Smooth-Automorphic Forms and Smooth-Automorphic Representations

This book provides a conceptual introduction into the representation theory of local and global groups, with final emphasis on automorphic representations of reductive groups G over number fields F.

Our approach to automorphic representations differs from the usual literature: We do not consider 'K-finite' automorphic forms, but we allow a richer class of smooth functions of uniform moderate growth. Contrasting the usual approach, our space of 'smooth-automorphic forms' is intrinsic to the group scheme G/F.

This setup also covers the advantage that a perfect representation-theoretical symmetry between the archimedean and non-archimedean places of the number field F is regained, by making the bigger space of smooth-automorphic forms into a proper, continuous representation of the full group of adelic points of G.

Graduate students and researchers will find the covered topics appear for the first time in a book, where the theory of smooth-automorphic representations is robustly developed and presented in great detail.

Contents:

• Local Groups:

  • Basic Notions and Concepts from Functional Analysis ('Local')
  • Representations of Local Groups — The Very Basics
  • Langlands Classification: Step 1 — What to Classify?
  • Langlands Classification: Step 2
  • Langlands Classification: Step 3
  • Special Representations: Part 1
  • Special Representations: Part 2

• Global Groups:

  • Basic Notions and Concepts from Functional Analysis ('Global')
  • First Adelic Steps
  • Representations of Global Groups — The Very Basics
  • Automorphic Forms and Smooth-Automorphic Forms
  • Automorphic Representations and Smooth-Automorphic
  • Cuspidality and Square-integrability
  • Parabolic Support
  • Cuspidal Support

Readership: PhD students and researchers in the fields of automorphic forms, representation theory of local groups (archimedean and non-archimedean) and, more generally, the Langlands Program.
Key Features:

• Our approach to local as well as to global representation theories are new and cannot be found, yet, in this form in a textbook