Topic Reductive Groups over Local Fields
topic admitted

Reductive Groups over Local Fields

A reductive group over a local field \(G/F\) (with \(F\) locally compact and non-discrete) carries a rich representation theory: locally compact topology, smooth representations, parahoric structure, local Langlands. This is the natural setting for local Langlands and the foundation of the global automorphic theory.

Setup

A local field is a locally compact non-discrete topological field. Up to isomorphism:

  • Archimedean: \(\mathbb{R}\) or \(\mathbb{C}\).
  • Non-archimedean mixed characteristic: finite extensions of \(\mathbb{Q}_p\) (\(p\)-adic fields).
  • Non-archimedean equal characteristic: finite extensions of \(\mathbb{F}_q((t))\) (Laurent series).

For \(F\) a local field, \(G(F)\) is a locally compact topological group; for the non-archimedean case it is a totally disconnected topological group.

Topology and structure

For \(G\) reductive over \(F\):

  • \(G(F)\) is a locally compact topological group.
  • For non-archimedean \(F\): \(G(F)\) is totally disconnected (the open subgroups form a basis of identity).
  • The Bruhat-Tits building \(\mathcal{B}(G, F)\) is a contractible metric space on which \(G(F)\) acts properly by simplicial isometries.

Representation theory

  • Smooth representations: representations on complex (or \(\bar{\mathbb{Q}}_\ell\)) vector spaces such that every vector has open stabiliser.
  • Admissible representations: smooth representations on which all finite-dim vector spaces of invariants are finite-dimensional.
  • Supercuspidal representations: admissible representations not appearing as subquotients of any parabolically induced representation from a proper Levi.

Parahoric structure

\(G(F)\) has a rich collection of compact open subgroups:

  • Hyperspecial \(K_v = G(\mathcal{O}_F)\) at hyperspecial vertices \(v\) of \(\mathcal{B}(G, F)\); exist iff \(G\) is unramified.
  • Iwahori \(I = K_C\) for an alcove \(C \subseteq \mathcal{B}(G, F)\).
  • Parahoric \(K_F\) for any facet \(F\); see parahoric subgroups.

These compact opens stratify the topology of \(G(F)\) and give the Bushnell-Kutzko "types" for parametrising representations.

Classification (Tits)

Reductive groups \(G\) over a non-archimedean local field \(F\) are classified by:

  • The quasi-split form over the maximal unramified extension \(\breve F\) (= a quasi-split type, like \({}^2 A_n\), \({}^3 D_4\), etc.);
  • An inner-form class in \(H^1(F, G_{\mathrm{ad}})\).

For the unramified case: the inner-form class is detected by Newton/ Kottwitz invariants on \(B(G)\).

Lang's theorem and connectedness

Lang's theorem. For \(G\) connected reductive over \(\mathbb{F}_q\): \(H^1(\mathbb{F}_q, G) = 0\); the Lang map \(g \mapsto g^{-1} F(g)\) is surjective.

Consequence: every reductive group over \(\mathbb{F}_q\) is split (= quasi-split form is itself); inner forms over a local field reduce to the residue-field case via the Tits index.

Archimedean vs.\ non-archimedean

Setting Archimedean \(\mathbb{R}, \mathbb{C}\) Non-archimedean \(\mathbb{Q}_p, \mathbb{F}_q((t))\)
Topology Connected Lie group Totally disconnected; locally profinite
Building Symmetric space \(G/K\) Bruhat-Tits building (CAT(0) simplicial complex)
Forms Classified by Vogan/strong real forms Classified by Tits index + \(H^1(F, G_{\mathrm{ad}})\)
Reps \((g, K)\)-modules, Harish-Chandra Smooth admissible reps, Bushnell-Kutzko types
L-parameters \(W_F = \mathbb{C}^\times\) or non-abelian for \(\mathbb{R}\) Weil-Deligne group \(W_F \times \mathrm{SL}_2(\mathbb{C})\)

Tate-Nakayama duality

For an algebraic torus \(T\) over a local field \(F\): Tate-Nakayama duality gives canonical isomorphisms \[ H^i(F, T) \;\cong\; H^{2-i}(F, X^*(T))^\vee, \] relating the cohomology of \(T\) to the Pontryagin dual of the cohomology of its character lattice.

For non-abelian reductive \(G\): Borovoi-Kottwitz abelianisation gives an analogue, foundational for stable conjugacy and endoscopy.

Role in Langlands

  • Local Langlands: \(G(F)\)-representations \(\leftrightarrow\) L-parameters \(W_F' \to {}^L G\).
  • Local L-functions: defined for L-parameter and finite-dim representation of \(\hat G\).
  • Local epsilon factors: defined for L-parameter; the local components of global epsilon factors.
  • Endoscopic transfer: of stable distributions across \(G\) and its endoscopic groups.

Mathlib

Partial infrastructure (\(\mathbb{Q}_p\), \(\mathbb{F}_q((t))\), local fields in general) exists; reductive groups over them not yet formalised.

References

  • Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).
  • Borel, Linear Algebraic Groups (1991), §16.
  • Bushnell–Henniart, The Local Langlands Conjecture for GL(2) (2006).
  • Bernstein–Zelevinsky, Induced representations of reductive p-adic groups I (Ann.\ Sci.\ ENS 1977).

Prerequisite Topics

Used by