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| GL_2(Q_p) | |
|---|---|
| Name | GL_2(Q_p) |
| Type | Reductive group |
| Field | Q_p |
GL_2(Q_p)
GL_2(Q_p) is the group of 2×2 invertible matrices with entries in the p-adic field Q_p. It is a non-compact, locally compact, totally disconnected topological group intimately connected with the arithmetic of P-adic number, the representation theory of Bernstein center, and the local components of Automorphic representation. The group plays a central role in the local Langlands correspondence for GL_n and in the study of Modular forms, Galois representations, and Lubin–Tate formal groups.
GL_2(Q_p) is defined as the set of 2×2 matrices with entries in Q_p and determinant in Q_p^×. It contains subgroups isomorphic to Borel-type matrices, maximal compact subgroups conjugate to GL_2(Z_p), and split tori isomorphic to Q_p^× × Q_p^×. The group admits the Cartan decomposition relative to a maximal compact subgroup and the Iwasawa decomposition relative to a fixed Borel; these decompositions are fundamental in harmonic analysis on the group and in studying spherical functions related to the Satake isomorphism. Conjugacy classes correspond to characteristic polynomials with coefficients in Q_p, linking to the theory of Weil group elements and local L-factors.
Standard matrix subgroups include the diagonal torus, the upper-triangular Borel, the unipotent subgroup of transvections, the normalizer of the torus, and compact open subgroups such as GL_2(Z_p), the principal congruence subgroups, and the Iwahori subgroup associated with the Bruhat–Tits building. The Bruhat decomposition expresses GL_2(Q_p) as a union of double cosets relative to the Borel and the Weyl group generated by a simple reflection; the Cartan decomposition uses the valuation on Q_p to parametrize double cosets by dominant cocharacters. These decompositions underlie the construction of parabolic induction from proper parabolic subgroups and the analysis of intertwining operators studied by Jacquet and Langlands.
As a p-adic Lie group, GL_2(Q_p) has a structure modeled on the analytic manifold of 4 dimensions over Q_p; compact open subgroups are profinite and admit filtrations by congruence subgroups used in the study of Tate modules and Fontaine-theory. The Bruhat–Tits building for GL_2 is a regular tree on which the group acts simplicially; this action is a key tool in the study of reduction theory related to Drinfeld upper half plane and Mumford curve phenomena. Local harmonic analysis employs the Haar measure normalized on GL_2(Z_p), and the topology interacts with the representation theory of smooth and admissible representations developed by Bernstein and Zelevinsky.
Smooth irreducible representations of GL_2(Q_p) over C were classified by the work of Bernstein, Zelevinsky, and Bushnell–Henniart; principal series, special (Steinberg), and supercuspidal representations appear as building blocks. The local Langlands correspondence for GL_2 relates two-dimensional Weil–Deligne representations of the local Weil group to irreducible admissible representations; this is central to the compatibility with global correspondences studied by Carayol and Harris–Taylor. Modular representations over finite fields and p-adic Banach space representations are studied in the context of the p-adic Langlands program initiated by Colmez and connected to (φ,Γ)-module theory of Fontaine.
Local factors for L-functions of automorphic forms are computed via local representations of GL_2(Q_p), influencing the study of Modular curves, Elliptic curve local factors, and conductor exponents in the Artin conductor formalism. The correspondence between two-dimensional Galois representations and admissible representations of GL_2(Q_p) informs the proofs of modularity lifting theorems by Wiles, Taylor–Wiles, and Kisin. Local newform theory for GL_2(Q_p) and the Atkin–Lehner theory play roles in explicit computations of p-adic L-functions associated with Hida familys and Coleman families.
Hecke algebras associated with compact open subgroups of GL_2(Q_p) act on spaces of automorphic forms on GL_2 over global fields; the spherical Hecke algebra for GL_2(Z_p) is commutative and described by the Satake isomorphism linking to representations of the dual group studied by Langlands. Iwahori–Hecke algebras model the action on Iwahori-fixed vectors and connect to affine Hecke algebras appearing in the work of Iwahori and Matsumoto; these structures are instrumental in the construction of local components of automorphic representations used in the proofs by Gelbart and Jacquet–Langlands.
Cohomology of arithmetic quotients with coefficients in smooth representations of GL_2(Q_p) relates to the etale cohomology of Shimura varietys and the p-adic local systems arising from Galois representations. The action on the Bruhat–Tits building yields cohomological descriptions used in the study of the Lubin–Tate tower and its link to the local Langlands correspondence via the work of Carayol, Harris–Taylor, and Boyarchenko–Weinstein. Moduli interpretations appear in the study of deformation spaces of one-dimensional formal groups, the reduction of modular curves at p studied by Deligne–Rapoport, and the geometry of Rapoport–Zink spaces investigated by Rapoport and Zink.