Deligne–Langlands correspondence

Deligne–Langlands correspondence
Name	Deligne–Langlands correspondence
Field	Number theory; Representation theory; Algebraic geometry
Introduced	1970s
Contributors	Pierre Deligne; Robert Langlands; George Lusztig; Kazhdan; Gelfand; Macdonald

Deligne–Langlands correspondence The Deligne–Langlands correspondence is a conjectural and partial theorem connecting representations of reductive p-adic groups with geometric and Galois-theoretic data, formulated in the context of the work of Pierre Deligne and Robert Langlands and developed by researchers such as George Lusztig, J. Bernstein, I. G. Macdonald, and D. Kazhdan. It lies at the intersection of representation theory of Reductive groups over local fields, the arithmetic of Local fields, and the geometry of Algebraic varietys, and has motivated advances in the theories of Weil group, Galois representation, Hecke algebra, and Perverse sheafs.

Introduction

The correspondence proposes a link between admissible representations of a reductive group over a non-archimedean local field, such as GL(n), SL(n), SO(n), and Sp(2n), and Langlands parameters given by certain homomorphisms from the Weil group or Weil–Deligne group into the L-group of the reductive group, influenced by constructions in Étale cohomology, ℓ-adic cohomology, and the theory of Perverse sheafs. Early formulations drew on ideas from the Langlands program, the Local Langlands correspondence for GL(n) proved in work by Michael Harris, Richard Taylor, and others, and from geometric techniques introduced by Deligne in the context of the Weil conjectures and Étale cohomology.

Historical background and motivations

The seeds of the correspondence appear in Langlands's letters and lectures linking automorphic representations with Galois groups, and in Deligne's use of algebraic geometry to study L-functions arising from Étale cohomology and the Weil conjectures. Work by Jacquet and Langlands on reductive groups over local fields, and later contributions by Kazhdan and Lusztig on character sheaves and representations of Hecke algebras, provided concrete cases and computational tools. Developments in the representation theory of p-adic groups, including the classification of irreducible representations for GL(2), and progress on the Local Langlands correspondence for GL(n), further motivated a Deligne-style reformulation linking geometric parameters to representation-theoretic objects.

Statement of the correspondence

Roughly stated, the correspondence attaches to an irreducible admissible representation of a reductive group G over a non-archimedean local field a finite set of Langlands parameters: semisimple homomorphisms from the Weil–Deligne group W' into the L-group ^LG, respecting Frobenius and monodromy, together with data accounting for component groups and central characters. For split groups like GL(n), the correspondence is bijective and compatible with local factors; for groups such as SL(n), SO(2n+1), and GSp(4), one must consider endoscopic transfer, stable distributions, and packets predicted by the Arthur conjectures. Compatibility properties tie the correspondence to notions from Representation theory of Lie groups, including parabolic induction, supercuspidal support studied by Bernstein, and character identities framed using Harish-Chandra theory.

Constructions and proof techniques

Constructions employ harmonic analysis on p-adic groups, the theory of Hecke algebra modules, and geometric approaches using sheaves on Flag varietys and moduli of local systems inspired by Deligne's use of perverse sheaves; key tools include the study of simple types by Bushnell and Kutzko, the method of quasi-characters developed by Harish-Chandra and Howe, and the use of fundamental lemmas proven by Ngô Bảo Châu in the context of the Hitchin fibration. Geometric Langlands techniques, drawing on ideas of Beilinson and Drinfeld, and the formulation of character sheaves by Lusztig give insight into packets and endoscopy, while ℓ-adic local monodromy and the work of Deligne on the Weil–Deligne group clarify the Galois side. Global-to-local arguments utilize automorphic lifting techniques from the work of Clozel, Harris, and Taylor.

Examples and special cases

The correspondence is fully established for GL(1) via local class field theory associated to Kummer theory and the Weil group, and for GL(n) through the proof of the local Langlands correspondence by work of Harris–Taylor and alternative approaches by Henniart and Scholze. Explicit descriptions exist for depth-zero supercuspidal representations studied by Moy and Prasad and for simple groups of low rank such as SL(2), PGL(2), and small classical groups where endoscopic phenomena first arise, as analyzed by Arthur and Shelstad. Examples constructed via Deligne’s ℓ-adic techniques illuminate how monodromy operators in the Weil–Deligne group correspond to nilpotent orbits classified by Dynkin diagrams and work of Kostant.

Relation to local Langlands and other correspondences

The Deligne–Langlands correspondence refines and complements the Local Langlands correspondence by emphasizing geometric and categorical constructions inspired by Pierre Deligne's methods, and it interacts with the Automorphic Langlands correspondence in global contexts through trace formulas developed by James Arthur and stabilization results proved using endoscopy pioneered by Robert Langlands and D. Shelstad. It connects to the Geometric Langlands correspondence as formulated by Beilinson and Drinfeld on curves over finite fields and to the categorical perspectives advanced by Gaitsgory and Ben-Zvi in the study of sheaves on moduli stacks of local systems.

Applications and consequences

Consequences include the classification of irreducible admissible representations of many p-adic groups, applications to the computation of local L-factors and ε-factors informing work on the Global Langlands correspondence, advances in the understanding of harmonic analysis on p-adic groups, and input to the proof of instances of the Ramanujan–Petersson conjecture via automorphic liftings of Galois representations. The framework has inspired progress in the theories of character sheaves by Lusztig, the proof of fundamental lemmas by Ngô, and ongoing developments relating categorical representation theory pursued by Bernstein, Kazhdan, and Bezrukavnikov.

Category:Langlands program