Higgs bundles and Hermitian symmetric spaces

Roberto Rubio N´u˜nez ICMAT CSIC-UAM-UC3M-UCM

Tesis presentada para la obtenci´ondel t´ıtulode Doctor en Matem´aticas por la Universidad Aut´onomade Madrid. Director: Oscar´ Garc´ıa-Prada.

24 de febrero de 2012 2 A mis padres y a mi hermano 4 Contents

Acknowledgementsi

List of Symbols iii

Introducci´on:resumen y conclusiones vii

1 Introduction: summary and main results1

2 Hermitian symmetric spaces 11 2.1 Basics on symmetric spaces and root theory...... 11 2.2 ...... 19 2.3 Restricted root theory...... 24 2.3.1 Classical results...... 24 2.3.2 More results on restricted roots...... 26 2.3.3 Sums of roots...... 28 2.4 The Toledo character...... 30 2.4.1 Definition of the Toledo character...... 30 2.4.2 structure...... 34 2.5 Parabolic subgroups...... 40 2.5.1 Basics on R-parabolic subgroups...... 40 2.5.2 Relevant parabolic subgroups...... 42 2.5.3 Subtubes and Levi factors...... 48 2.6 Normalizer of the maximal tube subdomain...... 51

3 Higgs bundles 56 3.1 Basics on twisted Higgs bundles and stability...... 56 3.2 HK correspondence and moduli spaces...... 59 3.3 G-Higgs bundles when G is of Hermitian type...... 60 3.4 α-Milnor-Wood inequality...... 62 3.5 The Toledo invariant and the topological class...... 66

4 Maximal Toledo invariant 68 4.1 Milnor-Wood inequality...... 68 4.2 Tube-type groups and Cayley correspondence...... 70 4.3 Non-tube groups and stabilization of a maximal tube...... 77

5 Surface group representations 83 5.1 Correspondence with Higgs bundles...... 83 5.2 Equivalent definitions of the Toledo invariant...... 87 5.3 Milnor-Wood inequality and rigidity of maximal representations... 89 5.4 Future directions...... 91

A Examples 93

B Additional remarks 104

C Tables 107

Bibliography 112 Acknowledgements

This thesis would not have been possible without the aid and support of countless people and many institutions during the past seven years. I must first express my gratitude towards my supervisor Oscar´ Garc´ıa-Pradafor suggesting me this problem and for his support, guidance and sponsoring during all this time. I would also like to express my deepest gratitude to Olivier Biquard, for hosting me during several visits to the University of Paris VI, and for his time and his commitment with the research project. He has indeed acted as an unofficial co-supervisor. I would also like to thank the people and institutions that have hosted other valu- able research stays: Ignasi Mundet at the University of Barcelona, Peter Gothen at the University of Porto and, in particular, Nigel Hitchin at the University of Oxford. I would wish to convey thanks to the institutions that have hosted this project, the Instituto de Ciencias Matem´aticasICMAT CSIC-UAM-UC3M-UCM and the Univer- sity Aut´onoma of Madrid, as well as to their members and staff. Many other mathematicians have helped me with useful conversations, as Luis Alvarez-C´onsul,Steve´ Bradlow, Marco Castrill´on,Vicente M´u˜noz,and in particu- lar Tom´asG´omez. I wish to give my thanks to some fellow graduate students and postdocs for their (not only mathematical) support and friendship. My gratitude to Paloma Bengoechea, Ang´elicaBenito, Javier Fres´an,Luis Emilio Garc´ıa, Car- los Gonz´alez-Guill´en,Luis Hern´andez-Corbato,Jes´usP. Moreno, Jos´eand Alberto Navarro, Mar´ıaPe, Ana Primo, Steve Rayan, Juanjo Ru´e,Laura Shaposnik, Car- los Vinuesa, Alfonso Zamora, especially to the 1001 people: Marta Aparicio, Emilio Franco, Fernando Jim´enez,Marina Logares, C´edricMart´ınez-Campos, Ana Pe´on,and very especially to Alvaro´ Ant´onand Mario Garc´ıa-Fern´andez. Tracking back in time, I do not forget my years as an undergraduate in the Uni- versity of Valencia. Thanks to professors Juan Climent, Rafael Crespo, Olga Gil, Antonio Mart´ınez-Naveira, Pepe Mar