Speh Representations

• Speh representations are defined as irreducible constituents obtained by parabolic induction from a square-integrable or cuspidal building block, serving as atomic units in automorphic form classification.
• They are characterized using degenerate (k,c) models and unique invariant functionals, facilitating the computation of Rankin–Selberg integrals and local L-factors.
• Their detailed structure via branching laws, Jacquet modules, and Eisenstein series residues provides key insights into the residual spectrum and unitary duals in both local and global settings.

A Speh representation is a distinguished member of the discrete, non-cuspidal automorphic or admissible spectrum of a general linear group over a local or global field, arising as an irreducible constituent—typically a unique submodule or quotient—of a parabolically induced representation built from a single square-integrable (or, in the global setting, cuspidal) building block. These representations serve as fundamental "atoms" in the classification of unitary duals, the structure theory of automorphic forms, and the functional-analytic theory of non-generic models.

1. Construction and Characterization

A classical Speh representation for GLₙ over a local or global field is constructed as follows: let $T$ be an irreducible unitary square-integrable (or automorphic cuspidal) representation of GLₖ; for an integer $m \geq 1$, form the normalized parabolic induction

$$u(T,m) = \operatorname{Ind}_P^{GL_{km}} \Big( \nu^{\frac{m-1}{2}} T \otimes \nu^{\frac{m-3}{2}} T \otimes \cdots \otimes \nu^{-\frac{m-1}{2}} T \Big),$$

where $\nu(g) = |\det g|$. The unique irreducible quotient (or submodule, depending on normalization) of $u(T,m)$ is the Speh representation, usually denoted $\operatorname{Sp}(T,m)$ or $A(T,m)$. In the global context, the construction involves taking multi-residues of Eisenstein series attached to such induced representations, isolating the "square-integrable" part of the residual spectrum (Ginzburg et al., 30 Oct 2024).

In terms of multisegments, Speh representations correspond to the ladder of segments $[v^{(1-d)/2} p, v^{(d-1)/2} p]$, iterated appropriately, and are characterized as ladder or "atomic" representations in the Zelevinsky/Tadić classification (Tadic, 2013).

Key Defining Properties (Local):

Property	Formulation	Reference
Induction datum	$$u(\delta,k) = L(\nu^{(k-1)/2} \delta \times \cdots \times \nu^{-(k-1)/2} \delta)$$	(Badulescu, 2011)
Unitarity	Irreducible/unitary except for endpoints (complementary series)	(Badulescu, 2011, Tadic, 2013)
Unique model	Admits unique (k,c) Whittaker model of type $(c^k)$ (degenerate model)	(Cai et al., 2021)
Duality	Aubert/Zelevinsky duality: $i(u(\delta, k)) = u(\tau, k)$, switching segment parameters	(Badulescu, 2011)

2. Models and Symmetric Function Formulas

Speh representations, being non-generic for $m > 1$, lack a classical Whittaker model. Instead, their analytic and combinatorial structure is captured via degenerate (k,c) models. These models are realized by unique functionals on the unipotent radical corresponding to the rectangular partition $(c^k)$, and are characterized by multiplicity one for this functional (Cai et al., 2021).

In the unramified case, the normalized spherical vector in the (k,c) model satisfies a Casselman-Shalika-type formula:

$$\mathscr{W}^{(k,c)}(\operatorname{diag}(g, I_{(k-1)c})) = \delta_{(k,c)}^{1/2} (\operatorname{diag}(g, I_{(k-1)c})) \cdot Q_\mu(q^{-z_1}, \ldots, q^{-z_k}; q),$$

where $Q_\mu$ is a modified Hall–Littlewood polynomial evaluated at Satake parameters, generalizing the Schur polynomial in the classical case (Zelingher, 10 Jul 2024).

The uniqueness of the (k,c) functional, both over local fields and finite fields, is critical to establishing the Eulerian property of integral representations of L-functions (see §5) (Cai et al., 2021, Zelingher, 3 Aug 2024).

3. Reducibility, Duality, and Jacquet Modules

Speh representations play a pivotal role in the structure and reducibility theory of induced representations:

• Products $u(\delta, k) \times u(\delta, k)$ and $u(\delta, k-1) \times u(\delta, k+1)$ are irreducible for all $k \geq 1$ (Badulescu, 2011).
• The dual of a Speh representation is again a Speh representation, with the segment length and thickness parameters exchanged (via the Moeglin–Waldspurger algorithm) (Badulescu, 2011).
• In the structure formula for Jacquet modules of normalized induced representations $u_1 \times \cdots \times u_r \rtimes \pi$ (with the $u_i$ essentially Speh on GL, and $\pi$ on a classical group), the Tadić formula expresses the Jacquet functor as $M^*(\pi) \rtimes \mu^*(\sigma)$, a powerful multiplicativity tool in the analysis of compositional structure (Bošnjak et al., 1 Apr 2025).

At special reducibility points (beyond the endpoint of the complementary series), induced products of Speh representations exhibit explicit composition series, with each irreducible subquotient classified by Langlands parameters corresponding to distinct multisegments; all these representations are multiplicity one (Tadic, 2013, Luo, 2021).

4. Branching Laws and Restriction

Speh representations demonstrate controlled branching upon restriction to subgroups:

• For real groups, the restriction of a Speh representation of $\operatorname{GL}(2n, \mathbb{R})$ to $\operatorname{GL}(2n-1, \mathbb{R})$ decomposes as a direct integral of unitarily induced representations from a maximal parabolic subgroup, with induction data involving a Speh representation of lower rank and a unitary character (Ditlevsen et al., 7 Aug 2025).
• For block-diagonal Levi subgroups (e.g., $$\operatorname{GL}_n \times \operatorname{GL}_{n+2l}$$), explicit nonzero morphisms exist from $\operatorname{Sp}(\pi,n+l)$ to representations with factors in $\operatorname{Sp}(\pi, l)$ and representations associated to the block factor, established via explicit zeta integrals in the Shalika model (Ito, 2021).
• These branching laws reflect and generalize the predictions of the theory of adduced representations and provide the local facets of global phenomena such as Miyawaki liftings.

5. Periods, Invariant Functionals, and Distinguishedness

Distinguished Speh representations, i.e., those supporting nontrivial invariant functionals under the action of a subgroup, are classified in terms of parities and distinguishedness of the inducing data:

• A Speh representation $\operatorname{Sp}(A,k)$ of $\operatorname{GL}_{2m}(F)$ is $H_{m,m}$-distinguished (has a nonzero linear period with respect to $\operatorname{GL}_m \times \operatorname{GL}_m$) if and only if $A$ is an essentially square-integrable representation of even degree $d$ and itself $H_{d/2, d/2}$-distinguished (Yang, 2020).
• For real groups, Speh representations of $\operatorname{GL}(2n, \mathbb{R})$ admit unique Siegel parabolic-invariant functionals for even $n$, but do not admit invariant functionals with respect to $U(n) \subset \operatorname{Sp}_{2n}(\mathbb{R})$ for odd $n$ (Gourevitch et al., 2014).
• Quaternionic or division algebra analogues admit precisely defined unique degenerate Whittaker models, with the twisted Jacquet module isomorphic to the central character composed with the reduced norm (Cai, 2021, Nadimpalli et al., 2023).
• In the p-adic symmetric space $$\operatorname{Sp}_{2n}(F) \backslash \operatorname{GL}_{2n}(F)$$, Speh representations of the form $U(\delta, 2)$ are relatively discrete—i.e., they appear in the $L^2$-spectrum of the space with no continuous spectrum contribution, as predicted by the relative Langlands correspondence (Smith, 2018).

6. Rankin–Selberg Integrals, Local and Global Models

Rankin–Selberg zeta integrals for Speh representations generalize the classical theory:

• For local non-generic (Speh) representations, explicit integral representations are constructed in Shalika or (k,c) models; these integrals compute local $L$-factors, with meromorphic continuation and functional equations paralleling the Whittaker case (Lapid et al., 2018, Atobe et al., 2021).
• The compatibility between the Zelevinsky/Whittaker and Shalika models is established for the computation of such integrals, demonstrating that the analytic invariants (local factors, newform theory) are preserved across models (Atobe et al., 2021).
• Over finite fields and their relation to level zero supercuspidal representations, there exists a commutative diagram connecting the finite field Speh representation and its local lift, including the compatibility of all (k,c) ψ–Whittaker models and associated integrals. Gamma factors computed via Ginzburg-Kaplan integrals match across the finite and local settings, even in the exceptional self-contragredient case, up to a sign and power of $q$ (Zelingher, 3 Aug 2024).

7. Analytic Theory, Eisenstein Series, and Residual Spectrum

Speh representations are key to the residual spectrum and the analytic structure of Eisenstein series:

• Eisenstein series induced from Speh representations on GL groups have precisely determined locations of their simple poles, occurring at values $s = (m_1 + m_2)/4 - i/2$ ($0 \leq i \leq \min(m_1, m_2)-1$) for induction from two Speh representations with parameters $m_1, m_2$ (Ginzburg et al., 30 Oct 2024).
• In the case $m_1 = m_2$, these Eisenstein series vanish at $s=0$. Analytically, these locations reflect linkage in the underlying segments and are determined via explicit constant term and "global derivative" computations.
• For classical groups (e.g., $\operatorname{Sp}_{2n}$ and its metaplectic covers), residual representations induced from Speh representations are parametrized by a unique maximal nilpotent orbit (the wavefront set), detectable by non-vanishing Fourier coefficients, computed by integrating the residue of the Eisenstein series against suitable unipotent subgroups. This relationship is pivotal in the theory of descent and functorial transfers (Ginzburg et al., 2020).

8. Broader Applications and Research Directions

Speh representations provide canonical "non-generic" building blocks across harmonic analysis, automorphic forms, and representation theory:

• They are central to the classification of the unitary dual and play a decisive role in the structure of residual and relative discrete spectra (Smith, 2018, Tadic, 2013, Badulescu, 2011).
• The (k,c) model theory unifies the analytic paper of generalized doubling integrals (ensuring Eulerian product structure of $L$-functions) and opens further connections with symmetric function theory (e.g., via Hall–Littlewood polynomials) (Zelingher, 10 Jul 2024, Cai et al., 2021).
• Quaternionic Speh and inner form analogues permit the extension of these concepts and integral representations to non-split forms, enriching the global Jacquet–Langlands correspondence and the analytic theory of L-functions (Cai, 2021).
• All the above illustrates the tight interplay between induced representations, analytic L-factors, nilpotent orbits, unique models, and the combinatorics of multisegments or symmetric functions—a theme that continues to motivate current research.

Speh representations thus form a bridge between the geometry of orbits, the combinatorics of ladder and multisegment classifications, the functional analysis of unitary representations, and the global analytic theory of automorphic forms and L-functions. Their analytic, algebraic, and arithmetic properties are central to deep results across the entire spectrum of automorphic and classical representation theory.