Fourier–Jacobi Models in Representation Theory

• Fourier–Jacobi models are linear representation-theoretic functionals defined as Hom-spaces that combine Weil representations with group representations, playing a key role in the study of automorphic forms and theta correspondences.
• They utilize explicit multiplicity formulas and uniform bounds, such as the 25^n (n!)^2 bound for symplectic groups, which clarify representation-theoretic invariants over finite and local fields.
• Applications include the computation of period integrals, L-functions, and the study of the local Gan–Gross–Prasad conjecture, underscoring the models' significance in modern harmonic analysis.

A Fourier–Jacobi model is a linear (or sesquilinear) representation-theoretic functional, typically constructed for a group $G$ with respect to a pair consisting of a unipotent subgroup and an oscillator (Weil) representation tied to a Heisenberg group, together with an auxiliary representation of a Jacobi-type subgroup. These models are essential in the harmonic analysis of classical groups over local and finite fields, with deep connections to the theory of automorphic forms, theta correspondences, and the Gan–Gross–Prasad conjectures.

1. General Definition and Abstract Setting

Let $G$ be a classical group (or its covering), $H$ a Jacobi or Fourier–Jacobi subgroup involving a unipotent radical and a Heisenberg group (arising from polarization or parabolic induction), and $\psi$ a nontrivial character on the center of the Heisenberg group. The corresponding Weil representation $\omega_\psi$ plays a crucial role in constructing models for irreducible (smooth or admissible) representations $\pi$ of $G$. Explicitly, the Fourier–Jacobi model is the space

$$\operatorname{Hom}_H(\pi, \psi \otimes \omega_\psi),$$

or, for external products,

$\operatorname{Hom}_H ( \pi_V \otimes \pi_W, \xi ),$

where $\xi$ encodes the character data and possibly a representation of a Jacobi quotient, with $V$ and $W$ spaces of compatible dimensions and types (e.g., Hermitian, symplectic). For finite and $p$-adic fields, explicit combinatorial or integral formulas for these spaces' dimensions are available, and for Archimedean fields, the concept extends naturally to Casselman–Wallach representations (Liu et al., 2022).

2. Construction in Finite and Local Fields

• Finite fields: For $G = \mathrm{Sp}_{2n}(\mathbb F_q)$ or $\mathrm{U}_n(\mathbb F_q)$, the standard setup involves a polarization $V = W \oplus W^-$ (with $W$ smaller rank), identification of a parabolic subgroup $P = MN$ with $N$ containing a Heisenberg group $H(W)$, and inflation of the Weil representation to $H = \mathrm{Sp}(W) \times N$. The Fourier–Jacobi multiplicity is defined as $$m(\pi, \sigma) := \dim\operatorname{Hom}_H( \pi|_H, \sigma \boxtimes \omega )$$, with $\pi$ and $\sigma$ irreducible representations of $\mathrm{Sp}_{2n}(\mathbb F_q)$, $\mathrm{Sp}_{2m}(\mathbb F_q)$ respectively (Shi, 2023).
• $p$-adic and Archimedean fields: Extensions rely on analogous Jacobi groups and oscillator representations defined via the Stone–von Neumann theorem, making use of parabolic induction, reductions to Schrödinger models, or Schwartz induction. The essence is the interplay between the unipotent and Heisenberg subgroups, with functionals intertwining the input representation and the Weil representation (sometimes twisted by another representation of a Jacobi quotient) (Boisseau, 2024, Chen, 2023, Liu et al., 2012).

3. Explicit Multiplicity Formulas and Uniform Bounds

In the finite field case, when irreducible representations $\pi$, $\sigma$ are expressed in terms of Deligne–Lusztig characters, powerful explicit multiplicity formulas exist. If $R_T^G(\theta)$ (resp. $R_S^G(\nu)$) is the Deligne–Lusztig character induced from maximal torus $T$ (resp. $S$), the multiplicity is given by finite Weyl group sums involving character values on semisimple elements and certain sign and cardinality factors: $$m(R_T^G(\theta), R_S^G(\nu)) = \sum_{J \in \mathcal J(S, T)} (-1)^{\mathrm{rk} T+\mathrm{rk} S+\ell(Z_J)} |W_{G,J}(T)^F|\,|W_G(S)^F|^{-1} \sum_{w, v} \theta(w z_J) \nu( v z_J ),$$ where $J$ runs over matching Levi-centralizers, $Z_J$ is a common torus, and $W_{G,J}(T)^F$ is a stabilizer subgroup (Liu et al., 2022).

A central theorem for finite symplectic groups states that for $$\pi, \pi' \in \operatorname{Irr}\mathrm{Sp}_{2n}(\mathbb F_q)$$ and $q > 2^{2n}$, the basic multiplicity satisfies

$m_0(\pi, \pi') \leq 25^n (n!)^2,$

and more generally $m(\pi, \sigma) \leq 25^n (n!)^2$, independent of $q$ (Shi, 2023). For type $A$ (general linear and unitary groups), the bound $C(n) = (2n!)^2$ suffices (Liu et al., 2022). These results settle conjectures on uniform bounds (e.g., Hiss–Schröer).

4. Multiplicity-One Theorems and Uniqueness

In the $p$-adic and Archimedean settings, Fourier–Jacobi models typically exhibit multiplicity-one: for generic representations, the dimension of the Hom-space is at most one. Explicitly, for $G = \mathrm{GL}_n(\mathbb R)$, $U(p,q)$, $\mathrm{Sp}_{2m}$, or metaplectic double cover $\widetilde{\mathrm{Sp}}_{2m}$, with properly compatible data,

$$\dim \operatorname{Hom}_{N_r \rtimes J_r'}( \pi, \psi \otimes \sigma ) = 1,$$

where $N_r$ is a maximal unipotent, $J_r'$ is the appropriate Jacobi cover, and $\psi$ is a generic character (Liu et al., 2012). This result extends to certain finite field cases for exceptional and classical groups, with exceptions occurring only for specific fully induced representations (Liu et al., 2018).

For Archimedean local fields, general multiplicity and preservation results ensure that for normalized induced representations, the Fourier–Jacobi multiplicity satisfies

$$m( I_P^{G_V}(\sigma), \pi_W ) = m( \pi_{V_0}, \pi_{W_0} ),$$

where the right side is the multiplicity in the lower rank (or “basic cell”) model (Chen, 2023).

In the context of the local Gan–Gross–Prasad (GGP) conjecture (e.g., for symplectic-metaplectic pairs), the unique (up to scalar) nontrivial Fourier–Jacobi model within a Vogan packet is characterized via distinguished characters constructed from local root numbers (the so-called $\varepsilon$-dichotomy) (Chen et al., 14 Jan 2026).

5. Fourier–Jacobi Descent and Explicit Local-Global Correspondence

For $p$-adic unitary groups, the technique of "Fourier–Jacobi descent" (a la Soudry and Tanay) transfers representation-theoretic invariants and models from larger to smaller unitary groups. The setup considers non-tempered representations, especially depth-zero supercuspidals of $G = U_{2n}(F)$, induced from distinguished types on a Siegel Levi. The local descent constructs, via Jacquet modules and Heisenberg embedding, generic supercuspidal representations of smaller $U_{2n-1}$ or $U_{2n}$:

• For unramified $m = 2n-1$, the descent is multiplicity-free over the generic supercuspidals of $U_{2n}(F)$.
• For ramified $m = 2n$, the top descent produces a unique depth-zero supercuspidal of $U_{2n}(F)$ associated to the inducing type (Liu et al., 2022).

This explicitly realizes the non-tempered branch of the local GGP correspondence, with exact matching controlled by poles of the Rankin–Selberg gamma-factors $\gamma(s, \pi \times \tau, \psi_F)$ at $s=1$.

6. Applications to Periods, $L$-functions, and Spherical Functions

Fourier–Jacobi models enter the explicit calculation of periods of automorphic forms, especially in relation to the global GGP and Ichino–Ikeda conjectures. For unitary groups, the Whittaker–Shintani functions associated to Fourier–Jacobi models are constructed explicitly, and their unramified values are expressed as sums over Weyl groups with explicit $L$-factors: $$W^\circ(\lambda) = \frac{\Delta_{U(W)} \cdot \Delta_{T_W}}{W^I(1)} \sum_{w \in W_G} b(w_V\chi, w_W\eta) d(w(\chi \otimes \eta)) (w(\chi \otimes \eta) \delta_{B^+}^{-1/2})(\lambda),$$ where each factor is determined by the group structure and the $L$-theory (Boisseau, 2024).

These explicit formulas allow direct comparison between nonvanishing periods and the arithmetic of central $L$-values, confirming the expected relationships of the form

$$\frac{ \mathcal P_H( \varphi^\circ, \varphi_\nu^\circ ) }{ (\varphi^\circ, \varphi^\circ)(\varphi_\nu^\circ, \varphi_\nu^\circ) } = \Delta_{U(V)} \frac{ L(1/2, \sigma \otimes \bar\mu) }{ L(1, \sigma, \mathrm{Ad}) },$$

under suitable unramified assumptions (Boisseau, 2024).

7. Open Problems, Extensions, and Future Directions

Several open directions and challenges remain:

• Optimal growth of multiplicities: While explicit uniform bounds, such as $25^n(n!)^2$ for symplectic groups, are established, it is open whether the true maximal multiplicity is polynomial in rank $n$ (Shi, 2023).
• Characteristics and small fields: Some results, particularly over finite fields, require $q$ odd and in certain arguments, $q > 2^{2n}$. It is believed (but not universally proven) that multiplicity bounds hold for all odd $q$.
• Non-tempered and positive-depth generalizations: For $p$-adic groups, the explicit descent and type theory for positive-depth supercuspidals and for non-supercuspidal discrete series remain to be fully developed.
• Extension to other spherical varieties: Extending explicit multiplicity formulas to other models, such as general Bessel or more general Fourier–Jacobi settings, is an ongoing area of research (Liu et al., 2022).
• Interactions with the Arthur–Langlands program: The explicit realization of Fourier–Jacobi models informs expected correspondences between local Arthur packets, types, and explicit root-number criteria.

These directions underscore the central role of Fourier–Jacobi models at the intersection of representation theory, automorphic forms, and arithmetic geometry, as clarified through the recent work of Liu–Ma–Shi and others (Liu et al., 2022).