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Wen-Wei Li

Publications and source records attributed to Wen-Wei Li.

30 records · Page 2Linked to original sources
arXiv2017

Towards generalized prehomogeneous zeta integrals

Let $X$ be a prehomogeneous vector space under a connected reductive group $G$ over $\mathbb{R}$. Assume that the open $G$-orbit $X^+$ admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz functions on $X$, and (ii) generalized matrix coefficients on $X^+(\mathbb{R})$ of Casselman-Wallach representations of $G(\mathbb{R})$, upon a twist by complex powers of relative invariants. This merges representation theory with prehomogeneous zeta integrals of Igusa et al. We show their convergence in some shifted cone, and prove their meromorphic continuation via the machinery of $b$-function together with V. Ginzburg's results on admissible $D$-modules. This provides some evidences for a broader theory of zeta integrals associated to affine spherical embeddings.

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arXiv2017

The Shimura-Waldspurger correspondence for Mp(2n)

We describe some recent developments and formulate some conjectures in the genuine representation theory and the study of automorphic forms of the metaplectic group $\mathrm{Mp}(2n)$, from the point of view of the theta correspondence as well as from the point of view of the theory of endoscopy and the trace formula.

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arXiv2016

Spectral transfer for metaplectic groups. I. Local character relations

Let $\widetilde{\mathrm{Sp}}(2n)$ be the metaplectic covering of $\mathrm{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\mathrm{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

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arXiv2014

La formule des traces stable pour le groupe métaplectique: les termes elliptiques

Let F be a number field and $\widetilde{\mathrm{Sp}}(2n)$ be the metaplectic covering of Weil of $\mathrm{Sp}(2n, \mathbb{A}_F)$. We stabilize the elliptic semi-simple terms of the genuine part of trace formula for $\widetilde{\mathrm{Sp}}(2n)$, by using the transfer of test functions and the fundamental lemma established in a previous article. We obtain therefore an expression in terms of stable orbital integrals along certain $F$-elliptic classes in $\mathrm{SO}(2n'+1) \times \mathrm{SO}(2n''+1)$, which we call equisingular, for pairs $(n', n'')$ satisfying $n'+n''=n$. A simple formula for the coefficients is also obtained.

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arXiv2014

La formule des traces pour les revêtements de groupes réductifs connexes. IV. Distributions invariantes

We establish the invariant trace formula (à la Arthur) for the adélic covers of connected reductive groups over a number field, under the hypothesis that the trace Paley-Wiener theorem is verified for all Levi subgroups at the real archimedean places. For instance, this hypothesis can be verified for the metaplectic covers of GL(n), or the twofold metaplectic cover of Sp(2n). We also give simple trace formulae when the test function satisfies certain cuspidality properties.

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arXiv2014

Basic functions and unramified local L-factors for split groups

According to a program of Braverman, Kazhdan and Ngô Bao Châu, for a large class of split unramified reductive groups $G$ and representations $ρ$ of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,π,ρ)$ can be expressed as the trace of $π(f_{ρ,s})$ for a suitable function $f_{ρ,s}$ with non-compact support whenever $\mathrm{Re}(s) \gg 0$. Such functions can be plugged into the trace formula to study certain sums of automorphic $L$-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functions; this is supposed to lead to a generalized Tamagawa-Godement-Jacquet theory for $(G,ρ)$. In this article, we derive some basic properties for the basic functions $f_{ρ,s}$ and interpret them via invariant theory. In particular, their coefficients are interpreted as certain generalized Kostka-Foulkes polynomials defined by Panyushev. These coefficients can be encoded into a rational generating function.

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arXiv2013

Dual R-groups of the inner forms of SL(N)

We study the Knapp-Stein R-groups of the inner forms of SL(N) over a non-archimedean local field of characteristic zero, by using restriction from the inner forms of GL(N). As conjectured by Arthur, these R-groups are then shown to be naturally isomorphic to their dual avatars defined in terms of L-parameters. The 2-cocycles attached to R-groups can be described as well. The proofs are based on the results of K. Hiraga and H. Saito. We also construct examples to illustrate some new phenomena which do not occur in the case of SL(N) or classical groups.

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arXiv2013

On a pairing of Goldberg-Shahidi for even orthogonal groups

Let σ\otimesπ be a supercuspidal representation of SO(2n) \times GL(2n) over a p-adic field with π selfdual, where SO(2n) stands for a quasisplit even special orthogonal group. In order to study its normalized parabolic induction to SO(6n), Goldberg and Shahidi defined a pairing R between the matrix coefficients of σ and π which controls the residue of the standard intertwining operator. The elliptic part of R is conjectured to be related to twisted endoscopic transfer. Based on Arthur's endoscopic classification and Spallone's improvement of Goldberg-Shahidi program, we will verify some of their predictions for general n, under the assumption that π does not come from SO(2n+1).

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arXiv2010

Le lemme fondamental pondéré pour le groupe métaplectique

We state a variant of Arthur's weighted fundamental lemma for the metaplectic group of Weil, which will be an essential ingredient of the stable trace formula. Over a local field of large enough residual characteristic, we give a proof using the method of descent, which is conditional upon the weighted nonstandard fundamental lemma on Lie algebras. In view of the works of Chaudouard and Laumon, this condition is expected to hold.

math.RT↗
arXiv2010

Transfert d'intégrales orbitales pour le groupe métaplectique

This paper develops a formalism of endoscopy for the metaplectic group. We define the notions of stable conjugacy, elliptic endoscopic groups, correspondence of semisimple geometric conjugacy classes and the transfer factors in this setting, then we establish the transfer of antigenuine orbital integrals. Under the hypothesis that the residual characteristic of F is sufficiently large, the fundamental lemma for the units of the spherical Hecke algebra also holds. This generalizes the prior works of J. Adams and D. Renard for real metaplectic groups. Our approach is based on Harish-Chandra's descent method and the non-standard endoscopy on Lie algebras.

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