Equivariant character bijections in groups of Lie type [Elektronische Ressource] / Johannes Maslowski

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• Mathematics

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• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2010
• Nombre de lectures: 24
• Langue: English

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Equivariant character bijections
in groups of Lie type
Johannes Maslowski
Vom Fachbereich Mathematik der Technischen Universit at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation.
1. Gutachter: Prof. Dr. Gunter Malle
2.hter: Prof. Dr. Gerhard Hi
Vollzug der Promotion: 13.07.2010
D386Table of contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I. Fundamentals and preparation 5
1. Simply connected algebraic groups . . . . . . . . . . . . . . . . . . . 5
2. Universal algebraic group . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4. Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5. Cli ord theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6. Dual fundamental weights . . . . . . . . . . . . . . . . . . . . . . . . 15
7. Dual universal group . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
0II. The p {characters of the normalizer 25
Fq
08. Parametrizing Irr (B ) . . . . . . . . . . . . . . . . . . . . . . . . . 25p u
9. Automorphisms and labels . . . . . . . . . . . . . . . . . . . . . . . . 29
F10. Parametrizing Irr 0(B ) . . . . . . . . . . . . . . . . . . . . . . . . . 32p u
F
011. Prizing Irr (B ) . . . . . . . . . . . . . . . . . . . . . . . . . 35p sc
F12. Corresponding characters of Z(G ) . . . . . . . . . . . . . . . . . . . 46sc
0 F FIII.The p {characters of G and G 47u sc
13. Deligne{Luzstig theory . . . . . . . . . . . . . . . . . . . . . . . . . . 47
14. Steinberg map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
15. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
16. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
IV.Appendix 59
17. List of Cartan matrices and their inverses . . . . . . . . . . . . . . . . 59
iIntroduction
A classical conjecture in the representation theory of nite groups, the McKay conjec-
ture, states that for any nite group G and for any primep, the numberjIrr 0(G)j ofp
0
0complex irreducible characters ofp {degree coincides with the numberjIrr (N (P ))jp G
0of irreducible c ofp {degree of the normalizerN (P ) of any SylowG
p{subgroup P of G.
Recently a reduction theorem for the McKay conjecture was proved by Isaacs, Malle
and Navarro [IMN07, Theorem B]: The general McKay conjecture is true if a certain
stronger version holds for all nite simple groups and their covering groups. A simple
group admitting the stronger conditions of [IMN07, Section 10] is called \good".
This result has sparked a lot of research in an attempt to prove that all nite simple
groups are good. Malle has considered the sporadic groups, alternating groups and
exceptional covering groups of Lie type in [Mal08a]. For the nite simple groups
of Lie type and p not the de ning characteristic there are partial results by Malle
and Sp ath [Mal07,Mal08b,Sp a09,Sp a10a, Sp a10b], in particular they show that the
exceptional groups of Lie type are good for those primes.
In this work the goal is to make progress towards proving that the nite simple
groups of Lie type are good for their de ning characteristic. It was shown in [Bru09a]
that most groups of Lie type that coincide with their universal cover are good for
their de ning characteristic, mainly by counting characters. This covers many of the
exceptional groups of Lie type. In the last months further results by Brunat [Bru09b,
Bru10] and Brunat and Himstedt [BH09] have appeared on the arXiv, extending the
earlier results and showing among other things that those groups of Lie type for
which the center of the universal cover is of order two or three, are good for their
de ning characteristic.
A major ingredient necessary to show that a simple groupG is good is the construc-
tion of an automorphism{equivariant bijection for the universal covering group of G
that respects central characters. We generalize a construction of Alexandre Turull
for SL (q) [Tur08, Section 4] to all types of nite reductive groups, making use of then
so called Steinberg map. This yields more structural bijections than those obtained
by Brunat and we obtain new results for groups with more complicated covering and
automorphism groups (e.g. types A and D ), as well as recovering many of then n
results of Brunat by far simpler methods.
1We don’t consider Suzuki and Ree groups and related exceptional automorphisms of
f f fthe untwisted groups B (2 ), F (2 ) and G (3 ), as well as certain very small cases2 4 2
2
given in Table 13.2. With the exception of D (2) and D (2) the groups that wen n
don’t consider have already been treated with other methods elsewhere, see [Mal08a],
[Bru09a] and [Cab08]. However, it is possible to generalize our method to the Suzuki
and Ree groups and the exceptional automorphisms, see Example 9.5. Combining
the earlier work on these special cases with our results, we obtain:
1Theorem 1. Let G be a nite simple group of Lie{type , G a simply connectedsc
algebraic group de ned over an algebraically closed eld of characteristic p and F a
F FFrobenius map of G , such that G is the universal central extension of G. Let Usc sc
F F F Fbe a Sylow p{subgroup of G , B =N F (U ) and Z := Z(G ). Then there existsGsc sc scsc
a bijection
F Ff : Irr 0(B )! Irr 0(G );p psc sc
F
0with the following properties for all 2 Irr (B ):p sc
Let denote the unique character of Z below , then =f() .Z Z Z
F d d For any diagonal automorphism d stabilizing B we have f( ) =f() .sc
Let N denote the orbit of under the action of the group of diagonal auto-
Fmorphisms, then f(N ) = f(N ) for an arbitrary automorphism of G sc
Fstabilizing B .sc
As a special case we recover the results of [Bru09a] and [Bru09b]: When G coincides
F Fwith G , i.e., if Z(G ) is trivial, the sets N contain exactly one character andsc sc
Fthe theorem then states that G must be \good" in most cases (see the proof ofsc
F
0Theorem 5 and Remark 2 in [Bru09a]). In [Bru09b] the numbersjIrr (G )j arep sc
computed explicitly for most types, we obtain them in Theorem 11.8. The identity
dubbed \relative McKay" of [Bru09b, Theorem A] given there only for trivial or
1prime order of H (F; Z(G )), namelysc

F F F 0 0jIrr (G )j =jIrr (B )j for 2 Irr(Z(G ))p psc sc sc
Fis an immediate consequence of Theorem 1. SupposejZ(G )j = 2; 3 and that anysc
eld or graph automorphism xes at least one character of every {invariant set
Ff(N ) (which is shown in this work only on the side of B , see Proposition 11.13), sc
then we can recover many of the results of [BH09] and [Bru10].
To show that all the nite simple groups of Lie type are good in their de ning
characteristic much work remains to be done. In Section 16 we provide an interesting
example in the group SL (F 3) concerning extension properties of characters and pose9 7
Fa conjecture on the action of Aut(G ) on the sets N .sc
21exceptD (2) or D (2)n n
2