
t 1. INTRODUCTION
a
m This paper belongs to the series started in [ACGa1], in which we intend to de-
[ termine all racks related to (twisted) conjugacy classes in simple groups of Lie

type which are of type D cf. (2.1), as proposed in [AFGaV2, Question 1]. This,
2
v although being mainly a group-theoretical question, is intimately related with the
8 classificationoffinite-dimensionalpointedHopfalgebrasovernon-abeliangroups,
3
see below. Inthis article wewill focus on racks which arise as non-trivial twisted
6
7 conjugacyclassesinPSLn(q)forq = pm,panoddprime.
0
Recall that a rack is a non-empty set X together with a binary operation (cid:3)
.
1
satisfying faithfulness and self-distributive axioms, see 2.1. The prototypical ex-
0
5 ample of a rack is a twisted conjugacy class Oxψ withrespect to an automorphism
1 ψ ∈ Aut(G)insideafinitegroupG,x ∈ G,with
:
v
(1.1) y(cid:3)z = yψ(zy−1), y,z ∈Oψ.
i x
X
Thisisinfactaquandle, asy(cid:3)y = y,∀y ∈ Oψ.
r x
a A rack X is said to be of type D when there exists a decomposable subrack
Y = R S ⊆ Xand elements r ∈ R, s ∈ S such that r (cid:3)(s(cid:3)(r (cid:3)s)) 6= s,
seeSection2.1. Theirstudyisdeeplyconnected withtheclassificationproblem of
F
finite-dimensional pointedHopfalgebras, asfollows.
2000MathematicsSubjectClassification.16W30.
1-DipartimentodiMatematica,TorreArchimede-viaTrieste63-35121Padova-Italy
2 - FaMAF-CIEM (CONICET), Universidad Nacional de Co´rdoba, Medina Allende s/n, Ciudad
Universitaria,5000Co´rdoba,Repu´blicaArgentina.
TheworkofA.G.I.waspartiallysupportedbyANPCyT-FONCyT,CONICET,MinisteriodeCiencia
yTecnolog´ıa(Co´rdoba),Secyt(UNC)andtheGNSAGAproject.Partofitwasdoneasafellowofthe
ErasmusMundusEADICIIprogrammeoftheEUintheUniversita`degliStudidiPadova.G.C.was
partiallysupportedbyProgettodiAteneoCPDA125818/12andbythebilateralagreementbetween
theUniversitiesofCo´rdobaandPadova.
1
2 CARNOVALE,GARC´IAIGLESIAS
LetH beafinitedimensionalpointedHopfalgebraoveranalgebraically closed
fieldkandassumethecoradicalofH iskG,forafinitenon-abelian groupG. Fol-
lowing [AG, Section 6.1], there exist a rack X and a 2-cocycle q with values in
GL(n,k)such that grH, the associated graded algebra with respect to the corad-
ical filtration, contains as a subalgebra the bosonization B(X,q)#kG. See loc.
cit. for unexplained notation. Therefore, it is central for the classification of such
Hopfalgebras toknowwhendimB(X,q) < ∞forgivenX,q. ArackX issaid
tocollapsewhenB(X,q)isinfinitedimensionalforanyq. Aremarkableresultis
that if X is of type D, then it collapses. This is the content of [AFGV1, Theorem
3.6],also[HS,Theorem8.6],bothofwhichfollowfromresultsin[AHS].
NoweveryrackX admitsarackepimorphism π : X → S withS simpleandit
followsthatX isoftypeDifS isso. Hence,determining allsimpleracksoftype
Dis adrastic reduction indeed for the classification problem, as many groups can
be discarded and only a few conjugacy classes in simple groups remain. Only for
such classes one needs tocompute the possible cocycles that yield afinite dimen-
sional Nichols algebras. Simple racks are classified into three classes [AG], also
[J], namely affine, twisted homogeneous and that of non-trivial twisted conjugacy
classesonfinitesimplegroups, see[AG]fordefinitions. Most(twisted)conjugacy
classes in sporadic groups are oftype D[AFGV2], [FV]. Thisis also the case for
non-semisimple classes in PSL (q) [ACGa1], for unipotent classes in symplectic
n
groups [ACGa2]andfor(twisted) classes inalternating groups [AFGV1]. Similar
results follow fortwisted homogeneous racks[AFGaV1]. Affineracksseem tobe
notoftypeD.
InthisarticlewebegintheanalysisoftwistedclassesoftypeDinPSL (q),for
n
qoddandautomorphismsinducedbyalgebraicgroupautomorphismsofSL (F ).
n q
RecallthattheautomorphismsinPSL (q)arecompositionsofautomorphismsin-
n
ducedbyconjugation inGL (q)(diagonalandinnerautomorphisms), powersofa
n
standard graph automorphism θ of the Dynkin diagram and powers of the Frobe-
niusautomorphism Fr . Inner automorphisms maybeneglected [AFGaV1,§3.1].
p
Diagonalandgraphautomorphismsareinducedbyalgebraicgroupautomorphisms
ofSL (F ),whereasFr isinducedbyanabstractgroupendomorphism. Theirbe-
n q p
haviour istherefore different [St,10.13]andthisisreflectedinthestructure ofthe
twisted classes. In addition, if the d◦θa-class of x in PSL (p) is of type D, d a
n
diagonal automorphism anda = 0,1, thentheFrm◦d◦θa-class ofxinPSL (q)
p n
is of type D for every m and every q. Thus, we will focus on twisted classes for
automorphisms ψ = d◦θa. The analysis of standard conjugacy classes in simple
groupsofLietype(corresponding toa = 0)hasbeenstartedin[ACGa1,ACGa2].
Forthese reasons the first twisted classes to look at are theψ-classes in PSL (q),
n
whereψ isacomposition of adiagonal automorphism dwithθ. In analogy tothe
case of standard conjugacy classes, it ispossible to reduce most ofthe analysis to
the study of classes whose behaviour resembles that of semisimple or unipotent
ones. However, in contrast to that case, the choices to be made depend on the
gcd of |ψ| and p cf. Subsection 3.1. Therefore, the cases of p even and odd must
be handled with different methods. The diagonal automorphisms always satisfy
(|ψ|,p) = 1sowerestricttothecase(|ψ|,p) = 1andwewillrequireptobeodd.
θ-SEMISIMPLETWISTEDCONJUGACYCLASSESOFTYPEDINPSLn(q) 3
Set G = PSL (q), ψ = d ◦ θ ∈ Aut(G), for d a diagonal automorphism.
n
Thestudy of(ψ,p)-regular classes inG,i. e.,ofthose classes replacing semisim-
ple ones, can be reduced to the study of (θ,p)-regular G-orbits of elements in
PGL (q). SuchclasseshavearepresentativeinamaximaltorusTFrq ofPGL (q),
n w n
for some w ∈ Wθ, where we can take w up to conjugation cf. Theorem 5.1.
It turns out that in most cases, the property of being of type D depends on n, q
and the conjugacy class of w in Wθ. Such classes are parametrized by a parti-
tion λ = (λ ,...,λ ) of h = n , with r ∈ N, h > 0, and a certain vector
1 r 2 i
ε∈ Zr. Henceourresultdependsonthenumberofcyclesrofλandonthevector
ε= (ε2 ,...,ε )∈ Zr. Let1stan(cid:2)df(cid:3)orthepartition (1,...,1).
1 r 2
Theorem1.1. Letq beasabove. Letx ∈ TFrq. ThentheclassOθ,G isoftypeD,
w x
withthepossible exceptions ofclassesfittingintothefollowingtable:
w n q x
r = 2 ε = (0,ε ) even 3,5 any
2
4 3,7 any
ε= (0)
λ 6= 1
r = 1 4 5,9 θ(x)6= x−1
4 3,7 θ(x)6= x−1
ε= (1)
2×odd any Oθ,G ≃ Oθ,G
x ν
any* 3,5 any*
3 7,13 any
λ = 1
4 ≡ 3(4) any
4 9 any
TABLE 1. Possibleexceptions; ν asin(5.5).
* Actually, some of the classes listed on the table are of type D, for instance
whenn ≥ 6,n 6= 7andε =(0,...,0),seeLemma5.6. SeealsoRemark5.12.
WepresentthisresultinthelanguageofNicholsalgebras,asapartialanswerin
thiscasesto[AFGaV2,Question2],seealso[AFGV1,Theorem3.6],andloc. cit.
forunexplained notation. Consider theclasses Oθ,G inTheorem 1.1asracks with
x
therackstructure(1.1). Thesearesimpleracks.
Corollary 1.2. Let X = Oθ,G, x ∈ TFrq. Then dimB(Oθ,G,q) = ∞ for any
x w x
cocycleqonX,withthepossible exceptions oftheclasses inTable1. (cid:3)
Also,anextractofTheorem1.1canberephrased asfollows.
Theorem 1.1’. Let p be an odd prime, m ∈ N, q = pm. Set G = PSL (q),
n
ψ = d◦θ ∈ Aut(G),fordadiagonal automorphism.
If n ≥ 5, q ≥ 7, then any (ψ,p)-regular class O is of type D with the possible
exceptionn = 2×odd,O ≃ OAd(ν−1)◦θ,G,ν asin(5.5). (cid:3)
1
4 CARNOVALE,GARC´IAIGLESIAS
Whenψ = θ,weobtainthefollowing forclasses withtrivial(θ,p)-regular part
(alsocalledθ-semisimplepart)whichisthecontentofPropositions 6.1and6.2:
Proposition 1.3. Let O be a θ-twisted conjugacy class with trivial θ-semisimple
part. ThenO isoftypeDprovided
(1) n > 2iseven,theunipotent partisnontrivial, andq > 3.
(2) n > 3 is odd and the Jordan form of its p-part in Gθ corresponds to the
partition (n). (cid:3)
The paper is organized as follows. In Section 2 we fix the notation and recall
some generalities about racks and the group PSL (q). In Section 3 we discuss
n
some general techniques to deal with twisted conjugacy classes in a finite group.
InSection4wefocusonPSL (q)andwebeginasystematicapproachtothestudy
n
ofitstwisted classes, thatincludes ananalysis oftheWeylgroup. InSection5we
concentrate on θ-semisimple classes and obtain the main results of the article. In
Section6wepresentsomeresultsonclasseswithtrivialθ-semisimplepart.
2. PRELIMINARIES
Let H be a group, ψ ∈ Aut(H). A ψ-twisted conjugacy class, or simply a
twisted conjugacy class, is an orbit for the action of H on itself by h · x =
ψ
hxψ(h)−1. WedenotethisclassbyOψ. IfK < H isψ-stable,wewillwriteOψ,K
h h
to denote the orbit of h under the restriction of the · -action to K. In particular,
ψ
O = Oid denotes the (standard) conjugacy class of h ∈ H. The stabilizer in
h h
K < H of an element x ∈ H for the twisted action willbe denoted by K (x) so
ψ
that H (x) isH , the usual centralizer of x. Forany automorphism ψ ofagroup
id x
H, we write Hψ for the set of ψ-invariants in H. The inner automorphism given
byconjugation byx ∈ H willbedenoted byAd(x). IfK ⊳H isnormal, thenwe
alsodenote byAd(x)theautomorphism induced from theconjugation byx ∈ H.
Z(H)will denote the center ofH. Recall that the group µ (F )of roots ofunity
n q
inafinitefieldFq isisomorphic toZd,ford := (n,q−1).
We denote by S , n ∈ N, the symmetric group on n letters. Wealso set I :=
n n
{1,2, ..., n}and(b) = 1+a+a2+···+ab−1,a,b ∈ N.
a
2.1. Racks. A rack (X,(cid:3)) is a non-empty finite set X together with a function
(cid:3) :X ×X → X suchthati(cid:3)(·) :X → X isabijection foralli ∈X and
i(cid:3)(j (cid:3)k) = (i(cid:3)j)(cid:3)(i(cid:3)k), ∀i,j,k ∈X.
Recallthatarack(X,(cid:3))isaquandle wheni⊲i= i,∀i∈ X.
WeshallwritesimplyX whenthefunction(cid:3)isclearfromthecontext.
If H is a group, then the conjugacy class O of any element h ∈ H is a rack,
h
with the function (cid:3) given by conjugation. More generally, if ψ ∈ Aut(H), any
twistedconjugacy classinH isarackwithrackstructure givenby(1.1),see[AG,
Theorem3.12,(3.4)]. Theseareindeedexamplesofquandles.
θ-SEMISIMPLETWISTEDCONJUGACYCLASSESOFTYPEDINPSLn(q) 5
A subrack Y of a rack X is a subset Y ⊆ X such that Y (cid:3)Y ⊆ Y. A rack
is said to be indecomposable if it cannot be decomposed as the disjoint union of
twosubracks. A rack X is said to be simple if cardX > 1 and for any surjective
morphismofracksπ : X → Y,eitherπ isabijection orcardY = 1.
2.1.1. Racks of type D. A rack X is of type D when there exists a decomposable
subrackY = R S ofX andelementsr ∈ R,s ∈ S suchthat
(2.1) r(cid:3)(s(cid:3)(r(cid:3)s))6= s.
F
If a rack X has a subrack of type D, or if there is a rack epimorphism X ։ Z
and Z is of type D, then X is again so. In particular, if X is decomposable and
X has a component of type D, then X is of type D. On the other hand, if X is
indecomposable, thenitadmitsaprojectionX ։ Z,withZ simple. Hence,inthe
questofracksoftypeDitisenoughtofocusonsimpleracks. Theclassificationof
simpleracksisgivenin[AG,Theorems3.9,3.12],seealso[J]. Abigclassconsists
oftwistedconjugacy classesinfinitesimplegroups.
Remark2.1. LetObeaψ-twistedconjugacyclass. ThenOisoftypeDifthereare
r,s ∈ O suchthatr ∈/ Oψ,L,forLtheψ-stable closureofthesubgroup generated
s
byr ands,and
(2.2) rψ(s)ψ2(r)ψ3(s) 6= sψ(r)ψ2(s)ψ3(r).
In fact, if the above conditions hold, we set S = Oψ,L and R = Oψ,L and then
s r
Y = R S isadecomposable subrackofO whichsatisfies(2.1).
If ψ = idthen the condition isalso necessary: if O is oftype D, then there are
r,s ∈ OF,r ∈/ OL,satisfying (2.2)[ACGa1,Remark2.3].
s
2.2. The group G = PSL (q). Fix n ∈ N. Let p ∈ N be a prime number and
n
let k = F . Fix m ∈ N, q = pm. Weassume throughout the paper that n > 2 or
p
q 6= 2,3.
Wefixonceandforallthefollowingnotation:
(2.3) G = SL (k), G = PSL (k), G := PSL (q).
n n n
Wealsofixπ: GL (k) → PGL (k) ≃ Gtheusualprojection. Weshallkeepthe
n n
name π := π|G: G → G for the restriction of π to G. We fix the subgroups of
diagonalmatrices
(2.4) T ≤ GL (k), T≤ G, T := π(T) ≤ G.
n
2.2.1. Generalproperties ofG. Considertheexactsequence:
(2.5) 1−→ Z(G) −→ G −π→ G −→ 1
and let F = Frm be the endomorphism of GL (k) raising every entry in X ∈
p n
GL (k)totheq-thpower. TakingF-points, (2.5)yields:
n
1−→ Z(SL (q)) −→ SL (q) −→ PGL (q).
n n n
ThenG ≤ PGL (q)istheimageofthelastarrow:
n
G = PSLn(q) ≃ SLn(q)/Z(SLn(q)) ≃ SLn(q)/Zd,
6 CARNOVALE,GARC´IAIGLESIAS
ford = (n,q−1). ThegroupGissimple1.
We will denote by B,U,U− ≤ G be the subgroups of G of upper triangular,
unipotent upper-triangular, unipotent lower-triangular matrices. Set
W := NG(T)/T ≃ NG(T)/T ≃ Sn.
Recall that [SL (q),SL (q)] = SL (q) and [PGL (q),PGL (q)] = G, for
n n n n n
n> 2orq 6= 2,3. Also,wehavetheidentifications:
GF = PGL (q) = TF[PGL (q),PGL (q)] = TFG
n n n
≃ GL (q)/Z(GL (q)) ≃ GL (q)/F×.
n n n q
2.2.2. AutomorphismsofG. RecallthatadiagonalautomorphismofGisanauto-
F
morphisminducedbyconjugation byanelementinT . Thegraphautomorphism
θ: GL (k)→ GL (k)isgivenbyx 7→ J tx−1J−1,for
n n n n
0 ... 0 1
0 ... −1 0
(2.6) Jn =  ... ... ... ....
(−1)n−1 ... 0 0
It induces a non-trivial automorphism of G for n ≥3 and it is unique up to inner
automorphisms2. ItalsoinducesautomorphismsofGL (q),SL (q),PGL (q)and
n n n
G. We will drop the subscript n and write J = J when it can be deduced from
n
thecontext.
By [MT, Theorem 24.24] every automorphism of G is the composition of an
inner,adiagonal,apowerofFr andapowerofθ,sotheelementsingroupofouter
p
automorphismsofGhaverepresentativesinOut(G):= hFr ,θ,Ad(t) :t ∈ TFi.
p
3. GENERAL ARGUMENTS
In this section we present some general techniques to deal with twisted conju-
gacyclassesinfinitegroups.
Westartwithawell-knownlemma.
Lemma 3.1. Let H be a finite group, ϕ ∈ Aut(H). Let K,N < H be ϕ-stable
subgroups, withN (cid:1)H. Fixx ∈ H.
(1) The set Oϕ,K is a subrack of Oϕ,H if and only if for every k ∈ K there is
x x
t ∈ H (x)suchthatxkx−1t ∈K.
ϕ
(2)[AFGaV1, §3.1] Assume ϕ = Ad(x)◦ψ, for some ψ ∈ Aut(H). Thenfor
everyg ∈ H thereareracksisomorphisms Oϕ,H ≃ Oψ,H andOϕ,N ≃Oψ,N.
g gx g gx
(3)Lety ∈ H withy ∈ Oϕ,H. ThenOϕ,N ≃ Oϕ,N.
x x y
Proof. (1)isstraightforward. In(2),wehavetheequalityofsetsOϕ,H =Oψ,Hx−1
g gx
and right multiplication by x defines the rack isomorphism. The second isomor-
phismfollowsbyrestriction. Asfor(3),letg ∈ H besuchthatg· x = y. Thenthe
ϕ
1RecallthatPSL (2)≃S ,PSL (3)≃A ≤S .
2 3 2 4 4
2Indeed,thisisnotthechoicemadein[ACGa1]butitis,however,moreadequateforoursetting.
θ-SEMISIMPLETWISTEDCONJUGACYCLASSESOFTYPEDINPSLn(q) 7
map T : Oϕ,N → Oϕ,N given by T(z) = g· z is a rack isomorphism. Observe
x y ϕ
thatifz = h· xthenT(z) = (ghg−1)· y. (cid:3)
ϕ ϕ
Remark3.2. Noticethattheassumptionin(1)inLemma3.1holdsifx ∈ N (K).
H
In particular, it always holds for K (cid:1) H. Also, (2) allows us to neglect inner
automorphisms ofH.
Thefollowingslightgeneralization of[FV,Lemma2.5]willbeveryuseful.
Lemma 3.3. Let H be a finite group and let K (cid:1)H. Let s ∈ H be a non-trivial
involution. Then OK is a rack of type D if and only if there is r in OK such that
s s
|rs|isevenandgreaterthan4.
Proof. By Lemma 3.1, Remark 3.2, OK is a rack. Observe first that, if r ∈ OK,
s s
thentheracksOhr,siandOhr,siaresubracksofOK. Indeed,ifr = k⊲s= ksk−1,
s r s
then a generic element of hs, ri has the form y = saksk−1s···ksk−1sb for
a,b
a,b ∈ {0,1}. Letsks = l ∈ K. Then,ifa = 1wehave
y ⊲s = y ⊲s= lk−1···lk−1⊲s ∈ OK,
1,b 1,0 s
y ⊲r = lk−1···lk−1sbksb⊲s ∈ OK,
1,b s
whereasifa = 0wehave
y ⊲s = y ⊲s= kl−1···kl−1⊲s ∈ OK,
0,b 0,1 s
y ⊲r = kl−1···kl−1sb−1ksb−1⊲s ∈ OK,
0,b s
so the racks Ohr,si,Ohr,si ⊂ OK. Now, if an r as in the statement exists, then
s r s
r⊲(s⊲(r⊲s)) 6= sandOhr,siandOhr,siaredisjoint,soOK isoftypeDbyRemark
s r s
2.1forψ = id. Conversely, ifthere isnosuch anr,then foreveryx ∈ OK either
s
|xs| ≤ 4 or it isodd, so either (xs)2 = (sx)2 or Ohs,xi = Ohs,xi and Remark 2.1
s x
forψ = idappliesoncemore. (cid:3)
Remark3.4. LetH beafinitegroup, φ∈ Aut(H),h ∈ H.
(1) Assume K = H is φ-stable. If k ∈ K, then Oφ,K = Oφ,Khas sets and
h kh k
rightmultiplication byh−1 givesarackisomorphism Oφ,K ≃ Oφ,K.
kh k
(2) Let L = H ⋊ hφi. Then, for x = gφ, we have the equality of sets:
Oφ,H = OLφ−1 andy 7→ yφinduces arackisomorphism Oφ,H ≃ OL.
g x g x
Remark 3.5. Let H be a finite group, φ ∈ Aut(H). Let A be a φ-stable abelian
subgroup ofH,a ∈ A.
(1) By Remark 3.4 (1), Oφ,A ≃ Oφ,A as racks. Moreover γ: A → A, b 7→
a 1
bφ(b−1),isagroupmorphism andOφ,A = Im(γ) ≃ A/Aφ asgroups.
1
(2) Ifφisaninvolution,thenOφ,AisoftypeDifandonlyifthereisb ∈ A/Aφ
a
suchthat|b|iseven,> 4byRemark3.4(2)andLemma3.3.
(3) Let p be a prime number dividing |H|. Let h = us = su ∈ H be the
(unique) decomposition ofhasaproductofap-elementuandap-regular
elements. IfOHs isoftypeD,thenO isagainso,asOHs identifieswith
u h u
asubrack ofOH.
h
8 CARNOVALE,GARC´IAIGLESIAS
Remark3.6. LetH beagroup,letφ,ψ ∈ Aut(H),withφψ = ψφ,andletN(cid:1)H
beφ-stableandψ-stable.
(1) If Oφ,N ∩ Hψ 6= ∅, then ψ(Oφ,N) = Oφ,N. Indeed, let x ∈ Oφ,N with
h h h t
ψ(x) = x. Now, if y = kxφ(k−1) ∈ Oφ,N = Oφ,N, k ∈ N, then ψ(y) =
x h
ψ(k)xφ(ψ(h)−1) ∈ Oφ,N.
h
(2) Conversely, if ψ(Oφ,N) = Oφ,N and the map N → N, given by x 7→
h h
x−1ψ(x), x ∈ N, is surjective, then Oφ,N ∩ Hψ 6= ∅. To see this, fix g ∈ N
h
such that ψ(h) = ghφ(g−1)and letx ∈ N be such that g−1 = x−1ψ(x). Then it
followsthatx· h∈ Hψ ∩Oφ,N.
φ h
3.1. (ψ,p)-elementsand(ψ,p)-regularelements. LetH beafinitegroup,pbea
primenumberdividing|H|andletψ ∈ Aut(H),withℓ := |ψ|. SetH = H⋊hψi.
Definition3.7. Anelementh ∈ H iscalled(ψ,p)-regular ifhψ isp-regularinH,
b
i. e. if (|hψ|,p) = 1. An element h ∈ H is called a (ψ,p)-element if hψ is a
p-elementinH,i.e. if|hψ| =pa forsomea ∈ N. b
Letψ = ψ ψ bethedecompositionofψasaproductofitsusualp-regularpart
r p
b
and its p-part in Aut(H). Then for every hψ in H we have hψ = sψ (u)ψ =
r
uψ (s)ψ wheresis(ψ ,p)-regular anduisa(ψ ,p)-elementinH.
p r p
In the quest of ψ-classes of type D, a first analybsis can be done by looking at
subracks given by the orbits with respect to Hψr or Hψp. For this reason, the
analysis shouldbeginwiththecasesinwhicheitherψ = 1,i.e. when(ℓ,p) = 1,
p
orwhenψ = 1,i.e. whenℓisapowerofp.
r
If (ℓ,p) = 1, then for every h ∈ H there is aunique decomposition h = us =
sψ(u) with u a p-element in H and s a (ψ,p)-regular element. In this case s is
(ψ,p)-regularifandonlyifthenormNorm (s) := sψ(s)···ψℓ−1(s)isp-regular
ψ
in H. Here, if C = H (s)and C′ = H , then Remarks 3.4 (2) and 3.5 (3)give
ψ sψ
therackinclusions
(3.1) Oψ,H ≃ OHbb⊃ OC′ ⊃ OC.
h hψ u u
So if OC is of type D, then Oψ,H is again so. Hence the first classes to be at-
u h
tacked are either standard conjugacy classes of p-elements in C ortwisted classes
of(ψ,p)-regular elementsinH. ThelatteraredealtwithinSection5.
Similarly, if ℓ = pb for some b > 0, then for each h ∈ H there is a unique
decomposition h = su = uψ(s) with s a usual p-regular element in H and u a
(ψ,p)-element. In this case u is a (ψ,p)-element if and only if Norm (u) is a
ψ
p-elementinH. Thefirstreduction istolookatclassesof(ψ,p)-elements andthe
standardp-regularclassesinH (u). Wewillnotpursuethisanalysisinthispaper.
ψ
Notice that, when dealing with twisted classes in simple groups of Lie type,
thereisaprivilegedchoiceforp,namely,thedefiningcharacteristic.
4. TWISTED CLASSES AND PSLn(q)
In this section we collect some results that contribute to establish a systematic
approachtotwistedclassesinPSL (q). Thisinparticularrequiresadetailedstudy
n
θ-SEMISIMPLETWISTEDCONJUGACYCLASSESOFTYPEDINPSLn(q) 9
oftheconjugacyclassesinthesubgroupofθ-invariantelementsoftheWeylgroup,
andofthecorresponding F-stablemaximaltoriinG,thatwedevelopin§4.2.
Recall the notation from §2.2, specially in (2.3), (2.4). Next proposition deals
F
withdiagonal automorphisms d = Ad(t),t ∈ T .
Proposition 4.1. Letx ∈ G,ϕ = Ad(t)◦ψ ∈ Aut(G),t ∈ TF. Lety = t−1x ∈
GF. ThenOϕ,G ≃ Oψ,G. If,inaddition, ψ ∈ Out(G)andz ∈ Oψ,PGLn(q),then
x y y
Oϕ,G ≃ Oψ,G.
x z
Proof. In this case, x = ty and Oϕ,G ≃ Oψ,G by Lemma 3.1 (2). The last
x y
assertion isLemma3.1(3). (cid:3)
Let ψ = Fra◦θb ∈ Aut(GL (q)) and let ℓ := |ψ|. Then ψ induces an auto-
p n
morphism of SL (q),PSL (q) and PGL (q) of the same order. Let H be either
n n n
GL (q),SL (q),PSL (q),orPGL (q),H = H ⋊hψi.
n n n n
If (ℓ,p) = 1, then the (ψ,p)-elements in H are the unipotent elements in H.
The (ψ,p)-regular elements are those g ∈bH such that Norm (g) is semisimple.
ψ
If, instead, ℓ = pb for some b > 0, then the (ψ,p)-regular elements in H are the
semisimple elements in H, while the (ψ,p)-elements are those g ∈ H such that
Norm (g)isap-element.
ψ
Wewillconcentrate onthecase(ℓ,p) = 1. Wehavethefollowingequivalence.
Lemma 4.2. Let ψ ∈ Aut(GL (q)) with (|ψ|,p) = 1. Then x ∈ GL (q) is
n n
(ψ,p)-regular ifandonlyifx = π(x) ∈ PGL (q)is(ψ,p)-regular.
n
Proof. Norm (x) is semisimple if and only π(Norm (x)) = Norm (x) is so.
ψ ψ ψ
(cid:3)
4.1. The case ψ = θ, p 6= 2. We intend to study twisted classes for automor-
phisms induced from algebraic group automorphisms. ByRemark3.2and Propo-
sition 4.1, we may reduce to the case ψ = θ. We will focus on the case of p odd
andweshallinvestigate (ψ,p)-regular classes.
Remark4.3. ItwaspointedtousbyProf. VinbergthatwhenthegroupisGL (F )
n q
and ψ = θ, then the map x 7→ xJ allows to identify the θ-twisted conjugacy
class of x with the equivalence classes of the non-degenerate bilinear form with
associated matrixxJ. Thus,theclassification oftwistedclassesinthiscasecanbe
deduced from the classification of bilinear forms on F n. The latter, in turn, goes
q
over in odd characteristic, as the classification in characteristic zero which is to
be found for instance in [HoP]. From this, SL (q)-orbits could be also classified.
n
However, since the action of the center by twisted conjugation is non-trivial, the
steptoPSL (q)-orbits ofelements inPGL (q)wouldneedslightcare. Themain
n n
reason for our apparently less natural approach is related to the general problem
of detecting twisted classes of type D in all finite simple groups. One of the aims
in this paper is to propose a general systematic approach that could be applied, at
least,toallfinitesimplegroupsofLietype.
Lemma4.4. Letx ∈ GL (q).
n
10 CARNOVALE,GARC´IAIGLESIAS
(1) xisθ-semisimpleifandonlyifthereisag ∈ GL (k)suchthatg· xlies
n θ
inaθ-stabletorusT inGL (k).
0 n
(2) x is θ-semisimple if and only if there is a g′ ∈ SL (k) ⊂ GL (k) such
n n
thatg′· x ∈ T.
θ
Proof. (1) is [Mo2, Proposition 3.4]. Following the construction in [Mo2, page
382]wecanmakesurethatT isF-stableandthatitiscontainedinT. For(2),let
0
Z := Z(GL (k)),henceGL (k) = ZGandθ actsasinversion onZ. Therefore,
n n
ifz ∈ Z,thenz· x = xz2. Letg = zg′ ∈ZGbesuchthatg· x = t ∈ T. Then
θ θ
g′·x = tz−2 ∈ T,asZ iscontained ineverymaximaltorus. (cid:3)
Thelemmaabovemotivatesthefollowingdefinition.
Definition4.5. Wesaythatanelementx ∈ PGL (q)isθ-semisimpleifitis(θ,p)-
n
regular.
4.2. F-stable maximal tori. In this section we collect preparatory material in
order to find suitable representatives of G-classes of θ-semisimple elements in
PGL (q). Unlessotherwisestated, pisarbitrary.
n
Let H denote either G, G or GL (k) and, consequently, set K = T, T or
n
T (= TZ(GL (k))). Let w ∈ W, w˙ ∈ wK and g = g ∈ H be such that
n w
g−1F(g) = w˙ (Lang-Steinberg’s Theorem). Weset
(4.1) K := gKg−1.
w
ThenK isanF-stablemaximaltorusofH andallF-stablemaximaltoriinH are
w
obtainedthisway[MT,Proposition25.1]. TwotoriK andK areHF-conjugate
w σ
if and only if σ and w are W-conjugate. We will provide a θ-invariant version of
thisfactinLemma4.7forK = TandT. Weset
(4.2) F := Ad(w˙)◦F, so(K )F =gKFwg−1.
w w
Theautomorphisms θ and F preserve T,hence they induce automorphisms onW
which we denote by the same symbol. The action of F on W is trivial, whereas
the action of θ is conjugation by the longest element w ∈ W, so Wθ = W .
0 w0
Observethat
(1, n)(2n−1)...(h, h+1) ifn = 2h,
w =
0
((1, n)(2, n−1)...(h, h+2) ifn = 2h+1.
Anyσ ∈ Wθ canbewrittenasσ = ωτ whereωpermutesthe2-cyclesinw andτ
0
isaproduct oftranspositions occurring inthecyclicdecomposition ofw . Infact,
0
Wθ ≃ S ⋊Zh,whereh = n ,the elements inS correspond toproducts cθ(c)
h 2 2 h
wherecisacycleinSIh ≤ Sn(cid:2),θ(cid:3)(c)= w0cw0 andtheelementsinZh2 areproducts
oftranspositions oftheform(i,n+1−i).
Remark 4.6. There is a set of representatives {σ˙} ⊂ NG(T) of W such that σ˙ ∈
NG(T)θ if σ ∈ Wθ, [St, 8.2, 8.3 (b)]. In addition, Gθ = Spn(k) if n is even,
Gθ = SO (k)ifnisoddandWθ isthecorresponding Weylgroup.
n