Bergische Universitat¨ Wuppertal Jordan Decomposition for the Alperin{McKay Conjecture Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich 4 der Bergischen Universit¨atWuppertal vorgelegt von Lucas Ruhstorfer The PhD thesis can be quoted as follows: urn:nbn:de:hbz:468-20200602-122838-0 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20200602-122838-0] DOI: 10.25926/pxey-hd44 [https://doi.org/10.25926/pxey-hd44] Contents 1 Representation theory 13 1.1 Modular representation theory . 13 1.2 Module categories . 14 1.3 The Brauer functor . 15 1.4 Brauer pairs and the Brauer category . 17 1.5 Morita equivalences and splendid Rickard equivalences . 19 1.6 First properties of splendid complexes . 21 1.7 Brauer categories and splendid Rickard equivalences . 23 1.8 Properties of splendid Rickard equivalences . 25 1.9 Lifting Rickard equivalences . 26 1.10 Descent of Rickard equivalences . 28 1.11 Morita equivalences and Clifford theory of characters . 30 1.12 Rickard equivalences for the normalizer . 31 1.13 The Brauer functor and Clifford theory . 33 1.14 The Harris–Kn¨orrcorrespondence . 36 1.15 Splendid Rickard equivalences and Clifford theory . 38 2 Deligne{Lusztig theory and disconnected reductive groups 41 2.1 Disconnected reductive algebraic groups . 41 2.2 `-adic cohomology of Deligne{Lusztig varieties . 42 2.3 Properties of Deligne{Lusztig varieties . 43 2.4 Godement resolutions . 47 2.5 Isogenies . 47 2.6 Duality for connected reductive groups . 48 2.7 Levi subgroups, isogenies and duality . 50 2.8 Rational Lusztig series for connected reductive groups . 52 2.9 Lusztig series for disconnected reductive groups . 54 2.10 Lusztig series and Brauer morphism . 56 2.11 Regular embedding and Lusztig series . 57 2.12 The Bonnaf´e{Dat{RouquierMorita equivalence . 58 3 3 On the Bonnaf´e,Dat and Rouquier Morita equivalence 62 3.1 A remark on Clifford theory . 62 3.2 Steinberg relation . 64 3.3 Notation . 65 3.4 Classifying semisimple conjugacy classes . 66 3.5 Computations in the Weyl group . 70 3.6 Representation theory . 74 3.7 Proof of Morita equivalence . 78 4 Equivariant Morita equivalence and local equivalences 81 4.1 Automorphisms of simple groups of Lie type . 81 4.2 Equivariance of Deligne{Lusztig induction . 83 4.3 Automorphisms and stabilizers of idempotents . 85 4.4 Generalizations to disconnected reductive groups . 89 4.5 Independence of Godement resolution . 91 4.6 Comparing Rickard and Morita equivalences . 93 4.7 Morita equivalences for local subgroups . 94 5 Extending the Morita equivalence 96 5.1 Disconnected reductive groups and Morita equivalences . 96 5.2 Local equivalences . 97 5.3 Restriction of scalars for Deligne{Lusztig varieties . 99 5.4 Duality in the context of restriction of scalars . 101 5.5 Comparing Weyl groups . 103 5.6 Restriction of scalars and Lusztig series . 106 5.7 Restriction of scalars and Jordan decomposition of characters . 108 5.8 Reduction to isolated series . 110 5.9 Jordan decomposition for local subgroups . 113 6 Application to the inductive Alperin{McKay condition 117 6.1 The inductive Alperin{McKay condition . 117 6.2 A criterion for block isomorphic character triples . 121 6.3 A condition on the stabilizer and the inductive conditions . 122 6.4 Extension of characters . 125 6.5 The case D4 ............................ 126 6.6 A first reduction of the iAM-condition . 128 6.7 Quasi-isolated blocks . 133 6.8 Normal subgroups and character triple bijections . 137 6.9 Application of character triples . 138 6.10 Jordan decomposition for the Alperin{McKay conjecture . 141 4 Acknowledgement First and foremost I would like to express my gratitude to my supervisor Britta Sp¨ath. Her guidance and constant support during the course of my PhD have been invaluable to me. I thank her for suggesting this interesting topic which I extremely enjoyed working with. I would like to thank Marc Cabanes for reading through this thesis and for many interesting discussions. I am deeply indebted to Gunter Malle for his suggestions and thoroughly reading some parts of my thesis. Moreover, I would like to thank Gabriel Navarro for helping me provide an explicit counterexample. I am grateful to C´edricBonnaf´eand Rapha¨elRouquier for insightful discussions on quasi-isolated elements and helpful suggestions. I thank Radha Kessar and Markus Linckelmann for answering my technical questions on Rickard equivalences at the MSRI. My thanks goes to all the people of the algebra department with whom I travelled along parts of this journey. Especially, I would like to thank Andreas B¨achle for being an excellent colleague and friend. I would like to thank Niamh Farrell for being an awesome person and collaborator. And finally, Julian Brough for countless conversations and for always being supportive. I would like to thank my family who have provided support and moral assistance throughout many years. Finally, Annika for her encouragement during the ups and downs of my thesis. This material is partly based upon work supported by the NSF under Grant DMS-1440140 while the author was in residence at the MSRI, Berkeley CA. The research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topol- ogy, which is funded by the DFG. 5 Introduction The Global-Local Conjectures The representation theory of finite groups is a field of mathematics which was introduced to study finite groups by means of linear algebra. This theory is concerned with the study of group homomorphisms X : G ! GLn(K) from a finite group G to a general linear group GLn(K) over a field K. From the very beginning of this topic, it became apparent that many properties of the group itself are encapsulated in its representations and there are theorems in group theory which can only be proved by using representation-theoretic methods. Motivated by the work of Brauer, the so-called local-global conjectures became an important area of research. These conjectures predict that for a prime number ` the information about a finite group G (global information) should relate to properties of `-local subgroups of G, that is, normalizers or centralizers of non-trivial `-subgroups of G (local information). One of the most simple yet extremely difficult conjectures is the so-called McKay conjecture, see [Mal17, Section 2]. Let ` be a prime and K a finite field extension of Q` large enough for all finite groups considered. Denote by Irr(G) the set of isomorphism classes of irreducible K-representations and by Irr`0 (G) the subset corresponding to irreducible representations X : G ! GLn(K) with ` - n. Conjecture (McKay). Let G be a finite group and P a Sylow `-subgroup of G. Then j Irr`0 (G)j = j Irr`0 (NG(P ))j. Later, Alperin [Alp76] refined this conjecture by taking into account the representation theory over a field of positive characteristic `. Denote by O the ring of integers of K over Z` and by k its residue field. For an `-block B of kG let Irr0(G; B) be the set of isomorphism classes of 6 height zero representations of kB, i.e., the set of irreducible representations X : G ! GLn(K) associated to the block B such that n` = jG : Dj`, where D denotes a defect group of B. Conjecture (Alperin{McKay). Let G be a finite group and b an `-block of G and B its Brauer correspondent in NG(D). Then j Irr0(G; b)j = j Irr0(NG(D);B)j: Both of these conjectures have been reduced (by Isaacs{Malle{Navarro, respectively Sp¨ath,see [IMN07, Theorem B] and [Sp¨a13,Theorem C]) to the verification of certain stronger versions of the same conjecture for finite quasi-simple groups. These stronger versions are usually referred to as the inductive conditions. This approach has turned out to be very fruitful in recent years. In their landmark paper [MS16], using this approach Malle–Sp¨athwere able to prove the McKay conjecture for the prime ` = 2 (proving the original conjecture of McKay). According to the classification of finite simple groups, many finite simple groups are groups of Lie type. These are finite groups which arise as fixed points GF of a simple algebraic group G under a Frobenius endomorphism F : G ! G. In this thesis, we focus on their representation theory and establish a new approach to the inductive Alperin{McKay condition for those groups. Representation theory of groups of Lie type In characteristic zero Deligne and Lusztig have constructed representations by means of `-adic cohomology groups of the so-called Deligne{Lusztig va- rieties. Let G be a connected reductive algebraic group with Frobenius F : G ! G and let L be an F -stable Levi subgroup of G contained in a parabolic subgroup P with Levi decomposition P = LnU. Then the variety G −1 YU = fgU 2 G=U j g F (g) 2 UF (U)g has a left GF - and a right LF -action. Recall that O denotes the ring of integers over Z` of a finite field extension K of Q`. The `-adic cohomology i G G groups Hc(YU; O) of the Deligne{Lusztig variety YU provide us with a map G F F X i i G RL⊆P : G0(OL ) ! G0(OG ); [M] 7! (−1) [Hc(YU; O) ⊗OLF M] i 7 on the respective Grothendieck groups, see Section 2.2. Let G∗ be the dual group of G with dual Frobenius F ∗ : G∗ ! G∗, see Section 2.6. Deligne{ Lusztig constructed a decomposition of the irreducible representations into rational Lusztig series a Irr(GF ) = E(GF ; s); (s) where (s) runs over the set of (G∗)F ∗ -conjugacy classes of semisimple el- ements of (G∗)F ∗ .