Twisted Jacquet Module in Representation Theory

• Twisted Jacquet module is a representation theory concept defined as the quotient of a smooth representation by the span of twisted unipotent actions using a nontrivial character.
• It isolates the ψ-isotypic part of a representation, enabling the identification of specific models such as Whittaker and Shalika models with precise dimension formulas.
• The construction underpins advanced techniques in harmonic analysis and automorphic forms, driving research in distinction, branching, and algorithmic classification.

A twisted Jacquet module is a fundamental object in the representation theory of reductive groups over local and finite fields, generalizing the classical Jacquet module by incorporating an explicit twist by a character of the unipotent radical of a parabolic subgroup. It serves as a key tool for isolating specific model spaces, such as Whittaker and Shalika models, and for investigating distinction and branching phenomena. The twisted Jacquet construction has far-reaching implications in both local harmonic analysis and the study of automorphic forms, and is central to numerous contemporary research programs (Nadimpalli et al., 2023, Harshitha et al., 31 Dec 2025, Balasubramanian et al., 2022, Matringe, 2020, Balasubramanian et al., 2024, Pandey et al., 2024).

1. Definition and General Properties

Let $G$ be a reductive group over a nonarchimedean local or finite field $F$, and $P=MN$ a parabolic subgroup with Levi $M$ and unipotent radical $N$. For a nontrivial character $\psi$ of $N$, the $\psi$-twisted Jacquet module $V_{N,\psi}$ associated to a smooth $G$-representation $(\pi,V)$ is defined as the quotient: $$V_{N,\psi} = V / \mathrm{Span}\{ \pi(n)v - \psi(n)v : n \in N, v \in V \}$$ This module carries an action of the stabilizer subgroup $$M_\psi = \{ m \in M : \psi(m n m^{-1}) = \psi(n) \ \forall n\}$$, isotypic to the model sought. In the case $\psi=1$, $V_N$ reduces to the ordinary Jacquet module. The twisted Jacquet module isolates the $\psi$-isotypic quotient, capturing harmonic analytic properties and multiplicity-one phenomena for models attached to $\psi$ (Nadimpalli et al., 2023, Harshitha et al., 31 Dec 2025, Matringe, 2020).

2. Explicit Computations in Classical and Non-Classical Settings

A significant case is $G = GL_{2n}(F)$, $P$ the maximal parabolic of type $(n,n)$, and $$N = \big\{ \begin{pmatrix} I_n & x \ 0 & I_n \end{pmatrix} : x \in M_n(F) \big\}$$, with $\psi(n(x)) = \psi_0(\mathrm{tr}\,Ax)$ for matrix $A$. For irreducible cuspidal $\pi$ of $GL_{2n}(F)$ over finite fields, dimension formulas for twisted Jacquet modules are established: $$\dim_\mathbb{C} \pi_{N,\psi_{A_k}} = (-1)^{n-1} (q;q)_{n-1} \prod_{i=k}^{n-1}(q^i-1)$$ where $A_k$ has rank $k$. The module vanishes for $k=0$ due to cuspidality, and for $k=n$ one recovers the nondegenerate Whittaker dimension (Balasubramanian et al., 2024).

For principal series representations $\pi = \mathrm{Ind}_P^G(\chi_1 \otimes \chi_2)$ over $GL_{2n}(F)$, with $N$ and $\psi$ the Shalika character, the twisted Jacquet module is always nonzero and explicitly identified as $\chi_1 \otimes \chi_2$ on the diagonal Levi $\Delta GL_n(F)$ (Harshitha et al., 31 Dec 2025).

In the context of division algebras ($G = GL_2(D)$, $D$ over $F$, nonarchimedean), which includes quaternionic cases, twisted Jacquet modules of Speh representations are shown in certain cases to be one-dimensional and precisely given by the central character of the inducing representation composed with the reduced norm. For depth-zero principal series, dimension and structure results involve computed formulas as in: $$\dim\,Sp(\tau)_{N,\psi} = \frac{d(d-1)}{2}, \quad \dim\,St(\tau)_{N,\psi} = \frac{d(d-1)}{2} + d$$ where $d = \dim(\tau)$ (Nadimpalli et al., 2023).

3. Algorithmic and Combinatorial Approaches

In the Zelevinsky or Langlands classifications for $GL_n(F)$, twisted Jacquet modules arise via precise algorithms acting on multisegments. The functor $J_{P,\sigma}(\pi)$ for essentially square-integrable $\sigma$ corresponds to a sequence of right derivatives $D_{\Delta_l}$, producing a multisegment that encodes the module: $$J_{P,\sigma}(\pi) = D_{\Delta_k} \circ \cdots \circ D_{\Delta_1} (\pi)$$ Explicit steps involve chain removal and truncation operations. Worked examples on multisegments illustrate how to compute the image of the functor for given input data, bringing twisted Jacquet module calculus into algorithmic territory (Chan et al., 2 Mar 2025).

4. Structural Theorems and Model Classification

The structure of twisted Jacquet modules is often governed by orbit decompositions and the Bernstein–Zelevinsky geometric lemma. Filtrations indexed by orbit or block parameters provide explicit descriptions for the module, with each quotient an induction from stabilizers of $\psi$. For symplectic groups ($Sp_4(F)$), the twisted Jacquet module along Siegel or Klingen parabolics is decomposed into explicit submodules, exhibiting short exact sequences or filtrations and connecting to contragredient or inflations of Jacquet modules from the Levi (Pandey et al., 2024).

Twisted Jacquet modules also directly classify the existence of Shalika models: for a smooth irreducible $\pi$ of $GL_{2n}(F)$, nonvanishing of $\pi_{N,\psi}$ is necessary and sufficient for the existence of a Shalika functional, with explicit examples constructed via the Langlands quotient of principal series (Harshitha et al., 31 Dec 2025).

5. Harmonic Analysis, Constant Term Maps, and Asymptotics

In the harmonic analytic approach, the twisted Jacquet module is realized through the behavior of constant term maps on spaces of generalized Whittaker functions. The Delorme constant term and Bushnell–Henniart isomorphism are proven to coincide on coinvariants, and the surjectivity yields structural decomposition and enables explicit asymptotic expansions for functions on $G$ in terms of the modules $J_P(\cdot)$ (Matringe, 2020). The eigencharacter expansion in these modules is essential for analyzing tempered, discrete, and cuspidal representation classes.

6. Conjectures and Open Problems

A central conjecture attributed to D. Prasad posits a connection between the vanishing of twisted Jacquet modules and poles of the adjoint $L$-function of Langlands parameters: $$\pi_{N,\psi} = 0 \iff L(s, \pi, \mathrm{Ad}) \text{ has prescribed poles at } s=1,2,\ldots,n$$ Evidence supports this for principal series and certain Langlands quotients, but a complete characterization remains open (Harshitha et al., 31 Dec 2025). Systematic theory for general supercuspidal and non-principal series representations is not fully established, especially for division algebra groups (Nadimpalli et al., 2023).

Recent developments suggest a growing landscape of "degenerate" and "unique" models for representations, with twisted Jacquet modules serving as the organizing principle for distinction, branching, and functoriality problems in automorphic representation theory.

7. Applications and Significance

Twisted Jacquet modules underpin the existence and uniqueness of Whittaker, Shalika, and other model spaces, with significance in the proof of multiplicity-one theorems, the structure of automorphic periods, and the classification of distinguished representations. Their dimension formulas and explicit realizations support comparative harmonic analysis between $p$-adic and finite field contexts, facilitate algorithmic computations in representation theory, and inform conjectures at the interface of harmonic analysis and number theory (Nadimpalli et al., 2023, Balasubramanian et al., 2024, Balasubramanian et al., 2022, Harshitha et al., 31 Dec 2025, Matringe, 2020, Pandey et al., 2024, Chan et al., 2 Mar 2025).