Langlands & Zelevinsky Classifications

• Langlands and Zelevinsky classifications are foundational frameworks that define irreducible representations of reductive groups and GLₙ(F) using standard modules and combinatorial multisegments.
• They employ normalized parabolic induction and explicit segment constructions to yield unique irreducible quotients, linking representation theory with the local Langlands correspondence.
• Extensions to Archimedean groups, metaplectic covers, and duality phenomena enhance their impact on automorphic forms, harmonic analysis, and arithmetic applications.

The Langlands and Zelevinsky classifications are fundamental frameworks in the representation theory of reductive groups over local fields, both Archimedean and non-Archimedean. The Langlands classification parametrizes all irreducible admissible representations of a reductive group in terms of standard modules and parabolic induction. The Zelevinsky classification, specific to $\mathrm{GL}_n(F)$ for a non-Archimedean local field $F$, refines this by exhaustively describing irreducibles in terms of combinatorial data—multisegments composed of cuspidal representations and explicit rules for construction and reducibility. Recent advances extend these frameworks to Archimedean groups and metaplectic covers, and clarify compatibility with the local Langlands correspondence and epsilon factors. These classifications are foundational for the structure and harmonic analysis of automorphic forms, explicit character computations, and arithmetic applications.

1. Langlands Classification for Reductive Groups

The Langlands classification provides a uniform method for constructing and parametrizing all irreducible admissible representations of a reductive group $G$ over a local field $F$ (either $p$-adic or real) via normalized parabolic induction. Given a standard parabolic $P=MN$ of $G$, an irreducible tempered representation $\sigma$ of $M(F)$, and a complex parameter $\nu$ in $(a_M)_{\mathbb C}^*$ with $\mathrm{Re}\,\nu$ dominant, one forms the (normalized) induced representation $$I_P^G(\sigma\otimes\nu) = \mathrm{Ind}_{P(F)}^{G(F)}(\sigma\otimes\nu\otimes\delta_P^{1/2})$$. The Langlands classification theorem asserts:

1. For each such $(P, \sigma, \nu)$, $I_P^G(\sigma\otimes\nu)$ has a unique irreducible quotient $J_P^G(\sigma,\nu)$.
2. Every irreducible admissible representation $\pi$ of $G(F)$ is isomorphic to exactly one $J_P^G(\sigma,\nu)$ modulo $G(F)$-conjugacy and twist-equivalence of data.

This framework translates representation-theoretic classification into the problem of identifying tempered representations of Levi subgroups and appropriate induction parameters, connecting to the classification of automorphic representations and the local Langlands correspondence (Kaletha, 2022).

2. Zelevinsky Classification for $\mathrm{GL}_n(F)$: Multisegment Theory

For $G = \mathrm{GL}_n(F)$ over a non-Archimedean local field $F$, Zelevinsky's classification describes irreducible admissible representations via segments and multisegments:

• A segment is a sequence $$\Delta = [\rho, \nu^k\rho] = \{\rho, \nu\rho, \ldots, \nu^k\rho\}$$ for an irreducible supercuspidal $\rho$ of a smaller $\mathrm{GL}_m(F)$.
• A multisegment $\mathfrak m$ is a finite, unordered collection of segments, with total length $n$.
• To each multisegment one attaches a standard module via parabolic induction: $$Z(\Delta_1,\ldots,\Delta_r) = \mathrm{Ind}_P^{\mathrm{GL}_n}( \delta(\Delta_1)\otimes\cdots\otimes\delta(\Delta_r) )$$, where $\delta(\Delta)$ is essentially square-integrable (the unique irreducible submodule of the segment product).
• The unique irreducible Langlands quotient $L(\Delta_1, \ldots, \Delta_r)$ of $Z(\Delta_1,\ldots,\Delta_r)$ is the corresponding irreducible representation.

Every irreducible admissible representation arises uniquely in this way, yielding a bijection between multisegments (subject to combinatorial ordering) and $\mathrm{Irr}(\mathrm{GL}_n(F))$. Reducibility and Jordan–Hölder factors of standard modules are controlled via the interaction of segments—linking, containment, and disjointness, with Zelevinsky's partial order reflecting inclusion or merge–split operations on multisegments (Kaletha, 2022, Mundy, 2023).

3. Archimedean and Real-Analogue Classifications

Prasad generalized the Bernstein–Zelevinsky paradigm to real groups, notably $\mathrm{GL}_n(\mathbb{R})$. In this context:

• Segments comprise either $[s]$ for GL$_1(\mathbb{R})$ (characters $t\mapsto |t|^s$) or $[v,u]$ for essentially discrete series of GL$_2(\mathbb{R})$.
• Multisegments encode standard modules similarly: $I(\Delta_1, \ldots, \Delta_t)$.
• The irreducible constituents and reducibility patterns mirror the $p$-adic theory via combinatorial rules (linking/containment/disjointness).
• Prasad formulates a real Zelevinsky classification conjecture: every irreducible admissible representation of $\mathrm{GL}_n(\mathbb{R})$ arises as $Z(\Delta)$ for a unique multisegment $\Delta$, and multiplicities/join relations follow Kazhdan–Lusztig–Vogan phenomena (Prasad, 2017).

The principal series, irreducibility, and Steinberg modules for real groups are similarly governed by these segment data, with explicit descriptions of reducibility, JH-factorization, and Langlands parameter correspondences.

4. Structural Features: Kazhdan–Lusztig Multiplicities and Dualities

Multiplicity phenomena in standard modules are conjecturally (and, in certain cases, provably) governed by Kazhdan–Lusztig type polynomials: the multiplicity of an irreducible $Z(\Delta')$ in the standard module $I(\Delta)$ equals $P_{\Delta, \Delta'}(1)$ for a suitable KL or Kazhdan–Lusztig–Vogan polynomial. In the non-Archimedean context, this is mirrored in the GL$_n$ Zelevinsky framework and analogues for real reductive groups; explicit computational recipes rely on combinatorial re-linking of segments.

Aubert–Zelevinsky duality, and its extension to classical groups via Moeglin–Waldspurger algorithms, gives a combinatorial involutive correspondence on multisegments: for GL$_n(F)$, Zelevinsky's involution exchanges the submodule and quotient realizations of irreducibles, while in classical groups a symmetric, signed multisegment structure is manipulated recursively to produce dual data reflecting Aubert duality. This algorithmic framework is explicit and entirely computable, with critical differences in duality behavior between generic and non-generic representations (Lanard et al., 16 Sep 2025).

5. Langlands Correspondence, Epsilon Factors, and Families

The local Langlands correspondence (LLC) asserts a bijection between irreducible admissible representations of $G(F)$ (notably GL$_n(F)$) and $L$-parameters, i.e., $n$-dimensional Frobenius-semisimple Weil–Deligne representations (for $p$-adic fields) or real (Weil group) parameters for Archimedean groups. For GL$_n$, the Zelevinsky multisegment data is mapped directly: $\Delta_i = [\rho_i, \nu^{k_i}\rho_i]$ (with $\rho_i$ corresponding to a $W_F$-representation $\tau_i$) maps to $$\phi = \bigoplus_i (\tau_i \otimes \mathrm{Sym}^{k_i}(\mathbb{C}^2))$$. This establishes a precise dictionary between combinatorial representation parameters and Galois-theoretic ones (Kaletha, 2022).

Analytic and rigidity properties in families—parametric variations of representations—display stability: in generically irreducible Hecke-theoretic families, the Bernstein–Zelevinsky multisegment data varies only by unramified character twists, ensuring analytic variation of associated arithmetic invariants, such as epsilon factors. By exploiting coverings (Bushnell–Kutzko types) and trace formulas on eigenvarieties, it is shown the local epsilon-factors are analytic (locally constant) on such families, linking p-adic geometry, type theory, and local–global compatibility (Mundy, 2023).

6. Extensions: Metaplectic Covers and Generalizations

Kaplan–Lapid–Zou establish a metaplectic (covering group) analogue of the Bernstein–Zelevinsky–Langlands framework for $\widetilde{\mathrm{GL}}_n(F)$:

• Genuine supercuspidals, segments, and multisegments must be defined in the metaplectic category, with a metaplectic tensor product and central character constraints.
• Standard modules are built using metaplectic parabolic induction and tensor products.
• The main classification (Theorem 7.5): every irreducible genuine representation is realized as the unique irreducible socle or cosocle of a standard module attached to a multisegment, exhaustively parametrizing the genuine spectrum.
• These constructions specialize to classical Zelevinsky theory in the split case and incorporate new cocycle and character phenomena unique to the covering setting (Kaplan et al., 2022).

7. Impact, Open Directions, and Broader Context

The Langlands and Zelevinsky classifications serve as the backbone for:

• Study of harmonic analysis on $p$-adic, real, and adelic groups.
• Explicit realization and computation of automorphic representations and their $L$- and $\epsilon$-factors.
• Geometric and $p$-adic variations (eigenvarieties, Hecke theory, rigid-analytic geometry).
• Duality operations and the structure of $L$-packets for both split and non-split groups.
• Algorithmic and combinatorial computation for both ordinary and covering groups.

A prevailing theme is the deep compatibility between combinatorial (segment, multisegment) representation structures, the Langlands parameterization by Galois data, and analytic invariants such as local constants. Major open directions include the full proof and refinement of the real-group Zelevinsky classification, explicit Kazhdan–Lusztig multiplicity formulas for general groups, and the extension of these frameworks to non-linear covering groups and beyond. Recent works utilize machine learning and algorithmic methods to further automate and elucidate duality and reducibility phenomena (Lanard et al., 16 Sep 2025).