Shalika Models for GL(2n) Representations

• Shalika models are specialized automorphic representations for GL(2n) defined via a subgroup combining diagonal and unipotent elements, exhibiting a unique (multiplicity one) property.
• They play a central role in the study of exterior-square L-functions and functoriality by linking the analytic behavior of L-functions to periods and automorphic lifts.
• Their stability under parabolic induction and explicit construction in local and global settings provide a foundation for developing p-adic L-functions and advancing arithmetic geometry.

A Shalika model is a special kind of automorphic or local model for representations of general linear groups, defined via equivariance with respect to the "Shalika subgroup"—a non-reductive subgroup combining diagonal and unipotent structures—and a character that is typically a product of a Hecke or central character and an additive character. Shalika models play a key role in the theory of automorphic $L$-functions, the classification of representations, and the geometry of eigenvarieties, and are linked to periods, multiplicity one theorems, functoriality, and arithmeticity. Their local and global theory is well-developed for $\mathrm{GL}_{2n}$, but generalizations also exist for metaplectic and other classical groups, as well as for unitary and symplectic groups.

1. Definition and Uniqueness of Shalika Models

Let $G = \mathrm{GL}_{2n}$ over a local or global field $F$. The Shalika subgroup $S$ is given by

$$S = \left\{ \begin{pmatrix} h & X \ 0 & h \end{pmatrix} \,:\, h \in \mathrm{GL}_n, X \in M_n \right\} \subset G.$$

Given a pair of characters $(\eta, \psi)$ on $F^\times$ and $F$, define the character on $S$ by

$$\chi_S\left( \begin{pmatrix} h & X \ 0 & h \end{pmatrix} \right) = \eta(\det h) \psi(\mathrm{tr} X).$$

A (local or global) representation $\pi$ of $G$ admits a Shalika model if there exists a nonzero linear form $\ell$ (termed Shalika functional) on $\pi$ such that for all $s \in S$, $\ell( \pi(s) v ) = \chi_S(s) \ell(v)$. Equivalently, there is an equivariant embedding

$$\pi \hookrightarrow \operatorname{Ind}_S^G\,(\chi_S).$$

For automorphic representations, the global Shalika period is given by

$$\Lambda_\psi(\varphi)(g) = \int_{Z_G(\mathbb{A}) S(F) \backslash S(\mathbb{A})} \varphi(sg)\eta^{-1}(s)\psi^{-1}(s)\,ds.$$

Shalika functionals are unique up to scalars (multiplicity one) in both local and global settings, provided the character is "generic"—this is a consequence of the Gelfand–Kazhdan method, Bruhat—Tits theory, and, for generalizations, Harish–Chandra descent and relative trace formula arguments (Chen et al., 2017, Naor, 2022, Cauchi et al., 2023).

2. Characterization via $L$-functions and Functoriality

A core result characterizes the existence of a Shalika model for a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_{2n}(\mathbb{A})$ in terms of the exterior-square $L$-function: $$\pi \text{ admits a Shalika model} \iff L^S(s, \pi, \wedge^2 \otimes \eta^{-1}) \text{ has a simple pole at } s=1.$$ This reflects that such $\pi$ are exactly those arising as functorial lifts from generic, globally generic, automorphic representations on GSpin$_{2n+1}$ or (in the split case) SO$_{2n+1}$ (Grobner et al., 2011, Gehrmann, 2015). The local analogue relates the local exterior-square $L$-factor: for generic $\pi_v$, nonvanishing of the Shalika period is equivalent to a simple pole of $L(s, \pi_v, \wedge^2)$ at $s=0$ (Matringe, 2015, Matringe, 2012, Beuzart-Plessis et al., 2018).

3. Structure Theory: Parabolic Induction, Multiplicity One, and Classification

Shalika models are stable under normalized parabolic induction: if $\pi_1$ and $\pi_2$ admit Shalika functionals, so does their induced product to $\mathrm{GL}(n_1+n_2)$, and the dimension of Hom spaces stays at most one (Matringe, 2015, Naor, 2022). Consequently, generic irreducible representations of $\mathrm{GL}_{2n}$ admitting a Shalika model are classified as those of the form

$$\pi \cong (\Delta_1 \times \Delta_1^\vee) \times \cdots \times (\Delta_s \times \Delta_s^\vee) \times \Delta_{s+1} \times \cdots \times \Delta_t,$$

where the $\Delta_i$ are discrete series or (for $i > s$) themselves admit Shalika models.

The connection to distinction problems is via periods: if $\pi$ is distinguished by $H = \mathrm{GL}_n \times \mathrm{GL}_n$, i.e., has a linear period (invariant under $H$), then under suitable integrability/square-integrability conditions, it also has a Shalika model (Matringe, 2012).

4. Explicit Constructions: Newforms, Test Vectors, and Zeta Integrals

Shalika newforms generalize the notion of local (newform) vectors in Whittaker models. For a generic irreducible representation over a non-archimedean local field, Jacquet–Piatetski–Shapiro–Shalika define a compact open subgroup $K$ with the fixed vector space $V^K$ one-dimensional, and the (canonically normalized) Shalika newform $J^{\mathrm{new}}$ (Okazaki, 2024). Okazaki provides a method to compute the values and support of Shalika newforms on the mirabolic subgroup, using combinatorics of double cosets, Gauss sums, and integral formulas. For ramified twisting characters, there exists an alternative "identity-nonvanishing" Shalika form $J_T$ playing the role of a true newform and realizing the full $L$-factor and root number through the associated local zeta integrals.

For $p$-adic analytic families and eigenvarieties, Shalika refinements (choices of eigenvectors in the local Shalika models with well-controlled Hecke eigenvalues) are key to the construction of $p$-adic $L$-functions (Salazar et al., 2022, Salazar et al., 2021, Dimitrov et al., 11 Aug 2025). The Friedberg–Jacquet local zeta integral,

$\zeta_v(s+\tfrac12, W_v, \chi_v) = \int_{\GL_n(F_v)} W_v\left( \begin{pmatrix} h & 0 \ 0 & 1 \end{pmatrix} \right)\chi_v(\det h) |\det h|^{s-\tfrac12}\,dh,$

with suitable test vectors, interpolates the local $L$-factors and is especially important at ramified and parahoric or Iwahori level.

5. Shalika Models and $p$-adic $L$-Functions in Families

Shalika models provide the structural input for the construction of multi-variable $p$-adic $L$-functions—functions varying over eigenvarieties and weight spaces—which interpolate critical values of standard $L$-functions for families of automorphic representations. The main method involves $p$-adic interpolation of classical evaluation functionals, attached to branching laws for $\mathrm{GL}_n \times \mathrm{GL}_n$ inside $\mathrm{GL}_{2n}$ and overconvergent cohomology (Salazar et al., 2022, Salazar et al., 2021, Dimitrov et al., 11 Aug 2025, Gehrmann, 2015).

The key results are as follows:

• For regular algebraic, symplectic-type cuspidal $\pi$ (i.e., essentially self-dual with symplectic twist), the eigenvariety is étale at classical points with non-critical Shalika refinements, and there is a Zariski-dense set of such points (Shalika families).
• The resulting $p$-adic $L$-functions interpolate classical special values (critical $L(\pi_y\otimes\chi, j+\tfrac12)$, up to explicit periods and $p$-adic Euler factors), and the analytic continuation to families rigorously mirrors the branching at Shalika points.
• Admissibility, slope bounds, and the uniqueness of these Shalika functionals are crucial in guaranteeing the growth, control, and arithmeticity of the constructed $p$-adic measures and their interpolation properties.

6. Generalizations, Metaplectic Variants, and Connections

Generalized Shalika models exist for groups $G = \mathrm{GL}_{n+m}$, with subgroup $H_{n,m}$ defined by a block structure, and the associated induced models are multiplicity-free (Naor, 2022). Metaplectic Shalika models (for double covers of GL$_{2n}$) are constructed by embedding the classical definition into the metaplectic context, yielding multiplicity one, explicit (Casselman–Shalika type) formulas, and new Godement–Jacquet style integral representations for $L$-value quotients (e.g., $\mathrm{Sym}^2/\wedge^2$) (Kaplan et al., 2016).

For groups of similitude or other classical types, Shalika models are also defined (e.g., for $\mathrm{PGU}_{2,2}$, where the model is characterized and realized via the theta correspondence, and a Casselman–Shalika formula is proven in terms of representations of the dual group of $\mathrm{PGSp}_4$) (Cauchi et al., 2023).

Connections to the geometry of eigenvarieties and Galois representations crucially use the existence, uniqueness, and analytic family structure of Shalika models, forming a bridge between analytic, representation-theoretic, and arithmetic aspects of automorphic forms.

7. Arithmeticity and Periods

Shalika models can be equipped with rational structures relating automorphic forms and (cohomological) Galois representations. For cohomological cuspidal automorphic representations $\Pi$ of $\mathrm{GL}_{2n}(\mathbb{A})$, a comparison of rational structures leads to canonical periods $\omega^\epsilon(\Pi_f)$, whose behavior under twisting matches the predictions for periods in Deligne's and Gross's conjectures. Explicit algebraicity and non-vanishing results for critical $L$-values, Rankin–Selberg, and symmetric cube $L$-functions for Hilbert modular and Siegel modular forms have been established via Shalika models (Grobner et al., 2011).

The construction of $p$-adic $L$-functions in families, including their interpolation properties, exceptional zero behavior, and explicit formulae in canonical cases, is fundamentally tied to the arithmetic and analytic qualities of the Shalika model and its associated periods (Gehrmann, 2015, Salazar et al., 2022, Salazar et al., 2021, Dimitrov et al., 11 Aug 2025).

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Selected Principal References:

Paper Title	arXiv ID	Main Contribution
"Shalika newforms for GL(n)"	(Okazaki, 2024)	Structure, uniqueness, newforms, explicit formulas
"Shalika periods and parabolic induction for GL(n)"	(Matringe, 2015)	Induction, classification, $L$-factor link
"On the arithmetic of Shalika models and the critical values of $L$"	(Grobner et al., 2011)	Arithmeticity, periods, algebraicity of critical values
"On Shalika models and p-adic L-Functions"	(Gehrmann, 2015)	$p$-adic $L$-functions via modular symbols, Shalika interpolation
"Shalika models for general linear groups"	(Naor, 2022)	Multiplicity one, generalizations to GL$_{n+m}$
"A Local Trace Formula for the Generalized Shalika Model"	(Beuzart-Plessis et al., 2018)	Trace formula, multiplicity on packets, link to $L$-functions
"On $p$-adic $L$-functions for $GL_{2n}$ Shalika families"	(Salazar et al., 2021)	Eigenvarieties, $p$-adic $L$-functions, geometric density
"On the GL(2n) eigenvariety: ... Shalika families and $p$-adic $L$"	(Salazar et al., 2022)	Shalika refinements, multivariable $p$-adic $L$-functions
"Parahoric level $p$-adic $L$-functions for ... Shalika models"	(Dimitrov et al., 11 Aug 2025)	Parahoric Shalika theory, refined local-global correspondences

These sources collectively form the modern theory of Shalika models and their applications across arithmetic, representation theory, and automorphic forms.