Topic Parahoric Group Schemes
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Parahoric Group Schemes

A parahoric group scheme \(\mathcal{G}_F\) is a smooth affine \(\mathcal{O}_F\)-group scheme whose generic fibre is the reductive group \(G\) and whose \(\mathcal{O}_F\)-points form a parahoric subgroup of \(G(F)\). They give the scheme-theoretic counterpart of the topological parahoric subgroups.

Definition

Let \(G\) be a connected reductive group over a non-archimedean local field \(F\) with valuation ring \(\mathcal{O}_F\). For each facet \(\mathcal{F} \subseteq \mathcal{B}(G, F)\) of the Bruhat-Tits building, the parahoric group scheme \(\mathcal{G}_\mathcal{F}\) is the unique smooth affine \(\mathcal{O}_F\)-group scheme such that:

  • \(\mathcal{G}_\mathcal{F} \otimes_{\mathcal{O}_F} F \cong G\) (generic fibre is \(G\));
  • \(\mathcal{G}_\mathcal{F}(\mathcal{O}_F) = \mathcal{P}_\mathcal{F}\) (parahoric subgroup).

Existence and uniqueness: Bruhat-Tits.

Structure

The special fibre \(\bar{\mathcal{G}}_\mathcal{F} := \mathcal{G}_\mathcal{F} \otimes \mathbb{F}_q\) is a connected smooth affine algebraic group over \(\mathbb{F}_q\) with:

  • Reductive quotient \(\bar{\mathcal{G}}_\mathcal{F}^{\mathrm{red}}\): a connected reductive \(\mathbb{F}_q\)-group, determined by the combinatorics of the facet (= "type" of \(\mathcal{F}\)).
  • Unipotent radical \(R_u(\bar{\mathcal{G}}_\mathcal{F})\): the reduction modulo the pro-unipotent radical of the parahoric.

Examples

\(\mathrm{GL}_n\), hyperspecial vertex

\(\mathcal{G}_v = \mathrm{GL}_{n, \mathcal{O}_F}\) (the standard \(\mathcal{O}_F\)-model); generic fibre = \(\mathrm{GL}_{n, F}\), special fibre = \(\mathrm{GL}_{n, \mathbb{F}_q}\) (reductive).

\(\mathrm{GL}_n\), Iwahori

For the Iwahori \(I\): \(\mathcal{G}_I\) has special fibre the Iwahori group scheme — block upper-triangular over \(\mathbb{F}_q\) modulo the strict block upper-triangular unipotent radical. Reductive quotient is \(\mathbb{G}_m^n\) (the standard torus mod \(\pi\)).

\(\mathrm{SL}_n\), general parahoric

For a composition \(n = n_1 + \cdots + n_k\) of \(n\) giving a parahoric \(P\): special fibre has reductive quotient \(\mathrm{SL}_{n_1} \times \cdots \times \mathrm{SL}_{n_k}\) (modulo central twist).

Construction (Bruhat-Tits)

The parahoric group scheme is constructed via:

  1. Glue affine charts indexed by root subgroups + Cartan.
  2. Each chart is \(\mathcal{O}_F\)-affine, smooth.
  3. Together they form a smooth \(\mathcal{O}_F\)-group scheme with generic fibre \(G\).

This is the "smooth model" of \(G\) at the facet \(\mathcal{F}\).

Functoriality

For \(\mathcal{F} \subseteq \mathcal{F}'\) (\(\mathcal{F}'\) in the closure of \(\mathcal{F}\)): there is a canonical morphism \(\mathcal{G}_\mathcal{F} \to \mathcal{G}_{\mathcal{F}'}\) (since the parahoric \(\mathcal{P}_\mathcal{F}\) contains \(\mathcal{P}_{\mathcal{F}'}\)).

These morphisms make the collection \(\{\mathcal{G}_\mathcal{F}\}\) into a "facet-indexed system" of smooth group schemes — a key algebro-geometric object.

Lattice models for \(\mathrm{GL}_n\) and \(\mathrm{SL}_n\)

For classical groups: parahoric group schemes have explicit descriptions as automorphism schemes of lattice chains: \[ \mathrm{GL}_n: \;\;\mathcal{G}_{\text{chain}} \;=\; \mathrm{Aut}(\Lambda_0 \supseteq \Lambda_1 \supseteq \cdots \supseteq \pi \Lambda_0), \] where \(\Lambda_i\) is a chain of lattices in \(F^n\) compatible with the chosen facet.

Smooth models and integral structure

Parahoric group schemes are the prototypical examples of:

  • Smooth integral models of reductive groups (in the sense of Bruhat-Tits / Conrad-Gabber-Prasad).
  • Néron models in the additive case (Néron blowing up).

These are the natural integral structures for studying mod-\(p\) phenomena (reduction of Shimura varieties, integral models of automorphic forms).

Role in Langlands

  • Local models of Shimura varieties: the parahoric group scheme appears as the local structure at each prime; Rapoport-Zink local models are based on parahoric structure.
  • Mod \(p\) representation theory: the special fibre's representation theory governs depth-zero supercuspidal parametrisation.
  • Integral / smooth Eisenstein series: rely on smooth integral models at each prime.

Mathlib

Not formalised; depends on \(p\)-adic infrastructure.

References

  • Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).
  • Tits, Reductive groups over local fields (Proc.\ AMS Symp.\ 1979).
  • Pappas–Rapoport, Local models in the ramified case I, II, III (Compositio 2003-2009).
  • Pappas–Zhu, Local models of Shimura varieties and a conjecture of Kottwitz (Invent.\ Math.\ 2013).

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