Reductive Quotients of Parahoric Group Schemes
The reductive quotient of a parahoric group scheme is the connected reductive \(\mathbb{F}_q\)-group obtained by killing the pro-unipotent radical of the special fibre. It is the natural target of reduction- mod-\(p\) for representations of \(G(F)\) and the bridge to finite reductive group theory.
Definition
Let \(\mathcal{G}_\mathcal{F}\) be the parahoric group scheme attached to a facet \(\mathcal{F} \subseteq \mathcal{B}(G, F)\), with special fibre \(\bar{\mathcal{G}}_\mathcal{F}\) over \(\mathbb{F}_q\). The special fibre is a connected smooth affine \(\mathbb{F}_q\)-group with:
- Unipotent radical \(R_u(\bar{\mathcal{G}}_\mathcal{F})\), a connected smooth unipotent group;
- Reductive quotient \[ \bar{\mathcal{G}}_\mathcal{F}^{\mathrm{red}} \;:=\; \bar{\mathcal{G}}_\mathcal{F} \big/ R_u(\bar{\mathcal{G}}_\mathcal{F}). \]
\(\bar{\mathcal{G}}_\mathcal{F}^{\mathrm{red}}\) is a connected reductive group over \(\mathbb{F}_q\).
Reduction map
The natural projection \[ \mathcal{P}_\mathcal{F} \;=\; \mathcal{G}_\mathcal{F}(\mathcal{O}_F) \;\twoheadrightarrow\; \bar{\mathcal{G}}_\mathcal{F}^{\mathrm{red}}(\mathbb{F}_q) \] is the reduction map. Its kernel is the pro-unipotent radical \(\mathcal{P}_\mathcal{F}^+\) of the parahoric, a pro-\(p\) group.
Examples
Hyperspecial vertex
For \(G\) unramified at a hyperspecial \(v\): \[ \bar{\mathcal{G}}_v^{\mathrm{red}} \;=\; G_{\mathbb{F}_q}, \] the reductive group "of the same type" as \(G\), now over \(\mathbb{F}_q\). E.g.\ for \(G = \mathrm{GL}_n\): \(\bar{\mathcal{G}}_v^{\mathrm{red}} = \mathrm{GL}_n(\mathbb{F}_q)\).
Iwahori
For the Iwahori \(I\) (= alcove stabiliser): \[ \bar{I}^{\mathrm{red}} \;=\; T_{\mathbb{F}_q}, \] a maximal torus over \(\mathbb{F}_q\) (the diagonal torus for \(\mathrm{GL}_n\)).
General parahoric (composition \(n = n_1 + \cdots + n_k\))
\[ \bar{\mathcal{P}}^{\mathrm{red}} \;=\; \mathrm{GL}_{n_1}(\mathbb{F}_q) \times \cdots \times \mathrm{GL}_{n_k}(\mathbb{F}_q), \] a Levi-like decomposition.
Ramified non-split case
For ramified \(\mathrm{SU}_n\): the reductive quotient may be a different reductive group of "lower type" than \(\mathrm{SU}_n\) — for example a product of \(\mathrm{GL}_a \times \mathrm{Sp}_{2b}\) for some \(a, b\) depending on the specific facet and the ramification structure.
Depth-zero representation theory
Theorem (Morris / Moy-Prasad). The set of irreducible smooth representations of \(G(F)\) at depth zero is in bijection with pairs: \[ (\mathcal{F}, \pi) \] where:
- \(\mathcal{F}\) is a facet of \(\mathcal{B}(G, F)\) modulo \(G(F)\)-action;
- \(\pi\) is an irreducible cuspidal representation of \(\bar{\mathcal{G}}_\mathcal{F}^{\mathrm{red}}(\mathbb{F}_q)\).
This is the depth-zero supercuspidal parametrisation — reducing representations of \(p\)-adic reductive groups to representations of finite reductive groups.
Functoriality
For \(\mathcal{F} \subseteq \mathcal{F}'\) (\(\mathcal{F}'\) in the closure of \(\mathcal{F}\)): there is a parabolic embedding \[ \bar{\mathcal{P}}_{\mathcal{F}'}^{\mathrm{red}} \;\hookrightarrow\; \bar{\mathcal{P}}_\mathcal{F}^{\mathrm{red}} \] as a Levi factor of a parabolic subgroup. This is the geometric basis of parabolic induction at the level of finite reductive groups, lifted to depth-zero representations of \(p\)-adic groups.
Galois descent for non-split groups
For \(G\) non-split: compute the reductive quotient of \(G_K\) at the corresponding facet over \(K = \widehat{F^{\mathrm{unr}}}\), then take \(\mathrm{Frob}\)-invariants (a finite group of Lie type).
The result is generally smaller than expected: ramification reduces the rank of the reductive quotient.
Role in Langlands
- Depth-zero supercuspidals: classified via cuspidal characters of the reductive quotient at vertices.
- Local Langlands at depth zero: the L-parameters of depth-zero representations are constructed via Deligne-Lusztig theory on the reductive quotient.
- Mod \(p\) representations: the special fibre's representation theory in characteristic \(p\) is the key to mod \(p\) local Langlands.
References
- Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).
- Moy–Prasad, Unrefined minimal K-types for p-adic groups (Inv.\ Math.\ 1994).
- Morris, Tamely ramified intertwining algebras (Invent.\ Math.\ 1993).
- Deligne–Lusztig, Representations of reductive groups over finite fields (Annals 1976).