Section
Langlands Letter
An annotated reading of Langlands' letter to Weil
Core idea
An annotated reading of Robert Langlands' famous 1967 letter to Andre Weil, which launched the Langlands program.
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Definitions
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Algebraic Groups
- Split reductive algebraic group
- Simply connected semisimple group
- Maximal torus and weight lattice
- Roots, Weyl group, and dominant weights
- Coroots and the weight–coroot pairing
- Root lattice, weight lattice, and isogeny forms
- Chevalley basis
- Chevalley lattice and integral model
- Pinning and pinned automorphisms
- Semisimple element and conjugacy class
- Characters separate semisimple conjugacy classes
- Borel–Mostow normalizer representative
Langlands Duality
- Langlands dual group
- Dual lattice
- L-group and Satake parameter
- Langlands functoriality and L-homomorphisms
- Contragredient (dual) representation
Number Fields and Local Fields
- Global and local fields; completions
- p-adic field
- Galois extension and Galois group
- Unramified extension of a p-adic field
- Unramified prime and Frobenius element
- Choosing embeddings Q̄ → Q̄ₚ
Adeles and Automorphic Forms
- Adeles and restricted products
- Ideles, Hecke characters, and Artin reciprocity
- Automorphic form and Hecke eigenvalues
- Eisenstein series on a reductive group
Hecke Algebras and L-functions
- Spherical Hecke algebra and Satake isomorphism
- Maximal compact and hyperspecial subgroup
- Group algebra of a lattice
- Euler product and local L-factor