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Dongwen Liu

Publications and source records attributed to Dongwen Liu.

At least 19 recordsLinked to original sources
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $Π_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $Π_{n+1}\times Π_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

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arXiv2026

Local Jacquet-Shalika integrals and modifying factors

Over any local field of characteristic zero, we develop a local theory of Jacquet-Shalika integrals that were originally introduced by Jacquet and Shalika to study exterior square local $L$-factors for $GL_m$. In particular, at least in the archimedean case, we establish a complete functional equation matching Artin local $\varepsilon$-factors. Central to the argument is a novel comparison strategy between Jacquet-Shalika integrals and open-orbit integrals associated to the Shalika subgroup.

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arXiv2026

Archimedean period relations for Rankin-Selberg convolutions

We formulate and prove the archimedean period relations for Rankin-Selberg convolutions of $\text{GL}(n)\times \text{GL}(n)$ and $\text{GL}(n)\times \text{GL}(n-1)$, for all generic cohomological representations. As a consequence, we prove the non-vanishing of the archimedean modular symbols. This extends the earlier results in [LLS24] for essentially tempered representations of $\text{GL}(n)\times\text{GL}(n-1)$.

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arXiv2026

Fourier positivity for spherical functions I: split tori and spherical principal series

We prove Fourier positivity for spherical functions on a semisimple linear algebraic group $G$ over a local field restricted to its split tori $A$ for unitary principal series parameters of $G$. For ${\rm SL}_n(F)$, where $F$ is a local field, we obtain an explicit recursive formula for the Fourier transform on the diagonal split torus in terms of local Rankin--Selberg factors for ${\rm GL}_n\times {\rm GL}_{n-1}$, together with uniform exponential lower bounds in the spectral parameters. The main input is a Plancherel expansion for the restriction of a ${\rm GL}_n(F)$-spherical function to ${\rm GL}_{n-1}(F)$. Its coefficients are spherical periods computed by Rankin--Selberg theory. Positivity of the Fourier transform for general semisimple groups with unitary principal series parameters is obtained by reduction to full-rank subgroups of type A. The results are motivated by variance non-vanishing problems for mixing abelian actions on homogeneous spaces.

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arXiv2026

Arithmetic Wavefront Set and Microlocal Structure of Harish-Chandra Character

In this paper, we establish in Theorem 1.2 the reciprocity of wavefront sets for irreducible admissible representations $π$ of classical groups $G$ over any local field $F$ of characteristic zero if $π$ has a generic local $L$-parameter. Over archimedean local fields, based on the progress made in our previous work (arXiv:2207.04700), we prove in Theorem 1.5 that for an irreducible Casselman--Wallach representation $π$ with a generic local $L$-parameter, the Wavefront Set Conjecture (arXiv:2207.04700, Conjecture 1.2) and its refinement (Conjecture 1.1) hold for the arithmetic wavefront set ${\mathrm{WF}}_{\mathrm{ari}}(π)$ as defined by the associated enhanced local $L$-parameter of $π$ and the wavefront set ${\mathrm{WF}}_{\mathrm{tr}}(π)$ defined by the Harish--Chandra distribution character $Θ_π$ of $π$. Hence the microlocal structure of $Θ_π$ is completely determined by the arithmetic information carried by the enhanced local $L$-parameter of $π$. The relations with the algebraic wavefront set ${\mathrm{WF}}_{\mathrm{wm}}(π)$ defined by the degenerate Whittaker models are extensively discussed by means of the composition law (Theorem 3.4) over all local fields of characteristic zero. Under Conjecture 1.3, the Wavefront Set Conjecture is fully established over archimedean local fields. As a consequence, we prove a refinement of Vogan's maximal-orbit principle (Theorem 1.7).

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arXiv2025

Arithmetic Wavefront Sets and Generic $L$-packets

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. Let $π$ be an irreducible admissible representation $π$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean. If $π$ has a generic local $L$-parameter, we define the arithmetic wavefront set ${\rm WF_{ari}}(π)$ of $π$, which is a subset of $F$-rational nilpotent orbits of the Lie algebra $\mathfrak{g}(F)$ of $G(F)$, by means of the arithmetic structures of the enhanced $L$-parameter $(φ,χ)$ of $π$. Those arithmetic structures are discovered by using our method of consecutive descents of enhanced $L$-parameters, based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. We study the basic structure of ${\rm WF_{ari}(π)}$ and prove that it is an invariant of $π$ (Theorem 5.10). Furthermore, those basic structures ${\rm WF_{ari}(π)^{max}}$ are expected to yield the precise $F$-rational structure of ${\rm WF_{ari}(π)^{max}}$, which has been realized, when $F$ is archimedean, in Theorems 1.3 and 1.4 (Theorems 7.10, 7.14, and 9.2). Based on the local Langlands reciprocity, the Wavefront Set Conjecture (Conjecture 1.2 and Conjecture 5.14) asserts that the wavefront sets on the L-parameter side should be closed related to those on the representation side, namely, \[ {\rm WF_{wm}(π)^{max}}={\rm WF_{ari}(π)^{max}}={\rm WF_{tr}(π)^{max}} \] when $π$ has a generic local $L$-parameter, where the algebraic wavefront set ${\rm WF_{wm}}(π)$ is defined by Moeglin and Waldspurger in [MW87], using generalized Whittaker models and the analytic wavefront set ${\rm WF_{tr}}(π)$ is defined by Howe [H81, Hd85] using distribution characters, and also by [H74, HC78, BV80]. Conjecture 1.2 is verified for families of interesting cases.

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arXiv2025

Relative completed cohomologies and modular symbols

Generalizing Emerton's completed cohomologies, we define relative completed cohomologies of arithmetic manifolds. We also define modular symbols for them, and show that the relative completed cohomology spaces interpolate the ``nearly ordinary part" of the classical automorphic cohomologies, and the modular symbols defined for them interpolate the classical modular symbols. As applications, we use these modular symbols to construct three families of nearly ordinary $p$-adic L-functions: (i) Rankin-Selberg $p$-adic L-functions for $\mathrm{GL}_n\times \mathrm{GL}_{n-1}$, (ii) Rankin-Selberg $p$-adic L-functions for $\mathrm{U}_n\times \mathrm{U}_{n-1}$, and (iii) Standard $p$-adic L-functions of symplectic type for $\mathrm{GL}_{2n}$. We define and calculate explicitly the modifying factors at $\infty$ and at $p$, and determine the exceptional zeros of the $p$-adic L-functions for these examples. The modifying factors at $\infty$ are consistent with the conjectures given by Deligne and Blasius, and the modifying factors at $p$ are consistent with the conjecture given by Coates and Perrin-Riou.

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arXiv2023

Irreducible representations of $\textrm{GL}_n(\mathbb{C})$ of minimal Gelfand-Kirillov dimension

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of $G=\textrm{GL}_n(\mathbb{C})$ possessing the minimal Gelfand-Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of $G$ of type $(n-1,1)$. We give the transition matrix between the two bases for the corresponding coherent families.

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arXiv2023

Arithmetic Branching Law and generic $L$-packets

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. For any irreducible admissible representation $π$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean, we extend the study of spectral decomposition of local descents in [JZ18] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field $F$. In particular, if $π$ has a generic local $L$-parameter, we introduce the spectral first occurrence index $\mathfrak{f}_{\mathfrak{s}}(π)$ and the arithmetic first occurrence index $\mathfrak{f}_{\mathfrak{a}}(π)$ of $π$ and prove in Theorem 1.4 that $\mathfrak{f}_{\mathfrak{s}}(π) = \mathfrak{f}_{\mathfrak{a}}(π)$. Based on the theory of consecutive descents of enhanced $L$-parameters developed in [JLZ22], we are able to show in Theorem 1.5 that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result ([JZ18, Theorem 1.7]) to the great generality.

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arXiv2023

Fourier-Jacobi models of Deligne-Lusztig characters and depth zero local descent for unitary groups

In this paper, we deduce explicit multiplicity formulas of the Fourier-Jacobi model for Deligne-Lusztig characters of finite symplectic groups, unitary groups, and general linear groups. We then apply these results to deduce the explicit depth zero local descent (à la Soudry and Tanay) for $p$-adic unitary groups. The result is a concrete example in the context of non-tempered Gan-Gross-Prasad program.

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arXiv2023

On the convergence of Kac-Moody Eisenstein series

Let $G$ be a representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We first consider Kac-Moody analogs of Borel Eisenstein series (induced from quasicharacters on the Borel), and prove they converge almost everywhere inside the Tits cone for arbitrary spectral parameters in the Godement range. We then use this result to show the full absolute convergence everywhere inside the Tits cone (again for spectral parameters in the Godement range) for a class of Kac-Moody groups satisfying a certain combinatorial property, in particular for rank-2 hyperbolic groups.

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arXiv2023

Strong integrality of inversion subgroups of Kac-Moody groups

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac--Moody algebra $\frak{g}$ over ${\mathbb Q}$. Let $V=V^λ$ be an integrable highest weight $\frak{g}$-module and let $V_{\mathbb Z}=V^λ_{\mathbb Z}$ be a ${\mathbb Z}Z$-form of $V$. Let $G$ be an associated minimal representation-theoretic Kac--Moody group and let $G({\mathbb Z})$ be its integral subgroup. Let $Γ({\mathbb Z})$ be the Chevalley subgroup of $G$, that is, the subgroup that stabilizes the lattice $V_{\mathbb Z}$ in $V$. For a subgroup $M$ of $G$, we say that $M$ is integral if $M\cap G({\mathbb Z})=M\cap Γ({\mathbb Z})$ and that $M$ is strongly integral if there exists $v\in V^λ_{\mathbb Z}$ such that, for all $g\in M$, $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G({\mathbb{Z}})$. We prove strong integrality of inversion subgroups $U_{(w)}$ of $G$ where, for $w\in W$, $U_{(w)}$ is the the group generated by positive real root groups that are flipped to negative roots by $w^{-1}$. We use this to prove strong integrality of subgroups of the unipotent subgroup $U$ of $G$ generated by commuting real root groups. When $A$ has rank 2, this gives strong integrality of subgroups $U_1$ and $U_2$ where $U=U_{1}{\Large{*}}\ U_{2}$ and each $U_{i}$ is generated by `half' the positive real roots.

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arXiv2023

Infinite dimensional Chevalley groups and Kac-Moody groups over $\mathbb{Z}$

Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$ analogous to a finite-dimensional Chevalley group over $R$. We use a $\mathbb{Z}$-form of the universal enveloping algebra of $\mathfrak{g}$ and a $\mathbb{Z}$-form of an integrable highest-weight module $V$. We construct groups $G_V(\mathbb{Z})$ analogous to arithmetic subgroups in the finite-dimensional case. We also consider a universal representation-theoretic Kac-Moody group $G$ and its completion $\widetilde{G}$. For the completion we prove a Bruhat decomposition $\widetilde{G}({\mathbb{Q}})=\widetilde{G}({\mathbb{Z}})\widetilde{B}({\mathbb{Q}})$ over $\mathbb{Q}$, and that the arithmetic subgroup $\widetildeΓ(\mathbb{Z})$ coincides with the subgroup of integral points $\widetilde{G}(\mathbb{Z})$

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