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Dongming She

Publications and source records attributed to Dongming She.

5 recordsLinked to original sources
arXiv2026

Stability of the exterior cube $\gamma$-factors for $\mathrm{GL}(6)$

We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.

math.RT↗
arXiv2025

Co-periods and central symmetric cube L-values

In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one result that there exist cuspidal automorphic forms with prescribed local components and non-vanishing central symmetric cube $L$-values.

math.NT↗
arXiv2023

Stability of Rankin-Selberg local $\gamma$-factors for split classical groups: the symplectic case

Given a split classical group of symplectic type and a split general linear group over a local field $F$, we use Langlands-Shahidi method to construct their Rankin-Selberg local $\gamma$-factors and prove the corresponding analytic stability for generic representations. The idea generalizes the work of J. Cogdell, F. Shahidi, T.-L. Tsai in 2017 and D. She in 2023 in the study of asymptotic behaviors of partial Bessel functions. Different from the known cases, suppose $P=MN$ is the maximal parabolic subgroup with Levi component $M\simeq \mathrm{GL}_r\times\mathrm{Sp}_{2m}$ that defines the local factors, the action of the maximal unipotent subgroup of $M$ on $N$ have non-trivial stabilizers, and the space of integration for the corresponding local coefficient is no longer isomorphic to a torus. We will separate its toric part out in our cases and show that it plays the same role as the torus over which the integral representing the local coefficient is taken in the known cases. This is a new phenomenon with sufficient generality and we believe that it may provide us with a possible direction towards a uniform proof of stability of Langlands-Shahidi $\gamma$-factors in our future work.

math.RT↗
arXiv2019

Local Langlands correspondence for the twisted exterior and symmetric square $\epsilon$-factors of $\textrm{GL}_n$

Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $\epsilon$-factors. This is also expected to be true under parallel compositions with a complex analytic representations of $\textrm{GL}_n(\mathbb{C})$. J.W. Cogdell, F. Shahidi, and T.-L. Tsai proved the equality of the symmetric and exterior square L- and $\epsilon$-factors [7] in 2017. But the twisted symmetric and exterior square L- and $\epsilon$-factor are new and very different from the untwisted case. In this paper we will define the twisted symmetric square L- and $\gamma$-factors using $\textrm{GSpin}_{2n+1}$, and establish the equality of the corresponding L- and $\epsilon$-factors. We will first reduce the problem to the analytic stability of their $\gamma$-factors for supercuspidal representations, then prove the supercuspidal stability by establishing general asymptotic expansions of partial Bessel function following the ideas in [7].

math.NT↗