Langlands dual group

Langlands dual group
Name	Langlands dual group
Type	Reductive algebraic group
Introduced	1960s
Founder	Robert Langlands
Area	Representation theory, Number theory, Algebraic geometry

Langlands dual group The Langlands dual group is a construction assigning to a connected reductive algebraic group over a field a new connected reductive algebraic group obtained by interchanging root and coroot data. It plays a central part in the Langlands program and bridges ideas from Robert Langlands, Harish-Chandra, Gel'fand-Graev methods, and the work of Chevalley, Cartan, Weyl, Borel, and Tits. The dual group appears in formulations linking automorphic representations of Adèle groups, arithmetic of Galois groups, and geometric objects studied by Grothendieck, Deligne, Drinfeld, and Laumon.

Definition and construction

Given a connected reductive algebraic group G over a field, one encodes G by its root datum constructed by Claude Chevalley and Tits; the root datum comprises a character lattice, a cocharacter lattice, a set of roots, and a set of coroots. The Langlands dual group G^ (often denoted by a superscript) is defined by swapping the root and coroot lattices and interchanging roots with coroots; this construction uses the classification of reductive groups by Dynkin diagrams, Cartan matrixs, and Satake isomorphism data. Constructions rely on choices of maximal torus and Borel subgroup as in the work of Borel and Bott, and are compatible with base change studied by Weil and Steinberg. Historical foundations connect to structural results by Kostant, Serre, Humphreys, and Springer.

Examples

For split groups one often encounters classical examples: the dual of GL_n is GL_n, the dual of SL_n is PGL_n, and the dual of Sp_{2n} is SO_{2n+1}. Exceptional groups yield examples such as duality between G_2 and itself, and self-duality phenomena for E_8, F_4, and E_7 in special forms; the dual of Spin_{2n+1} relates to Sp_{2n} and the dual of SO_{2n} splits into forms related to SO_{2n}, Spin_{2n}, and Pin group considerations in the classification of Lie algebras by Élie Cartan and Dynkin diagrams. Non-split inner forms produce duals influenced by Kottwitz invariants and Tamagawa number considerations studied by Langlands and Kottwitz–Shelstad.

Properties and functoriality

The Langlands dual group respects exact sequences and central isogenies investigated by Matsumoto and Steinberg. It is functorial for Levi subgroups and parabolic induction studied by Bernstein, Zelevinsky, and Casselman. Dual groups appear in endoscopic transfer studied by Kottwitz, Shelstad, and Arthur and feed into trace formula comparisons by Selberg, Sarnak, and Waldspurger. Duality interacts with Satake isomorphism for unramified representations as in work by Satake, Gross, and Kottwitz, and with local Langlands correspondences developed by Harris, Taylor, Henniart, Fargues, and Vignéras.

Relation to root data and Cartan duality

Root data and Cartan duality underlie the definition: the duality operation sends the root system of G classified by Bourbaki-type lists and Dynkin labels to the coroot system, reflecting classical Cartan duality studied by Cartan and Killing. This correspondence links weight lattices of representations as in Weyl character formula work by Weyl and Witt to coweight lattices used in studies by Kostant and Lusztig. The interplay with Affine Weyl groups and Iwahori subgroups appears in geometric representation theory by Kazhdan and Lusztig and in the formulation of dual groups for p-adic groups by Tits and Bruhat–Tits theory.

Role in the Langlands program

In the global and local Langlands conjectures of Robert Langlands, the dual group acts as the target for conjectural Langlands parameters from Weil groups, Weil–Deligne groups, or Galois groups; this framework unites results of Tate, Deligne, Drinfeld, Lafforgue, and Harris–Taylor. The dual group governs the parametrization of L-packets and statements of functoriality between automorphic representations of different groups as envisioned by Langlands and pursued by Arthur, Ngô Bao Châu, Gelbart, and Jacquet. The geometric Langlands program developed by Beilinson and Drinfeld recasts the dual group as a structural symmetry acting on moduli of G-bundles over algebraic curves studied by Atiyah and Bott.

Applications in representation theory and number theory

Dual groups are central to classification of admissible representations for p-adic groups via local correspondence work of Bushnell and Kutzko, and to description of automorphic L-functions defined by Langlands–Shahidi and Godement–Jacquet. They appear in arithmetic trace formulas of Kottwitz and in reciprocity laws extending Class field theory by Artin and Chebotarev. Geometric applications connect to the study of perverse sheaves by Beilinson and Bernstein and to categorification efforts by Gaitsgory and Frenkel. Dual groups also influence computations in Hodge theory contexts by Griffiths and Schmid and in the study of motives as framed by Grothendieck and Deligne.

Category:Langlands program