Representations of p-adic Groups

• Representations of p-adic groups are smooth representations of the F-points of a reductive algebraic group, encoding key arithmetic and harmonic analysis properties.
• Key constructions such as parabolic and compact induction provide explicit models and classifications, with roles played by supercuspidal supports and types.
• Bernstein decomposition and Hecke algebra equivalences break down the category into manageable blocks, linking representation theory with the local Langlands correspondence.

Representations of $p$-adic Groups are a central object in non-Archimedean harmonic analysis, automorphic forms, and the Langlands program. A $p$-adic group, usually denoted $G = \mathbf{G}(F)$, is formed by the $F$-points of a reductive algebraic group $\mathbf{G}$ over a non-Archimedean local field $F$ (for instance, a finite extension of $\mathbb{Q}_p$). The representation theory of $G$ seeks to understand its (complex or modular) smooth representations, their classification, associated module structures, and connections to arithmetic and geometry.

1. Structure of $p$-adic Groups and Smooth Representations

Let $G$ be the group of $F$-points of a reductive group $\mathbf{G}$. The group is locally compact, totally disconnected, and admits a basis of open compact subgroups, including Iwahori, parahoric, and pro-$p$ subgroups.

A representation $(\pi, V)$ of $G$ over a coefficient ring $R$ (commonly $\mathbb{C}$, but also $\mathbb{Q}_\ell$, $\mathbb{F}_\ell$, or more general rings) is called smooth if for every $v \in V$, the stabilizer subgroup $\mathrm{Stab}_G(v)$ is open. The collection $\mathrm{Mod}_R(G)$ of smooth $R$-representations forms an abelian category, whose objects typically encode arithmetic information.

Admissible representations. A smooth $R$-representation $V$ is called admissible if for every compact open $K \subset G$, the space $V^K$ of $K$-fixed vectors is finitely generated over $R$. This notion is crucial for harmonic analysis, and in characteristic zero, all irreducible smooth representations of reductive $p$-adic groups are admissible (Vignéras, 2022).

Principal constructions.

• Parabolic induction (from a Levi $M$ in a parabolic $P = MU$): The normalized parabolic induction $\mathrm{Ind}_P^G(\sigma)$, for a smooth $M$-representation $\sigma$, provides prototypical families such as the principal series and their generalizations. The G-action on $f \in \mathrm{Ind}_P^G(\sigma)$ is $[g \cdot f](x) = f(xg)$, compatible with modulus factors.
• Compact induction: For $K \subset G$ compact (modulo center) and $\rho$ a finite-dimensional $K$-representation, $c$-$\mathrm{Ind}_K^G(\rho)$ carries smoothness by construction (Fintzen, 9 Oct 2025).

2. Classification: Supercuspidal Representations, Bernstein Blocks, and Types

Supercuspidal representations are irreducible smooth $G$-representations not appearing as subquotients of any proper parabolically induced representation. They form the building blocks of all irreducibles: every irreducible $\pi$ embeds in some $\mathrm{Ind}_P^G(\sigma)$, for $\sigma$ supercuspidal on a Levi $M$ (Fintzen, 9 Oct 2025, Aubert, 2023).

Bernstein decomposition: The category $\mathrm{Rep}(G)$ splits (non-canonically) into full subcategories, each corresponding to a Bernstein block, labeled by inertial equivalence classes $[M, \sigma]$ where $M$ is a Levi subgroup and $\sigma$ a supercuspidal $M$-representation: $$\mathrm{Rep}(G) \simeq \prod_{[M,\sigma]} \mathrm{Rep}(G)_{[M,\sigma]}$$ Each block $\mathrm{Rep}(G)_{[M,\sigma]}$ contains all smooth $G$-representations whose irreducible subquotients have cuspidal support in $[M,\sigma]$ (Fintzen, 9 Oct 2025, Vignéras, 2022).

Types and Hecke algebras: Each Bernstein block admits a type $(K, \rho)$—a compact open subgroup $K$ and irreducible $K$-representation $\rho$ such that $c$-$\mathrm{Ind}_K^G(\rho)$ generates the block. The associated spherical Hecke algebra $\mathcal{H}(G, \rho) = \mathrm{End}_G(c$-$\mathrm{Ind}_K^G(\rho))$ controls the block: $$\mathrm{Rep}(G)_{[M,\sigma]} \simeq \mathrm{Mod}(\mathcal{H}(G, \rho))$$ Depth-zero blocks are modeled on affine Hecke algebras (possibly semidirect products with finite groups), generalizing the Iwahori case (Fintzen, 9 Oct 2025).

Explicit constructions of supercuspidals are achieved via compact induction. For $\mathrm{GL}_n$ and classical groups, the Bushnell–Kutzko and Stevens approaches, along with Yu’s construction for general tamely ramified groups (including recent results for $p = 2$), provide explicit models (Fintzen, 9 Oct 2025).

3. Parabolic Induction, Jacquet Modules, and Reducibility

Normalized parabolic induction preserves smoothness and interacts tightly with the structure of Bernstein components. For $P = MN$ and a smooth $M$-representation $\sigma$,

$$\mathrm{Ind}_P^G(\sigma) = \{ f : G \to \sigma \mid f(mng) = \delta_P^{1/2}(m) \sigma(m) f(g) \}$$

with right translation action and normalization by $\delta_P$.

Admissible irreducible representations are characterized as unique subquotients of certain parabolic inductions from supercuspidal blocks. In characteristic $p$, subquotient lattices and irreducibility loci are controlled explicitly via supersingularity and the structure of the pro-$p$ Iwahori Hecke algebra (Henniart et al., 2017, Abe et al., 2017).

Jacquet module and adjointness: To $V \in \mathrm{Mod}_R(G)$, the Jacquet module $V_N$ (the $N$-coinvariants for $N =$ unipotent radical) reflects the structure of parabolic subgroups. Jacquet functors serve as both left and right adjoints to induction. In modular cases and for admissible representations, Emerton’s ordinary part functor coincides with the right adjoint (Abe et al., 2017).

Reducibility loci: For classical groups, the reducibility of induced representations $\mathrm{Ind}_P^G(\pi \otimes \sigma)$ depends on separation conditions between the supercuspidal supports. The main theorem is: If $\sigma$ is separated from the support $\mathrm{supp}(\pi)$, reducibility of induction depends only on $\pi$ and is identical for all such $\sigma$ (Ciubotaru et al., 2017). If $\sigma$ is not separated, induction is always reducible (rigorously proved by results using Hecke algebra arguments and Kato’s exotic geometry (Ciubotaru et al., 2017)).

4. Affine Hecke Algebra Equivalences and Functoriality

For each set $I$ of inertial classes of supercuspidal representations of general linear groups, and a fixed supercuspidal $\sigma$ of a classical group, the category of all smooth complex representations of type $(I, \sigma)$ admits an equivalence with modules over tensor products (direct sums) of affine Hecke algebras of types $A$, $B$, and $D$: $$\mathcal{C}_{I, \sigma} \simeq \left\{ \text{f.g. right modules over } \bigoplus_{r \geq 1} \bigoplus_{O_1,\dots,O_r \in I} \left( \bigotimes_{i=1}^r H_{O_i,\sigma} \right) \right\}$$ where the $H_{O_i,\sigma}$ are extended affine Hecke algebras with generators and relations tailored to the root data, central parameters $q_s$, and Bernstein cross-relations (Ciubotaru et al., 2017). This equivalence allows categorical reduction of problems about classical group representations to combinatorial and geometric questions about Hecke module categories.

Functorial properties manifest as tensor product functors. If $I$ and $J$ are disjoint, there is a compatible tensor functor at the categorical and Hecke algebra level: $$\mathcal{C}_{I, \sigma} \times \mathcal{C}_{J, \sigma} \to \mathcal{C}_{I \cup J, \sigma}$$ inducing isomorphisms on Hecke algebras and compatibility with parabolic and Jacquet functors (Ciubotaru et al., 2017).

5. Hecke Algebra Arguments and Exotic Geometric Techniques

The proof of reducibility results, exact category equivalences, and lattice isomorphisms utilizes deep results about affine Hecke algebras and "exotic geometry" (Kato). For real positive separated central characters, the corresponding categories of equivariant perverse sheaves on geometric models ("exotic Springer fibers") are canonically independent of certain parameters, and their module structures coincide with those of the specialized affine Hecke algebras.

By Lusztig’s reduction, this lowers the analysis to smaller Hecke algebras with real central characters, showing that for separated cases, submodule lattices depend only on the general linear part, not on the classical group parameters (Ciubotaru et al., 2017). These geometric reductions, combined with Bernstein–Zelevinsky progenerator theory and Morita equivalence, yield explicit module-theoretic and combinatorial models for representation categories.

6. Modular Representations and General Coefficient Rings

For representations over general coefficient rings $R$ ($$\mathbb{Q}_\ell,\,\mathbb{F}_\ell,\,W(\mathbb{F}_\ell)$$, or characteristic $p$), new phenomena emerge:

• Admissibility may fail; induction and coinvariants need not be exact.
• Classification for irreducible admissible representations and simple Hecke modules proceeds via supersingularity in the modular case; each irreducible arises via unique triples $(P, \omega, Q)$ with supercuspidal $\omega$ for a Levi, as per (Henniart et al., 2017, Vignéras, 2022).
• Scalar extension and restriction functoriality are controlled via generalized Satake transforms and block centers.

7. Local Langlands Correspondence and Component Groups

Under the tempered local Langlands correspondence, irreducible representations are parametrized by enhanced $L$-parameters $(\varphi, \eta)$, where $\varphi$ is a morphism $W_F \times SL_2(\mathbb{C}) \to {}^LG$, and $\eta$ is an irreducible representation of the component group $S_\varphi$. Multiplicities in restriction to subgroups with common derived groups are equated to multiplicities in restriction of component group representations, with precise formulas via Clifford theory (Choiy, 2013, Aubert, 2023).

The internal structure of $L$-packets, their connections to Springer correspondence, and block decompositions are fundamental in arithmetic and geometric representation theory, with explicit formulas for multiplicities and functorial compatibilities.

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In summary, representations of $p$-adic groups comprise a unified system encoding arithmetic, geometry, and harmonic analysis. Classification via supercuspidal supports, Bernstein blocks, and Hecke algebra equivalences, reducibility via separation criteria and geometric techniques, and interaction with the local Langlands paradigm form the core technical and conceptual framework of the field (Fintzen, 9 Oct 2025, Ciubotaru et al., 2017, Choiy, 2013, Vignéras, 2022).