p-adic groups

p-adic groups
Name	p-adic groups
Type	Topological group
Field	Number theory; Representation theory; Harmonic analysis

p-adic groups are groups formed by the points of algebraic groups over non-Archimedean local fields that inherit a totally disconnected locally compact topology. They arise naturally in the study of Local fields, Adèles, and the Langlands program, and play a central role in the theories of Automorphic forms, the Representation theory of reductive groups, and the classification of arithmetic objects such as Galois representations and L-functions. Work by mathematicians including John T. Tate, Robert Langlands, Harish-Chandra, André Weil, and Pierre Deligne established foundational links between p-adic groups and major results in modern Number theory.

Definition and basic examples

A basic source of examples is an algebraic group G defined over a non-Archimedean local field K (for example Q_p or finite extensions studied in Local class field theory); the group G(K) equipped with the topology from K is a p-adic group. Typical instances include GL_n(K), SL_n(K), SO_n(K), Sp_{2n}(K), and inner forms such as unitary groups arising from Hermitian forms over quadratic extensions. Compact open subgroups such as GL_n(Z_p) and maximal compact subgroups linked to Bruhat–Tits buildings furnish concrete examples; finite groups like GL_n(F_q) appear as reductive quotients of compact subgroups and connect to the Deligne–Lusztig theory. Historical constructions involve contributions from Iwahori and Matsumoto and later formalization by Bruhat and Tits.

Topological and algebraic structure

As totally disconnected locally compact groups, these groups admit Haar measure (a concept used by Haar and by Von Neumann in operator contexts) and possess a rich supply of compact open subgroups studied by Moy and Prasad. The structure theory uses the Bruhat decomposition, Iwahori subgroups, and the combinatorial geometry of Bruhat–Tits buildings associated to reductive groups; these buildings are linked to work of Tits and Serre. Important algebraic tools include maximal tori, Borel subgroups, and root data as in Chevalley and Cartan frameworks, while [Kottwitz] and Steinberg contributions clarify forms and isogeny classes. Local notions of ramification relate to Weil–Deligne group representations studied by Deligne and Grothendieck.

Representation theory

The smooth complex representations of p-adic groups were systematized by Bernstein, Zelevinsky, Casselman, and Bernstein–Zelevinsky theory, leading to classifications of irreducible admissible representations for GL_n via segments and multisegments. The local Langlands correspondence conjectured by Langlands and established in many cases by Harris–Taylor, Henniart, and Moy–Prasad matches irreducible admissible representations with Weil–Deligne or Galois representation parameters. Supercuspidal representations, parabolic induction, and Jacquet modules (developed by Jacquet and Langlands) are central; types and covers in the sense of Bushnell and Kutzko produce explicit constructions. Notions of unitarity, Kirillov models, and the role of spherical representations studied by Macdonald connect to special functions and Satake isomorphism results by Satake.

Harmonic analysis and Hecke algebras

Harmonic analysis on p-adic groups employs convolution algebras of compactly supported functions and Hecke algebras introduced in contexts like Iwahori–Hecke algebras. The spherical Hecke algebra relates via the Satake isomorphism to representation rings and to Langlands dual group structures used in the formulation of the Langlands correspondence by Satake and Kottwitz. The theory of characters, Plancherel measure, and harmonic analysis was developed by Harish-Chandra for real groups and adapted to p-adic settings by Harish-Chandra's school, Bernstein, and Deligne. Hecke operators appear in the study of Modular forms, Shimura varietys, and in trace formulae due to Arthur and Selberg which connect to stable trace formula work by Labesse and Langlands.

Examples and classification (reductive p-adic groups)

Reductive p-adic groups are classified by root data, Galois action, and inner forms as in the algebraic classification by Chevalley, Steinberg, and Tits. Important families include general linear groups GL_n, special linear groups SL_n, symplectic groups Sp_{2n}, orthogonal groups SO_n, and quasi-split unitary groups arising in Hermitian symmetric space contexts and in the work of Ramakrishnan. Classification results for representations often treat split groups first, then extend to quasi-split and inner forms using methods of Kottwitz, Shelstad, and Arthur; endoscopic classification and the stabilization of trace formulas are central in modern advances.

Applications in number theory and automorphic forms

p-adic groups underpin local components of automorphic representations appearing in the Langlands program and in the proof of reciprocity results like those in Harris–Taylor and Wiles's approach to Modularity theorem. Local factors of L-functions, epsilon factors of Tate and Deligne, and local-global compatibility in the cohomology of Shimura varietys depend on detailed properties of p-adic group representations. The trace formula of Arthur leverages harmonic analysis on p-adic groups to compare automorphic spectra, leading to applications in functoriality conjectures and explicit reciprocity laws explored by Taylor, Clozel, and Kisin.

Category:Representation theory