arXiv subjects
Publications and source records attributed to Baiying Liu.
We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.
We prove a conjecture of the first-named author ([J14]) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of all classical groups over any number field. This conjecture generalizes the global version of the local tempered $L$-packet conjecture of F. Shahidi ([Sh90] and [Sh10]).
Optical fiber with high numerical aperture (NA) can efficiently relieve the degeneracy of higher order linearly polarized modes. The degeneracy relief is investigated in two types of high NA fibers, i.e., low index cladding fiber and high-index-core fiber. A naked core fiber, as with low index cladding, can be used theoretically to generate the orbital angular momentum mode (OAMM) HE21 and the cylindrically symmetric modes (CSMs) TM01 and TE01. A high index core fiber incorporated with high contrast index structure can be used similarly to obtain OAMM HE31. The generation of both CSMs and OAMMs required tilted optical gratings to couple the fundamental core mode HE11 into these modes. The tilt angle and modulation period of the grating fringes can be calculated simply and visually with the method proposed in this article.
Let $G_n$ denote either the group $SO(2n+1, F)$, $Sp(2n, F)$, or $GSpin(2n+1, F)$ over a non-archimedean local field of characteristic different than two. We determine all composition factors of degenerate principal series of $G_n$, using methods based on the Aubert involution and known results on irreducible subquotiens of the generalized principal series of particular type.
We complete the proof of Proposition 5.3 of [GJR04].
Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let \begin{align*} Y \subset \prod_{i=1}V_i \end{align*} be the closed subscheme consisting of $(v_1,v_2,v_3)$ on which $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. Motivated by conjectures of Braverman and Kazhdan and related work of Lafforgue, Ngô, and Sakellaridis we prove an analogue of the Poisson summation formula for certain functions on this space.
In this paper, we study top Fourier coefficients of certain automorphic representations of $\mathrm{GL}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of $\mathrm{GL}_n(\mathbb{A})$ of form $$ Δ(τ_1, b_1) \boxplus Δ(τ_2, b_2) \boxplus \cdots \boxplus Δ(τ_r, b_r)\,, $$ where $Δ(τ_i,b_i)$'s are Speh representations in the discrete spectrum of $\mathrm{GL}_{a_ib_i}(\mathbb{A})$ with $τ_i$'s being unitary cuspidal representations of $\mathrm{GL}_{a_i}(\mathbb{A})$, and $n = \sum_{i=1}^r a_ib_i$. Endoscopic lifting images of the discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $π$ of $G$, determine the occurrence of an irreducible representation $σ$ of $H$ in the restriction of $π$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $σ$ of $H$, find an irreducible representation $π$ of $G$ such that $σ$ occurs in the restriction of $π$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
In this paper, we prove certain multiplicity one theorems and define twisted gamma factors for irreducible generic cuspidal representations of split $G_2$ over finite fields $k$ of odd characteristic. Then we prove the first converse theorem for exceptional groups, namely, $GL_1$ and $GL_2$-twisted gamma factors will uniquely determine an irreducible generic cuspidal representation of $G_2(k)$.
In this paper, we prove the uniqueness of certain Fourier-Jacobi models for the split exceptional group $G_2$ over finite fields with odd characteristic. Similar results are also proved for $Sp_4$ and $U_4$.
We prove an analogue of Jacquet's conjecture on the local converse theorem for \ell-adic families of co-Whittaker representations of GL_n(F), where F is a finite extension of Q_p and \ell does not equal p. We also prove an analogue of Jacquet's conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \ell-adic family. These two theorems are closely related to the local Langlands correspondence in \ell-adic families.
Braverman and Kazhdan introduced influential conjectures generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands $L$-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As evidence for their conjectures, Braverman and Kazhdan considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. In this paper we consider a special case of the setting of the later paper, and prove a refined Poisson summation formula that eliminates the restrictive assumptions of loc. cit. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan.
We show that Fourier coefficients of automorphic forms attached to minimal or next-to-minimal automorphic representations of ${\mathrm{SL}}_n(\mathbb{A})$ are completely determined by certain highly degenerate Whittaker coefficients. We give an explicit formula for the Fourier expansion, analogously to the Piatetski-Shapiro-Shalika formula. In addition, we derive expressions for Fourier coefficients associated to all maximal parabolic subgroups. These results have potential applications for scattering amplitudes in string theory.
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.
In this paper, we reprove a global converse theorem of Cogdell and Piatetski-Shapiro using purely global methods.
The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].
We study model transition for representations occurring in the local theta correspondence between split even special orthogonal groups and symplectic groups, over a non-archimedean local field of characteristic zero.