Parahoric Subgroups

A parahoric subgroup of a reductive group \(G\) over a local field \(F\) is a compact open subgroup of \(G(F)\) attached to a facet (simplex) of the Bruhat-Tits building. Parahorics interpolate between the Iwahori subgroup (stabiliser of a chamber) and the hyperspecial maximal compact subgroup (stabiliser of an "optimal" vertex), and they index the natural categorification of Hecke algebra theory.

Definition

Let \(G\) be a connected reductive group over a non-archimedean local field \(F\) with valuation ring \(\mathcal{O}_F\) and residue field \(\mathbb{F}_q\). Let \(\mathcal{B}(G, F)\) be the Bruhat-Tits building.

For a facet \(F \subseteq \mathcal{B}(G, F)\) (a non-empty open simplex), the parahoric subgroup \(\mathcal{P}_F\) is the connected component of the identity in the stabiliser \(\mathrm{Stab}_{G(F)}(F)\) (in the category of group schemes via the Bruhat-Tits smooth model).

Equivalently: \(\mathcal{P}_F = G_F(\mathcal{O}_F)\) where \(G_F\) is a specific smooth affine \(\mathcal{O}_F\)-group scheme with generic fibre \(G\) (the parahoric group scheme of \(F\)).

Types of parahorics

Facet type	Parahoric type	Special features
Chamber (alcove)	Iwahori subgroup \(I\)	smallest parahoric; pro-\(p\) Iwahori is its maximal pro-\(p\) subgroup
Vertex (optimal)	Hyperspecial	\(\cong G(\mathcal{O}_F)\) for a smooth reductive model \(G/\mathcal{O}_F\); exists iff \(G\) is unramified
Vertex (general)	Special	maximal compact at that vertex; reductive quotient is "as large as possible"
General facet	Standard parahoric	between Iwahori and hyperspecial

For \(G = \mathrm{SL}_2(\mathbb{Q}_p)\):

Iwahori \(I\) = matrices that are upper-triangular mod \(p\);
Hyperspecial \(K_0 = \mathrm{SL}_2(\mathbb{Z}_p)\).

There are two conjugacy classes of hyperspecial vertices in the tree; each fixes a different end-class.

Reductive quotient

Each parahoric \(\mathcal{P}_F\) is a smooth affine group scheme over \(\mathcal{O}_F\). Its special fibre \(\bar{\mathcal{P}}_F\) (over \(\mathbb{F}_q\)) is a connected smooth affine group; its reductive quotient is \[ \bar{\mathcal{P}}_F^{\mathrm{red}} \;:=\; \bar{\mathcal{P}}_F \big/ R_u(\bar{\mathcal{P}}_F), \] a connected reductive group over \(\mathbb{F}_q\).

The reduction map \[ \mathcal{P}_F \;\twoheadrightarrow\; \bar{\mathcal{P}}_F^{\mathrm{red}}(\mathbb{F}_q) \] is the key tool for transferring representation theory of \(G(F)\) to representation theory of finite reductive groups.

Example: \(G = \mathrm{GL}_n\), hyperspecial vertex

\(\mathcal{P}_v = \mathrm{GL}_n(\mathcal{O}_F)\), with reductive quotient \(\mathrm{GL}_n(\mathbb{F}_q)\).

Example: Iwahori \(I \subseteq \mathrm{GL}_n(F)\)

\(I\) = matrices in \(\mathrm{GL}_n(\mathcal{O}_F)\) that are upper- triangular mod \(\pi\). Its reductive quotient is the diagonal torus \((\mathbb{F}_q^\times)^n\).

Example: Standard parahoric in \(\mathrm{GL}_n\)

For a composition \(n = n_1 + \cdots + n_k\), the corresponding standard parahoric \(P\) consists of block upper-triangular matrices (mod \(\pi\)) with block sizes \(n_1, \ldots, n_k\). Reductive quotient is \(\mathrm{GL}_{n_1}(\mathbb{F}_q) \times \cdots \times \mathrm{GL}_{n_k}(\mathbb{F}_q)\).

The pro-unipotent radical

The kernel \(\mathcal{P}_F^+\) of the reduction map is the pro-unipotent radical: \[ \mathcal{P}_F^+ \;=\; \ker(\mathcal{P}_F \twoheadrightarrow \bar{\mathcal{P}}_F^{\mathrm{red}}(\mathbb{F}_q)), \] a pro-\(p\) (where \(p\) = residue characteristic) normal subgroup. It is the \(p\)-adic analogue of the unipotent radical of a finite reductive parabolic.

Bruhat decomposition relative to parahoric

For an Iwahori \(I\): \[ G(F) \;=\; \bigsqcup_{\tilde w \in \tilde W} I \tilde w I. \]

For a general parahoric \(\mathcal{P}\) with Weyl subgroup \(W_{\mathcal{P}} \subseteq \tilde W\): \[ G(F) \;=\; \bigsqcup_{w \in W_{\mathcal{P}} \backslash \tilde W / W_{\mathcal{P}}} \mathcal{P} w \mathcal{P}. \]

Role in representations

Depth zero representations: irreducible reps of \(G(F)\) that have an invariant vector under some pro-unipotent radical \(\mathcal{P}_F^+\). Their classification reduces to representations of finite reductive groups \(\bar{\mathcal{P}}_F^{\mathrm{red}}(\mathbb{F}_q)\) (Moy-Prasad).
Types of Bushnell-Kutzko: each Bernstein component of \(\mathrm{Rep}(G(F))\) contains a "type" \((\mathcal{P}, \rho)\) with \(\mathcal{P}\) parahoric and \(\rho\) a representation; for \(\mathrm{GL}_n\) this gives the complete parameterisation of supercuspidals.
Hecke algebras: \(\mathcal{H}(\mathcal{P} \backslash G(F) / \mathcal{P})\) is a convolution algebra; for \(\mathcal{P} = I\), the Iwahori-Hecke algebra; for \(\mathcal{P}\) maximal compact, the spherical Hecke algebra (= Satake transform = \(\mathrm{Rep}(\hat G)\)).
Affine flag variety: \(G(F) / \mathcal{P}\) is the partial affine flag variety; the case \(\mathcal{P} = I\) gives the full affine flag variety, the case \(\mathcal{P}\) hyperspecial gives the affine Grassmannian.

Moy-Prasad filtration

For \(x \in \mathcal{B}(G, F)\) and \(r \ge 0\), Moy and Prasad defined a filtration \(\mathcal{P}_{x,r}\) by depth: \[ \mathcal{P}_{x,0} \;\supseteq\; \mathcal{P}_{x, 0^+} \;\supseteq\; \mathcal{P}_{x, r} \;\supseteq\; \cdots \] with successive quotients direct sums of finite-dimensional representations of finite reductive groups. The depth-zero subgroup recovers the parahoric; positive-depth quotients support types for positive-depth representations.

Mathlib

Not formalised. Even the basic Iwahori subgroup \(\mathrm{SL}_2(\mathbb{Z}_p)\) mod \(p\) would require local-field infrastructure beyond what is currently available.

References

Bruhat–Tits, Groupes réductifs sur un corps local I, II (Publ.\ IHES 1972, 1984).
Tits, Reductive groups over local fields (Corvallis 1979).
Moy–Prasad, Unrefined minimal K-types for p-adic groups (Inv.\ Math.\ 1994) and Jacquet functors and unrefined minimal K-types (Comment.\ Math.\ Helv.\ 1996).
Bushnell–Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups (Annals 1993).
Yu, Smooth models associated to concave functions in Bruhat-Tits theory (in: Autour des schémas en groupes, 2015).