Smooth-Automorphic Forms and Smooth-Automorphic Representations

✍ Scribed by Harald Grobner

• Publisher: World Scientific
• Year: 2023
• Tongue: English
• Leaves: 262
• Series: Series on Number Theory and Its Applications 17
• Category: Library

✦ Table of Contents

Contents
Preface
Acknowledgements
Part 1: Local Groups
1. Basic Notions and Concepts from Functional Analysis ("Local")
1.1 Seminormed spaces
1.2 Three constructions with seminormed spaces
1.2.1 Subspaces and -quotients
1.2.2 Direct sums
1.2.3 Tensor products
2. Representations of Local Groups – The Very Basics
2.1 Local fields and local groups
2.2 Local representations
2.3 Smooth representations
2.4 Admissible representations
3. Langlands Classification: Step 1 – What to Classify?
3.1 The relationship of smoothness and admissibility for irreducible representations – a detour to (g;K)-modules
3.2 Infinitesimal equivalence – advantages and disadvantages
3.3 Interesting representations
3.4 Unitary representations
4. Langlands Classification: Step 2
4.1 The contragredient representation
4.2 Square-integrable and tempered representations
4.3 Parabolic induction
4.3.1 Parabolic subgroups and attached data
4.3.2 Smooth induction
4.3.3 L2-induction and questions of unitarity
4.3.4 (g;K)-induction
4.3.5 Induction from unitary representation of GLn(D)
5. Langlands Classification: Step 3
5.1 The result – tempered vs. square-integrable formulation
5.2 Some remarks on square-integrable representations
6. Special Representations: Part 1
6.1 Supercuspidal representations
6.2 A specification of the general theory: The Bernstein–Zelevinsky classification for GLn
6.2.1 A step beyond GLn(F): General D
6.2.2 Local isobaric sums
7. Special Representations: Part 2
7.1 Generic representations
7.2 Unramified representations
Part 2: Global Groups
8. Basic Notions and Concepts from Functional Analysis ("Global")
8.1 Inductive limits
8.1.1 Inductive limits and strict inductive limits
8.1.2 LF-spaces
8.2 Tensor products involving an infinite number of semi-normed spaces
8.2.1 The inductive tensor product
8.2.2 The restricted Hilbert space tensor product
9. First Adelic Steps
9.1 Global fields and global groups
9.2 Parabolic subgroups and attached data
10. Representations of Global Groups – The Very Basics
10.1 Global representations
10.2 Smooth representations
10.3 Admissible representations
10.4 (g1;K1;G(Af))-modules
11. Automorphic Forms and Smooth-Automorphic Forms
11.1 Spaces of smooth functions with predetermined growth
11.2 The LF-space of smooth-automorphic forms
11.3 Smooth-automorphic forms as representation of G(A)
11.4 Automorphic forms
11.5 Remarks on the full space of all smooth-automorphic forms
12. Automorphic Representations and Smooth-Automorphic Representations
12.1 First facts and definitions
12.2 A dictionary between smooth-automorphic representations and automorphic representations
12.3 The restricted tensor product theorem
12.3.1 The algebraic restricted tensor product
12.3.2 The restricted tensor product theorem for irreducible smooth-automorphic representations
13. Cuspidality and Square-integrability
13.1 Unitary representations on spaces defined by square-integrable functions
13.2 (g1;K1;G(Af))-modules on spaces defined by square-integrable functions
14. Parabolic Support
14.1 Restricting to smooth-automorphic forms on [G]
14.2 LF-compatible smooth-automorphic representations
14.3 The parabolic support of a smooth-automorphic form
15. Cuspidal Support
15.1 Decomposing spaces of cuspidal smooth-automorphic forms
15.2 Associate classes of cuspidal smooth-automorphic representations
15.2.1 Review of Eisenstein series
15.3 The cuspidal support of a smooth-automorphic form
References
Index

✦ Subjects

Representation theory; Langlands Program