arXiv subjects

Wen-Wei Li

Publications and source records attributed to Wen-Wei Li.

At least 19 recordsLinked to original sources

arXiv2026

Harish-Chandra D-modules for bi-Whittaker reduction

Let $D(G)$ be the algebra of algebraic differential operators on a complex reductive group $G$. Denote by $\mathbb{W}$ the bi-Whittaker quantum Hamiltonian reduction of $D(G)$, also known as the quantum Toda lattice. In this article we define the admissible $\mathbb{W}$-modules and the special case of Harish-Chandra modules, the latter being bi-Whittaker variants of the invariant holonomic systems of Hotta-Kashiwara. We then study their torsion properties up to completion relative to the Kazhdan filtration, which in general is unbounded in both directions, through the geometry of universal centralizer. In the case of regular infinitesimal characters, the corresponding $D(G)$-module is shown to be the minimal extension of an irregular connection on the open Bruhat cell. Certain ring-theoretic properties of the completion of $\mathbb{W}$ are also obtained.

math.RT↗

arXiv2026

Variation of additive characters in the transfer for Mp(2n)

Let $\mathrm{Mp}(2n)$ be the metaplectic group of rank $n$ over a local field $F$ of characteristic zero. In this note, we determine the behavior of endoscopic transfer for $\mathrm{Mp}(2n)$ under variation of additive characters of $F$. The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan-Savin, which describes how the local Langlands correspondence for $\mathrm{Mp}(2n)$ depends on the additive characters.

math.RT↗

arXiv2025

Decomposed Levi subgroups in BD-covers of classical groups

For finite topological central extensions of $p$-adic classical groups, Heiermann and Wu introduced the notion of decomposed Levi subgroups in their study of intertwining algebras. In this note, we show that for symplectic and special orthogonal groups over local fields, except the split $\mathrm{SO}(4)$, all Levi subgroups are decomposed if the central extension arises from the Brylinski-Deligne construction. Discussions for certain unitary groups, $\mathrm{SO}(4)$ and $\mathrm{GSpin}(2n+1)$ are also given.

math.RT↗

arXiv2025

Spectral transfer for metaplectic groups. II. Hecke algebra correspondences

Let $\mathrm{Mp}(2n)$ be the metaplectic group over a local field $F \supset \mathbb{Q}_p$ defined by an additive character of $F$ of conductor $4\mathfrak{o}_F$. Gan-Savin ($p \neq 2$) and Takeda-Wood ($p=2$) obtained an equivalence between the Bernstein block of $\mathrm{Mp}(2n)$ containing the even (resp. odd) Weil representation and the Iwahori-spherical block of the split $\mathrm{SO}(2n+1)$ (resp. its non-split inner form), by giving an isomorphism between Hecke algebras. We revisit this equivalence from an endoscopic perspective. It turns out that the L-parameters of irreducible representations are preserved, whilst the difference between characters of component groups is governed by symplectic local root numbers.

math.RT↗

arXiv2025

Some Relations on Paratopisms and An Intuitive Interpretation on the Parastrophes of a Latin Square

This paper will present some intuitive interpretation of the parastrophe transformations of arbitrary Latin square. With this trick, we can generate the parastrophes of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and parastrophe transformations in composition will also be shown. It will solve the problem that when F1*I1=I2*F2 how can we obtain I2 and F2 from I1 and F1, where I1 and I2 are isotopisms while F1 and F2 are parastrophe transformations and "{*}" is the composition of transformations. These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares. This will improve the efficiency apparently in computation for some related problems.

math.CO↗

arXiv2024

Arthur packets for metaplectic groups

For metaplectic groups over a local field of characteristic zero, we define the Arthur packet attached to any Arthur parameter $ψ$ as a multi-set of unitary genuine irreducible representations, characterized by endoscopic character relations. Over number fields, we obtain a multiplicity formula for the genuine discrete $L^2$-automorphic spectrum in terms of global Arthur parameters and $ε$-factors, by leveraging the trace formula for metaplectic groups. This confirms a conjecture of Gan, and extends earlier results of Gan-Ichino on the Shimura-Waldspurger correspondences, whereas their works play a critical role in our proof. Furthermore, all these are shown to be compatible with existing results in rank one (Waldspurger) and two (Gan-Ichino).

math.RT↗

arXiv2024

La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale

We establish some results in local harmonic analysis which are necessary for Arthur's invariant trace formula for coverings of connected reductive groups. More precisely, for local coverings we will study (1) the Plancherel formula and its preparations, (2) the normalization of intertwining operators subject to Arthur's conditions, (3) the local behavior of characters of admissible representations in the nonarchimedean case, and (4) the genuine part of the invariant local trace formula. As a byproduct of the invariant local trace formula, we deduce the density of tempered characters for coverings.

math.RT↗

arXiv2024

Intertwining operators in the Takeda-Wood isomorphism

Over any non-Archimedean local field of characteristic not equal to $2$, Takeda and Wood constructed types for the two blocks containing the even and odd Weil representations of the metaplectic group $\tilde{G}$, and identified the resulting Hecke algebras $H_ψ^{\pm}$ with the Iwahori-Hecke algebras of odd orthogonal groups $G^{\pm}$ of the same rank. We describe normalized parabolic induction and Jacquet modules in terms of Hecke modules using a suitable variant of Bushnell-Kutzko theory. Furthermore, we match the standard intertwining operators of $\tilde{G}$ and $G^{\pm}$ by proving a variant of Gindikin-Karpelevich formula for $\tilde{G}$. As an application, we describe the behavior of normalized intertwining operators of $\tilde{G}$ in these blocks under Aubert involution, reducing everything to the $G^{\pm}$ side. This is mainly motivated by Arthur's local intertwining relations.

math.RT↗

arXiv2024

Stabilization of the trace formula for metaplectic groups

We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.

math.RT↗

arXiv2022

Higher localization and higher branching laws

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting using the formalism of h-complexes due to Beilinson-Ginzburg, and show that the localizations of Harish-Chandra $(\mathfrak{g}, K)$-modules onto $X = H \backslash G$ have regular holonomic cohomologies when $H, K \subset G$ are both spherical reductive subgroups. The relative Lie algebra homologies and $\mathrm{Ext}$-branching spaces for $(\mathfrak{g}, K)$-modules are interpreted geometrically in terms of equivariant derived localizations. As direct consequences, we show that they are finite-dimensional under the same assumptions, and relate Euler-Poincaré characteristics to local index theorem; this recovers parts of the recent results of M. Kitagawa. Examples and discussions on the relation to Schwartz homologies are also included.

math.RT↗

arXiv2022

On the Number of Conjugate Classes of Derangements

The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in an pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments.

math.CO↗

arXiv2022

Counting abelian varieties over finite fields via Frobenius densities

Let $[X,λ]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $ν_v([X,λ])$ for each place $v$ of $\mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,λ]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.

math.NT↗

arXiv2020

Stable conjugacy and epipelagic L-packets for Brylinski-Deligne covers of Sp(2n)

Let $F$ be a local field of characteristic not $2$. We propose a definition of stable conjugacy for all the covering groups of $\mathrm{Sp}(2n,F)$ constructed by Brylinski and Deligne, whose degree we denote by $m$. To support this notion, we follow Kaletha's approach to construct genuine epipelagic $L$-packets for such covers in the non-archimedean case with $p \nmid 2m$, or some weaker variant when $4 \mid m$; we also prove the stability of packets when $F \supset \mathbb{Q}_p$ with $p$ large. When $m=2$, the stable conjugacy reduces to that defined by J. Adams, and the epipelagic $L$-packets coincide with those obtained by $Θ$-correspondence. This fits within Weissman's formalism of L-groups. For $n=1$ and $m$ even, it is also compatible with the transfer factors proposed by K. Hiraga and T. Ikeda.

math.RT↗

arXiv2020

On the regularity of D-modules generated by relative characters

Following the ideas of Ginzburg, for a subgroup $K$ of a connected reductive $\mathbb{R}$-group $G$ we introduce the notion of $K$-admissible $D$-modules on a homogeneous $G$-variety $Z$. We show that $K$-admissible $D$-modules are regular holonomic when $K$ and $Z$ are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups $H_1$ and $H_2$, provided that the twisting character $χ_i$ factors through the maximal reductive quotient of $H_i$, for $i = 1, 2$; (ii) localization on $Z$ of Harish-Chandra modules; (iii) the generalized matrix coefficients when $K(\mathbb{R})$ is maximal compact. This complements the holonomicity proven by Aizenbud--Gourevitch--Minchenko. The use of regularity is illustrated by a crude estimate on the growth of $K$-admissible distributions which based on tools from subanalytic geometry.

math.RT↗

arXiv2020

Variations on themes of Sato

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation. In the second part, we prove a generalization that accommodates certain non-spherical spaces. As an application, the resulting theory applies to the prehomogeneous vector space underlying Bhargava's cubes, which is also considered by F. Sato and Suzuki-Wakatsuki in their study of toric periods.

math.RT↗

arXiv2020

Zeta integrals, Schwartz spaces and local functional equations

According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the $L^2$-spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of $p$-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of $L$-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the constructions of Braverman and Kazhdan (2002). In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals.

math.RT↗

arXiv2019

Generalized zeta integrals on certain real prehomogeneous vector spaces

Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions on $X$ against generalized matrix coefficients of admissible representations of $G(\mathbb{R})$, twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation as well as a functional equation in terms of abstract $γ$-factors. This subsumes the Archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case), and the previous works of Bopp-Rubenthaler. The proof of functional equations is based on Knop's results on Capelli operators.

math.RT↗

arXiv2019

Contragredient representations over local fields of positive characteristic

It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verify a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parameterization of Genestier-Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of V. Lafforgue's excursion operators.

math.RT↗