Modern Langlands foundations index

Parameter input

• Langlands dual group
• $L$-group
• Weil group
• Weil–Deligne group
• Weil–Deligne representation
• Satake parameter

Smooth representation theory

• Smooth representation of a totally disconnected group
• Admissible representation of a $p$-adic group
• Supercuspidal representation
• Unramified representation
• Normalized parabolic induction
• Whittaker model

Local Langlands

• Local $L$-parameter
• Basic local Langlands correspondence
• $L$-packet
• Component group of an $L$-parameter
• Whittaker datum
• Refined local Langlands correspondence
• Local Langlands for $\operatorname{GL}_n$
• Local Langlands compatibilities

Automorphic spectrum

• Automorphic form
• Automorphic representation
• Restricted tensor-product factorization
• Cuspidal automorphic representation
• Discrete automorphic spectrum
• Residual automorphic spectrum
• Eisenstein series

Global parameters and reciprocity

• Global Langlands parameter
• Global Langlands reciprocity
• Langlands functoriality
• Automorphic $L$-factor and Euler product

Algebraicity and Galois representations

• Algebraic automorphic representation
• $C$-algebraic automorphic representation
• $L$-algebraic automorphic representation
• Cohomological automorphic representation
• Regular algebraic cuspidal automorphic representation
• Compatible system of Galois representations
• Local–global compatibility
• Automorphic–Galois correspondence

Orbital geometry

• Strongly regular semisimple element
• Orbital integral
• Stable conjugacy
• Stable orbital integral
• $\kappa$-orbital integral

Transfer and stabilization

• Arthur–Selberg trace formula
• Endoscopic datum
• Endoscopic transfer factor
• Endoscopic transfer
• Fundamental lemma
• Stable trace formula

Arthur classification

• Arthur parameter
• Arthur packet
• Arthur multiplicity formula

• Shtuka
• $G$-shtuka
• Partial Frobenius on shtukas
• Coalescence of shtuka legs
• Excursion operator
• Excursion algebra
• Vincent Lafforgue global parameterization

• Fargues–Fontaine curve
• Kottwitz set $B(G)$
• $G$-bundle on the Fargues–Fontaine curve
• Local shtuka
• Stack of local $L$-parameters
• Spectral Bernstein center
• Fargues–Scholze spectral action
• Fargues–Scholze parameter map

• Local packets: Kaletha's survey, beginning with the basic local correspondence and then its refined form.
• Algebraicity: Buzzard–Gee, read with the separate $C$- and $L$-normalizations above.
• Trace formula and endoscopy: Arthur's trace-formula introduction, Ngô's fundamental-lemma survey, and Arthur's endoscopic classification.
• Function fields: Vincent Lafforgue's introduction, followed by the full excursion-operator construction.
• Local geometrization: Imai's survey, then Fargues–Scholze.
• Categorical geometric Langlands: the geometric collection and its cited theorem series.