Ordinary Parts Functor in Representation Theory

• Ordinary Parts Functor is defined on smooth representations of p‑adic groups, isolating 'ordinary' components via N-invariants and smooth induction.
• It is inherently left exact, with mod‑p adaptations achieving full exactness through contracting elements and specialized Hecke actions.
• The functor underpins key adjunctions with parabolic induction and extends to categorical frameworks, advancing studies in automorphic forms and arithmetic geometry.

The ordinary parts functor is a foundational tool in the paper of representation theory for $p$-adic reductive groups, the theory of automorphic forms, and in the categorification of set-theoretic constructions. It arises in contexts ranging from the decomposition of representations over local fields to categorical models for relations and direct images, and its paper has included deep analyses of exactness, adjunctions, derived functors, and dualities. In recent research, the functor's structural properties have been clarified using techniques from algebraic geometry, category theory, and the cohomology of locally symmetric spaces.

1. Definitions and Algebraic Framework

The ordinary parts functor, often denoted $\operatorname{Ord}_P$ (or with further decorations indicating group $G$ and subgroup $P$), is defined in the setting of smooth representations of a reductive $p$-adic group $G$ with respect to a parabolic subgroup $P = MN$ ($M$ Levi, $N$ unipotent radical).

Given a smooth $R$-representation $\pi$ of $G$, where $R$ is usually an Artinian local ring with $p$ nilpotent, one first fixes $N_0$ (a compact open subgroup of $N$) and $M^+ \subset M$ (an open submonoid of "contractive" elements). The functor is given by: $$\operatorname{Ord}_P(\pi) := \left(\operatorname{Ind}_{M^+}^M \pi^{N_0}\right)^{Z_M-\mathrm{loc.\,fin}}$$ where $\pi^{N_0}$ denotes $N_0$-invariants, $\operatorname{Ind}_{M^+}^M$ is smooth induction from $M^+$ to $M$, and then one takes the vectors locally finite for the action of the center $Z_M$.

This is closely analogous to the Jacquet functor but differs, especially in the mod-$p$ case, due to the behavior of cohomology and Hecke module actions. The functor is (right) adjoint to parabolic induction $\operatorname{Ind}_{\bar{P}}^G$ on admissible $G$-representations.

2. Exactness and Derived Functors

Traditionally, the ordinary parts functor is left exact (preserves injections), but it need not be right exact. In mod-$p$ settings (with $R$ Artinian, $p$ nilpotent, and $\operatorname{char}(F)=p$), the functor becomes exact, as shown in (Hauseux, 2017): $$\operatorname{Ord}_P: \operatorname{Mod}_G^{\mathrm{adm}}(R) \rightarrow \operatorname{Mod}_M^{\mathrm{adm}}(R) \quad\text{is exact}$$ Exactness follows from vanishing of higher $N_0$-cohomology after localization at certain Hecke operators (contracting elements $z \in Z_M^+$). A typical formula under these conditions is: $$\operatorname{Ord}_P(\pi) \cong R[z^{\pm1}] \otimes_{R[z]} \pi^{N_0}$$ The right derived functors $R^n \operatorname{Ord}_P^G$ are identified in (Hoff et al., 20 Aug 2025) as: $$R^n \operatorname{Ord}_P^G(V) \cong \operatorname{ind}_{M^+}^M H^n(N_0, V)$$ for any admissible smooth $G$-representation $V$. This realizes the derived ordinary parts as cohomological invariants induced up from $M^+$.

3. Adjunction Properties and Induction Comparisons

A central aspect is the adjunction between ordinary parts and parabolic induction. The ordinary parts functor provides a right adjoint to induction; on admissible objects, this adjunction is exact under the mod-$p$ hypotheses: $$\operatorname{Hom}_M(\sigma, \operatorname{Ord}_P(\pi)) \cong \operatorname{Hom}_G(\operatorname{Ind}_{\bar{P}}^G \sigma, \pi)$$ The derived functors respect Ext-groups as in: $$\operatorname{Ext}_M^n(\sigma, \operatorname{Ord}_P(\pi)) \cong \operatorname{Ext}_G^n(\operatorname{Ind}_{\bar{P}}^G \sigma, \pi)$$ Comparison theorems (Hoff et al., 20 Aug 2025) show that two induction types—small induction from compact subgroups and parabolic induction from proper Levi subgroups—are isomorphic after appropriate localization: $$\operatorname{ind}_{A_M^+}^{A_M}(\operatorname{ind}_K^G W) \cong \operatorname{Ind}_{\bar{P}}^G(\operatorname{ind}_{K \cap M}^M W)$$ where $K$ is a compact open subgroup with Iwahori factorization and $W$ a smooth $K$-representation.

4. Representation-Theoretic and Cohomological Applications

In mod-$p$ representation theory, the ordinary parts functor is used to extract ordinary (non-supersingular) components whose cohomological behavior is tractable. In the $p$-adic Langlands correspondence (Banerjee et al., 2022), ordinary local Galois representations correspond to automorphic representations containing principal series subrepresentations visible in the ordinary parts of completed cohomology.

Vanishing results for Ext-groups (e.g., $\operatorname{Ext}^1_G(\pi_1, W)=0$ for principal series $\pi_1$ and supersingular $W$) ensure a separation of ordinary and supersingular submodules via the functor.

In global Langlands–type settings (Hevesi, 2023), the "Q-ordinary" parts functor isolates slope-zero components relative to chosen parabolic $Q$. This is vital for verifying local–global compatibility in the cohomology of Shimura varieties, as associated Galois representations admit filtrations reflected by the ordinary decomposition under $Q$: $$\operatorname{Ord}_Q^{\mathrm{lalg}}(\Pi) := (\Pi^{N^0})_0$$ where $(\Pi^{N^0})_0$ refers to the slope-zero piece for Hecke operators.

These constructions have direct impact on local–global compatibility theorems for automorphic Galois representations, e.g., the upper-triangular form matching the blocks against the ordinary pieces in the decomposition.

5. Functorial and Categorical Extensions

In categorical settings, the ordinary parts functor concept is generalized as the direct image under the covariant power set functor $P_0$ and its categorified analogue $\mathcal{P}_1$ (Chand et al., 2015). Here, $\mathcal{P}_1$ is defined on the category $Cat_1$ (categories with compatible ordered hom-sets), sending each object to the collection of down-closed subsets and each morphism to the set of down-closed morphism sets.

In this setting, the direct image functorder $F_{(\rightarrow)}$ acts analogously to ordinary parts: $$F_{(\rightarrow)}(\mathcal{C}_0) = \{F(C): C \in \mathcal{C}_0\},\quad F_{(\rightarrow)}(\mathfrak{c}) = \downarrow\{F(c): c \in \mathfrak{c}\}$$ This abstraction recovers the classical relationship between functions, direct images, and relations as the Kleisli category of a monad.

Order-enriched solid functors and their ordinary counterparts (Sousa et al., 2018) generalize the notion further, ensuring transferability of limits and colimits in ordered algebraic settings (frames, vector spaces, ordered monoids), with exactness and faithfulness criteria extended by adding order-epicness and order-faithfulness.

6. Dualities and Adjointness: Bernstein’s Second Adjointness

The functorial adjunction principles culminate in mod-$p$ variants of Bernstein's Second Adjointness ((Hoff et al., 20 Aug 2025), appendix by Heyer). On complexes of admissible cohomology, the derived right adjoint to parabolic induction ($\mathcal{R}_{\bar{P}}^G$) and the left adjoint ($\mathcal{L}_P^G$) are isomorphic up to a cohomological shift and twist by a smooth character $\omega$: $$\omega \otimes_R^{\mathbb{L}} \mathcal{R}_{\bar{P}}^G V^{\bullet} \cong \mathcal{L}_P^G V^{\bullet}$$ where the shift is by $-\dim_{\mathbb{Q}_p} N$.

This duality both strengthens the structural understanding of the ordinary parts functor and clarifies its behavior in derived and categorical contexts.

7. Implications, Generalizations, and Open Questions

The structural analysis of the ordinary parts functor has clarified its role in mod-$p$ and $p$-adic representation theory, automorphic forms, Langlands correspondence, and enriched category theory. Its exactness permits transfer of extension structures and adjunctions, enables the computation of higher Ext-groups, and supports cohomological decomposition in arithmetic geometry.

Key open problems include criteria for order-faithfulness in enriched solid functors (Sousa et al., 2018), the explicit behavior of derived and ordinary functors for special classes of groups, and the full generalization of local–global compatibility beyond self-dual cases and to arbitrary parabolic subgroups.

The ordinary parts functor continues to serve as a bridge between the internal structure of representations, their cohomological properties, and their categorical analogues, underpinning critical advances in modern representation theory and categorical algebra.