Fachbereich Mathematik + The generic character table of Spin8 (q) Emil Rotilio 1. Gutachter: Prof. Dr. Gunter Malle 2. Gutachter: Prof. Dr. Meinolf Geck Datum der Disputation: 07. Mai 2021 Vom Fachbereich Mathematik der Technischen Universit¨atKaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation D 386 Acknowledgments These four years of doctoral studies have been an amazing period of my life. I want to thank everyone that made this work possible and enjoyable. First of all, I thank my supervisor professor Gunter Malle, who accepted me as doctoral student although I was coming from physics. I also thank him for all the support and availability when I was stuck or needed to talk. I thank professor Donna Testerman that accepted to supervise me for a minor in mathe- matics. She transmitted to me her passion for algebras and groups. Without her I wouldn't have chosen this path, and she introduced me to professor Malle. A big thank goes to Chantal. She followed me in this German adventure. Until the end she was supportive and understanding. Also, she made me father. Thank you, Chantal, for these two beautiful children! I would like to thank all the mathematicians that I met during this years. Thank you Ruwen, Carlo, Mikael, Alessandro, Caroline, Niamh, Olivier and Jay for helping me enter this field of mathematics. Thank you all for giving me pointers in the basics of algebraic geometry and representation theory that I was missing from physics. And with the other people of the working group, Patrick, Bernhard, Laura, Birte, Johannes, Dario, Emily, Ulrike, Julian and Inga, I thank you for all the social activities outside of work. A special thank to Ruwen, Carlo, Bernhard, Laura and Birte for an amazing office ambience (in particular Bernhard, you are the best!!!). A great thank you goes to Frank L¨ubeck for helping me understand what I had to do, how to do it, what I was doing and how to do it better. Thanks for all the discussions in Aachen and online. I wish to thank professor Meinolf Geck for useful discussions and for grading my thesis. A special thank you goes to Jean Michel and C´edricBonnaf´efor clarifications of crucial points of this thesis (regarding the Gel'fand{Graev characters). It was great living in Neustadt, especially for the amazing people that we met there. Thank you Daniel and Alena for all the nice weekends together. Thank you Elena, Clarissa, Max and Flo for your sympathy. Thank you Sebastian and Eva because the world is so small that we had to meet again in Neustadt! Thanks Ugur for running with me, even in winter. Thanks Sergio for all the ice cream! A huge thank you goes to my family for their support during all my years of study. In particular to my parents that allowed me to follow my dreams and always believed in me! Finally, I would like to thank again the proofreaders of this thesis (professor Malle, Laura and Birte) because I'm a terrible writer, but thanks to them, you (the reader) might never know it! Thank you everyone!!! 2 Abstract Deligne{Lusztig theory allows the parametrization of generic character tables of finite groups of Lie type in terms of families of conjugacy classes and families of irreducible characters \independently" of q. Only in small cases the theory also gives all the values of the table. For most of the groups the completion of the table must be carried out with ad-hoc methods. The aim of the present work is to describe one possible computation which avoids Lusztig's theory of \character sheaves". In particular, the theory of Gel'fand{Graev characters and + Clifford theory is used to complete the generic character table of G = Spin8 (q) for q odd. As an example of the computations, we also determine the character table of SL4(q), for q odd. In the process of finding character values, the following tools are developed. By explicit use of the Bruhat decomposition of elements, the fusion of the unipotent classes of G is determined. Among others, this is used to compute the 2-parameter Green functions of every Levi subgroup with disconnected centre of G. Furthermore, thanks to a certain action of the centre Z(G) on the characters of G, it is shown how, in principle, the values of any character depend on its values at the unipotent elements. + It is important to consider Spin8 (q) as it is one of the \smallest" interesting examples for which Deligne{Lusztig theory is not sufficient to construct the whole character table. The reasons is related to the structure of G = Spin8, from which G is constructed. Firstly, G has disconnected centre. Secondly, G is the only simple algebraic group which has an outer group automorphism of order 3. And finally, G can be realized as a subgroup of bigger groups, like + E6(q), E7(q) or E8(q). The computation on Spin8 (q) serves as preparation for those cases. Zusammenfassung Die Deligne{Lusztig Theorie ist ein wichtiges Konstrukt in der Darstellungstheorie, mit welcher die Parametrisierung generischer Charaktertafeln endlicher Gruppen vom Lietyp durchgef¨uhrt werden kann. Diese Parametrisierung erfolgt durch Familien von Konjugiertenklassen und Familien irreduzibler Charaktere, welche “unabh¨angig"von q sind. Allerdings ergeben sich aller Werte einer Charaktertafel nur in kleinen Gruppen durch diese Theorie. F¨urdie meisten Gruppen muss die Vervollst¨andigung der Charaktertafel mithilfe von Ad- hoc-Methoden durchgef¨uhrtwerden. Das Ziel dieser Arbeit ist es, eine m¨ogliche Rechnung zu beschreiben, welche Lusztigs Theorie von \character sheaves" vermeidet. Insbesondere wird + die generische Charaktertafel der Gruppe G = Spin8 (q) f¨urungerade Werte von q mithilfe von Gel'fand{Greav Charakteren und der Clifford Theorie vervollst¨andigt.Wir bestimmen die Charaktertafel von SL4(q), mit ungeradem q, um ein Beispiel f¨urdie Rechnungen zu geben. Um die Charakterwerte zu berechnen, werden im Laufe der Arbeit verschiedene Werkzeuge entwickelt werden. So wird zum Beispiel durch die explizite Nutzung der Bruhat-Zerlegung von Gruppenelementen die Fusion unipotenter Klassen in G festgelegt. Dies wird unter anderem verwendet, um die 2-Parameter Green-Funktionen jeder Leviuntergruppe von G mit unzusam- menh¨angendemZentrum zu berechnen. Dank einer bestimmten Operation des Zentrums Z(G) auf den Charakteren von G, kann weiterhin gezeigt werden, dass die Werte jedes Charakters im Prinzip nur von seinen Werten auf den unipotenten Elementen abh¨angen. + Die Gruppe Spin8 (q) ist hier von besonderem Interesse, da diese Gruppe eines der \klein- sten" interessanten Beispiele ist, f¨urwelches die Deligne{Lusztig Theorie nicht gen¨ugtum die ganze Charaktertafel zu berechnen. Dies l¨asstsich auf die Struktur der Gruppe G = Spin8 zur¨uckf¨uhren,von welcher G konstruiert wird. Zum einen hat G ein unzusammenh¨angendes Zentrum. Andererseits ist G die einzige einfache algebraische Gruppe, die einen Gruppenauto- morphismus der Ordnung 3 besitzt. Schließlich kann G als eine Untergruppe gr¨oßererGruppen + wie E6(q), E7(q) oder E8(q) aufgefasst werden. Die Berechnung f¨urSpin8 (q) in dieser Arbeit wird als Vorbereitung f¨urdiese F¨alledienen. 4 Contents Acknowledgments 2 Abstract/Zusammenfassung 4 List of Tables 7 Introduction 10 I Background theory 14 1 Finite groups of Lie type 14 1.1 Connected reductive groups . 14 1.2 Root system, Weyl group and structure of connected reductive groups . 16 1.3 Classification and isogenies of semisimple groups . 19 1.4 BN-pair, Bruhat decomposition and Chevalley relations . 20 1.5 Steinberg maps and finite groups of Lie type . 26 1.6 Regular embeddings . 31 1.7 Dual group and geometric conjugacy . 32 2 Basic ordinary representation theory of finite groups 34 3 Character theory of finite groups of Lie type 39 4 About 2-parameter Green functions 46 4.1 General observations and known facts . 46 4.2 Groups with non-connected centre . 51 4.3 The split case . 53 4.4 A note on the method of [Lue20] . 55 5 (Modified) Gel'fand{Graev characters 56 5.1 Regular unipotent elements . 56 5.2 Gel'fand{Graev characters . 58 5.3 Modified Gel'fand{Graev characters . 65 5.4 Some partial Gauss sums . 69 6 Outline of the computation 76 6.1 The starting point: conjugacy classes and almost characters of finite groups of Lietype ........................................ 76 6.2 Construction of the groups, the special case of simply connected groups . 78 6.3 Fusion of unipotent classes . 80 6.4 2-parameter Green functions . 82 6.5 Gel'fand{Graev, regular and semisimple characters . 82 6.6 Decomposition of almost characters . 83 II The case of SL4(q) for q ≡ 1 (mod 4) 88 7 The simply connected group of type A3 and the finite groups SL4(q) 88 7.1 Roots, Chevalley generators and Chevalley relations . 89 7.2 The finite special linear groups . 91 5 8 Fusion of unipotent classes 92 9 Levi subgroups with disconnected centre 95 9.1 Split Levi subgroup . 95 9.2 Twisted Levi subgroup . 97 10 The 2-parameter Green functions 99 11 Modified Gel'fand{Graev characters 101 12 Decomposition of almost characters 104 12.1 Decomposition of R15 and R18 ............................ 104 12.2 Decomposition of R20 and R21 ............................ 107 12.3 Decomposition of R33, R35 and R41 ......................... 107 12.4 Decomposition of R39 ................................. 109 12.5 Decomposition of R37 ................................. 109 12.6 Decomposition of R43, R47 and R51 ......................... 110 12.7 Closing remarks . 111 + III The case of Spin8 (q) for q odd 114 + 13 The simply connected group of type D4 and the finite groups Spin8 (q) 114 13.1 Explicit construction of the algebraic group . 115 13.1.1 Roots and weights .