GL(n,Q_p)

GL(n,Q_p)
Name	GL(n,Q_p)
Type	Matrix group
Field	p-adic numbers
Related	SL(n,Q_p), Mat_n(Q_p), O(n), GL(n,R)

GL(n,Q_p)

GL(n,Q_p) is the group of invertible n×n matrices with entries in the p-adic numbers Q_p, considered as a locally compact, totally disconnected topological group and as a p-adic analytic Lie group over Q_p. It plays a central role in the representation theory of reductive groups, in the formulation of the local factors in the Langlands program, and in arithmetic applications to automorphic forms and number theory. The group connects deep work of figures such as John Tate, Pierre Deligne, Robert Langlands, Harish-Chandra, and Igor Bernstein.

Definition and basic properties

As a set GL(n,Q_p) = {A ∈ Mat_n(Q_p) : det(A) ≠ 0 in Q_p}, with group law given by matrix multiplication and inverse given by the adjugate divided by det. Basic algebraic properties reflect its nature as a reductive algebraic group defined over Q_p, sharing links to SL(n,Q_p), Tate module considerations, and the theory of algebraic groups developed by Armand Borel and Jacques Tits. The determinant map det: GL(n,Q_p) → Q_p^× is a continuous surjective group homomorphism; its kernel is SL(n,Q_p). Compact open subgroups such as GL(n,Z_p) arise from the p-adic integers Z_p and are maximal compact subgroups in the sense of Bruhat–Tits theory as studied by François Bruhat and Jacques Tits.

Topology and p-adic analytic structure

Endowed with the topology induced by the product topology on n^2 copies of Q_p, GL(n,Q_p) is locally compact, totally disconnected and second countable. It admits the structure of a p-adic analytic manifold in the sense of Jean-Pierre Serre and Michel Lazard, giving it a Q_p-analytic Lie algebra isomorphic to Mat_n(Q_p). Neighborhood bases at the identity are provided by congruence subgroups 1 + p^k Mat_n(Z_p), tied to deformation-theoretic studies by Barry Mazur and to the theory of formal groups in the work of Alexander Grothendieck. The Bruhat decomposition organizes topology via the Weyl group W ≅ S_n, connecting to combinatorial structures investigated by James Humphreys and George Lusztig.

Subgroups and decompositions (Borel, Iwahori, parahoric)

Standard subgroups include Borel subgroups of invertible upper triangular matrices, maximal tori of diagonal matrices, and unipotent radicals; these mirror the algebraic structure emphasized by Armand Borel and Tits building geometry of Bruhat–Tits building. Iwahori subgroups, introduced in the work of Nagayoshi Iwahori and Hitoshi Matsumoto, are compact open subgroups analogous to Borel subgroups over finite fields and correspond to chambers in the Bruhat–Tits building as developed by Bruhat and Tits. Parahoric subgroups generalize Iwahori subgroups and connect to models of reductive groups over discrete valuation rings studied by Jean-Pierre Serre and Robert Kottwitz. Cartan and Iwasawa decompositions describe GL(n,Q_p) as double coset unions relative to GL(n,Z_p) and diagonal tori, with Cartan decomposition parameters given by dominant cocharacters studied in the context of Satake isomorphism by Ichirô Satake.

Representations and harmonic analysis

Smooth complex representations of GL(n,Q_p) were systematically developed by Harish-Chandra, Joseph Bernstein, and Allan Moy with Gopal Prasad extending to types and Hecke algebras. Principal series representations induced from characters of the diagonal torus, supercuspidal representations constructed via compact induction and Bushnell–Kutzko types, and the classification results of Colin Bushnell and Philip Kutzko are central. The harmonic analysis of GL(n,Q_p) involves spherical functions, Hecke algebras, and the Plancherel formula connected to work of Ilya Piatetski-Shapiro and Frederic Shahidi. Local zeta integrals and Rankin–Selberg convolutions, building on techniques of Gelfand and Ilya Graev, relate representations to local L-factors and epsilon-factors.

Arithmetic applications and local Langlands correspondence

Local arithmetic applications include the parametrization of n-dimensional Weil–Deligne representations of the local Weil group of Q_p and the formulation of the local Langlands correspondence for GL(n), proved by Michael Harris, Richard Taylor, Guy Henniart, and aggregated via work of Bernstein and Deligne. This correspondence matches irreducible admissible representations of GL(n,Q_p) with Frobenius-semisimple Weil–Deligne representations, playing a key role in proofs of modularity lifting theorems by Andrew Wiles, Richard Taylor, and Mark Kisin. Connections to Shimura varieties studied by James Milne, to Galois deformation rings in the work of Barry Mazur and Andrew Wiles, and to the global Langlands program of Robert Langlands make GL(n,Q_p) indispensable in modern arithmetic geometry.

Measures, Haar measure and integration on GL(n,Q_p)

As a locally compact group, GL(n,Q_p) admits a Haar measure unique up to scalar, used to define Haar integrals and orbital integrals central to the trace formula of James Arthur and to character theory pioneered by Harish-Chandra. Haar measure normalization choices (e.g., vol(GL(n,Z_p)) = 1) influence local factors in Tamagawa number computations associated with Tamagawa number conjecture contexts and integrals in the Rankin–Selberg method employed by Jacquet and Shalika. Integration theory on GL(n,Q_p) interacts with harmonic analysis via convolution algebras, Hecke operators studied by Iwahori and Satake, and with the study of p-adic L-functions promoted by Kazuya Kato and Robert Coleman.

Category:Reductive groups Category:p-adic groups