---
title: Mod p Principal Series Representations
url: https://www.emergentmind.com/topics/mod-p-principal-series-representations
type: topic
---

# Mod p Principal Series Representations

The theory of mod $p$ principal series representations occupies a central position in the modular representation theory of reductive groups and their covers over local and finite fields. These representations, defined over coefficient fields of characteristic $p$, exhibit unique structural, homological, and categorical properties that distinguish them from their characteristic zero analogues and have deep connections with questions in the mod $p$ local Langlands program, cohomology of arithmetic groups, and $p$-adic Hodge theory.

## 1. Definitions and Construction

Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ a connected split reductive group over $F$. Fix a Borel subgroup $B \subset G$ with unipotent radical $N$ and Levi torus $T=B\cap B^-$. For a coefficient field $k_E$ of characteristic $p$, and a smooth character $\chi:T(F)\rightarrow k_E^\times$, the smooth mod $p$ principal series $\operatorname{Ind}_{B^-(F)}^{G(F)} \chi$ is the space of locally constant functions $f:G(F)\to k_E$ satisfying $f(b^-g)=\chi(b^-)\,f(g)$ for all $b^-\in B^-(F)$, $g\in G(F)$, with compact support modulo $B^-(F)$. This is a smooth admissible $G(F)$-module over $k_E$ [1307.1818, 1708.03014]. For groups over finite fields, e.g., $G_q=\mathrm{GL}_2(\mathbb{F}_q)$, analogous constructions yield $\operatorname{PS}(\chi_1,\chi_2):=\operatorname{Ind}_{B_q}^{G_q}(\chi_1\otimes\chi_2)$ [2506.14207]. 

The principal series concept generalizes to various contexts:
- Principal series for finite rings $G_n=\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^n)$ [2511.04378]
- Genuine principal series for metaplectic covers $\widetilde{G}$ of $G$ [1603.04692, 2208.12478, 1412.0741]

In the metaplectic or covering group context (split, type $C_n$, $p\neq 2$), a smooth character $\sigma:T\to\Bbbk^\times$ is "genuine" after twisting by a certain character $\chi_\psi$ arising from the Weil index and additive character of $F$, and the principal series is defined via induction from the lifted torus character to the cover [1603.04692].

## 2. Filtrations, Bruhat Theory, and Homological Tools

Mod $p$ principal series are analyzed using several filtrations and functorial constructions:
- **Bruhat Filtration:** The classical Bruhat stratification of $\operatorname{Ind}\,U$ as a $B(F)$-representation enables precise control of submodule structure and explicit calculation of Ext-groups. If $d = \dim N$, the filtration $0=I_{-1}\subset I_0\subset \dots\subset I_d = \operatorname{Ind}\,U$ has graded pieces $I_r/I_{r-1} \cong \bigoplus_{\ell(w)=r} C_c(N_w(F), U)$, where $N_w=N\cap w^{-1}N w$ [1307.1818].
- **Ordinary Parts and $\delta$-Functor Techniques:** Emerton's derived ordinary parts functor $H^\bullet \mathrm{Ord}_{B(F)}$ provides a delta-functor from admissible $G(F)$-representations to those of $T(F)$, yielding spectral sequences that express higher Ext-groups between principal series in terms of the torus and its derived functors. This equips the category of mod $p$ representations of $G(F)$ with powerful homological control [1307.1818].
- **Socle and Radical Filtrations:** For principal series over finite rings, explicit "types" (e.g., for $G_2=\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^2)$) and carry-set theory parameterize submodule lattices, and adjacency in the poset of types determines the socle and radical gradations [2511.04378].

These tools enable the computation of projective resolutions (e.g., via the Schneider–Stuhler coefficient systems) and the deduction of vanishing results for higher Ext and cohomology [1408.3679].

## 3. Irreducibility Criteria and Submodule Structure

The irreducibility properties of mod $p$ principal series differ sharply from characteristic zero:

- **Genericity Conditions:** A character $\chi$ of $T(F)$ is weakly generic if for every simple root $\alpha$, $s_\alpha(\chi)\neq\chi$; strongly generic if $w(\chi)\neq\chi$ for all nontrivial $w\in W$ [1307.1818]. Failure of (strong) genericity leads to additional (often self-) extensions and to "accidental" non-split submodules.
- **Explicit Classification Over Finite Rings:** In $\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^2)$, the submodule lattice and Jordan–Hölder factors are determined combinatorially by types, with linear Hasse diagrams in the totally-ramified case and infinite lattices in the non-totally-ramified case [2511.04378].
- **Metaplectic Covers:** In $\widetilde{\mathrm{Sp}}_{2n}(F)$, the length of a genuine mod $p$ principal series is $2^{m}$, where $m$ is the number of short simple roots killed by the character, and is irreducible precisely when the underlying character is nontrivial on each short coroot-lift [1603.04692]. This extends the irreducibility in the $\widetilde{\mathrm{SL}}_2(F)$ case, where all (genuine) principal series are irreducible [1412.0741].

## 4. Extension Groups and Homological Results

Extensions between mod $p$ principal series representations are controlled by Weyl group combinatorics and derived functor calculations:

- **Yoneda Ext-groups:** For $G$ split with simple roots $\Delta$ and characters $\chi,\chi'$, one has $\operatorname{Ext}^1_{G(F)}(\operatorname{Ind}\chi',\operatorname{Ind}\chi)\neq0$ only if $\chi'=\chi$ or $\chi'=s_\alpha(\chi)$ for some $\alpha\in\Delta$, and the corresponding extension space is one-dimensional for generic $\chi$ [1307.1818].
- **Spectral Sequence Realization:** The spectral sequence from the (Ind,Ord) adjunction allows identification of extensions in terms of ordinary parts of the induced representation. In degree one,
  $$
  0 \to \operatorname{Ext}^1_{T(F)}(U, \operatorname{Ord} V) \to \operatorname{Ext}^1_{G(F)}(\operatorname{Ind}U, V) \to \operatorname{Hom}_{T(F)}(U, H^1\operatorname{Ord}_{B(F)}(V))
  $$
  and the ordinary parts can be computed explicitly using the Bruhat filtration [1307.1818].
- **Connection with Mod $p$ Langlands:** These extension classes are expected to coincide with those predicted by (modular) local Langlands correspondences for generic principal series parameters, and the unique non-split extensions given by simple reflections correspond to extensions within blocks associated to the same parameter [1307.1818]. For $\mathrm{GL}_n$, higher cohomology of pro-$p$ Iwahori invariants reveals supersingular summands of categorical significance for local Langlands conjectures [1708.03014].

## 5. Functors and Cohomological Constructions

The structure of mod $p$ principal series is intricately tied to various functorial constructions:

- **Schneider–Vignéras Functor:** This functor associates to a $B$-representation a module over the Iwasawa algebra $\Lambda(N_0)$ (with $N_0$ a compact open subgroup of the unipotent radical). For irreducible principal series, the Schneider–Vignéras module is controlled by the top Bruhat stratum, and only this stratum supports an étale $(\phi,T)$-module structure relevant for $p$-adic Galois representations [1409.8045].
- **Resolutions via Bruhat–Tits Buildings:** Over arbitrary characteristic, the principal series admits a finite-length explicit projective resolution by coefficient systems on the semisimple building, a property not shared by (most) supercuspidal mod $p$ representations [1408.3679]. This is crucial for computation of derived functors and for the realization of principal series as $H_0$ of the building complex.
- **Cohomology of Pro-$p$-Iwahori Subgroups:** The cohomology $H^1(I_1, \pi)$, for $\pi$ principal series and $I_1$ the pro-$p$-Iwahori, exhibits a filtration whose graded pieces reflect the modular representation theory of Levi subgroups and reveal occurrences of supersingular modules [1708.03014].

## 6. Restriction, Branching Laws, and Finiteness

Branching rules for restriction of mod $p$ principal series exhibit rich combinatorial patterns:

- **Finite Groups:** The restriction of $\operatorname{Ind}_{B_q}^{G_q}\chi_r$ from $G_q=\mathrm{GL}_2(\mathbb{F}_q)$ to $G_p=\mathrm{GL}_2(\mathbb{F}_p)$ decomposes into a direct sum of principal series, toral, and Steinberg-twisted summands, with multiplicities dependent on the parity of the degree $f$ of the extension and explicit formulas given via Mackey theory and analysis of orbits in projective space [2506.14207].
- **Multiplicity and Irreducibility:** The unique principal series summand occurs with multiplicity one; other summands (Steinberg-twisted, toral) have explicit multiplicities, and outside $\mathbb{F}_p$ the restriction is never irreducible [2506.14207].
- **Metaplectic and Covering Groups:** Classification of irreducible admissible genuine mod $p$ representations for covers such as $\widetilde{\mathrm{Sp}}_{2n}(F)$ or $\widetilde{\mathrm{GL}}_2(\mathbb{Q}_p)$ proceeds via categorical equivalences between genuine modules and modules over appropriate Hecke algebras, with principal series and supersingular objects corresponding to distinct blocks [1603.04692, 2208.12478, 1412.0741].

## 7. Connections to Mod $p$ Langlands Correspondence and Socle Structure

Mod $p$ principal series are essential components in the emerging mod $p$ local Langlands correspondences and in the understanding of socle and extension structures of completed cohomology and Galois representations:

- **Blocks and Socle Gradations:** For $G=\mathrm{GL}_n$ and generic principal series, extension classes and socle gradations realized in completed cohomology match the predictions of topological and diagrammatic models of mod $p$ Langlands [1307.1818, 2511.04378].
- **Supersingular Constituents:** For $n\geq3$, cohomological calculations show that supersingular constituents are unavoidable in the $H^1$ of pro-$p$-Iwahori, supporting the expectation that principal series blocks "see" the entire range of modular phenomena necessary for derived mod $p$ local Langlands [1708.03014].
- **Functorial Lifts and Galois Parameters:** The Schneider–Vignéras module of a principal series provides the input for constructing étale $(\phi,\Gamma)$-modules, thus situating mod $p$ principal series at the interface of representation theory and arithmetic geometry [1409.8045].

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This synthesis covers the construction, structural theory, homological, and categorical aspects of mod $p$ principal series representations and their centrality in modern approaches to the modular representation theory of reductive groups over local fields, finite rings, and their metaplectic covers.

Source: https://www.emergentmind.com/topics/mod-p-principal-series-representations