Bruhat–Tits Theory

Tasho Kaletha is Professor of Mathematics at the University of Michigan. He is an expert on the Langlands program, and has studied arithmetic and representation-theoretic aspects of the local Langlands correspondence for p-adic groups. Gopal Prasad is Raoul Bott Professor Emeritus of Mathematics at the University of Michigan. He is a leading expert on real and p-adic Lie groups and algebraic groups. Together with Ofer Gabber and Brian Conrad, he published the complete classification and structure theory of pseudo-reductive groups in the books Pseudo-reductive Groups (2010, 2015) and Classification of Pseudo-reductive Groups (2015).

Introduction; Part I. Background and Review: 1. Affine root systems and abstract buildings; 2. Algebraic groups; Part II. Bruhat–Tits theory: 3. Examples: Quasi-split groups of rank 1; 4. Overview and summary of Bruhat–Tits theory; 5. Bruhat, Cartan, and Iwasawa decompositions; 6. The apartment; 7. The Bruhat–Tits building for a valuation of the root datum; 8. Integral models; 9. Unramified descent; Part III. Additional Developments: 10. Residue field f of dimension ≤ 1; 11. The buildings of classical groups via lattice chains; 12. Component groups of integral models; 13. Finite group actions and tamely ramified descent; 14. Moy–Prasad filtrations; 15. Functorial properties; Part IV. Applications: 16. Classification of maximal unramified tori (d'après DeBacker); 17. Classification of tamely ramified maximal tori; 18. The volume formula; Part V. Appendices: A. Operations on integral models; B. Integral models of tori; C. Integral models of root subgroups; References; Index.