Dongwen Liu, Yongchang Zhu
In this paper we study the theta lifting for loop groups and extend the classical tower property established by S. Rallis to the loop setting. As an application we obtain cusp forms on loop groups, and we give the first example where the cusp forms constructed using this method are nonvanishing.
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Casselman-Wallach property for homological theta lifting
本文研究了同调θ提升的Casselman-Wallach性质，利用corank-one的parabolic稳定滤波器，证明了欧拉-皮卡尔特征在格罗滕迪克群中的well-defined性。
Full theta lifting for type II reductive dual pairs
本文研究非阿基米德局部域上第二类约化对偶对的完全theta提升，确定了GL(2)×GL(2)的不可约表示提升结构，并证明GL(n)×GL(n+1)的不可约温表示提升仍不可约且温。
We study the generalized theta lifting between the double covers of split special orthogonal groups, which uses the non-minimal theta representations constructed by Bump, Friedberg and Ginzburg. We focus on the theta liftings of non-generic representations and make a conjecture that gives an upper bound of the first non-zero occurrence of the liftings, depending only on the unipotent orbit. We prove both global and local results that support the conjecture.
Picard groups of Siegel modular threefolds and theta lifting
证明了Humbert曲面有理生成Siegel模三重的Picard群.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
In this paper, we give an explicit determination of the non-vanishing of the theta liftings for unitary dual pairs (U(p,q), U(r,s)). Assuming the local Gan-Gross-Prasad conjecture, we determine when theta lifts of tempered representations are nonzero in terms of the local Langlands correspondence.
We study the theta lifting for real unitary groups and completely determine the theta lifts of tempered representations. In particular, we show that the theta lifts of (limits of) discrete series representations can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
In his paper 'Theta lifting for representations with non zero cohomlogy', Jian-Shu Li proved that a certain kind of cohomological representations of $U(a,b)$ is automorphic. In this paper, this result is generalized to a more general class of cohomological representations of this group. It comes from the fact that these cohomological representations are the image by the theta correspondance of some representations obtained by cohomological induction. In proving this theorem, we also prove some cases of Adams conjecture.
On Irreducibility and Multiplicity One Property for Non-Archimedean Type-II Theta Lifting
本文建立了非阿基米德II型theta提升的重数一性质，结合Godement-Jacquet zeta积分计算Arthur型表示的大theta提升，并提出其不可约性猜想。
Theta liftings for $(\mathrm{GL}_n,\mathrm{GL}_n)$ type dual pairs of loop groups
该研究针对圈群$(\n\thrm{GL}_n,\n\thrm{GL}_n)$型对偶对的相关问题，证明了由经典尖点形式诱导的圈群$\n\thrm{GL}_n$上尖点形式的theta提升是艾森斯坦级数，为圈群表示论相关研究提供了新的结论。
Theta lifting of nilpotent orbits for symmetric pairs
本文考虑一个在稳定域中的厄米对称型约化对偶（G，G '），其中G'是较小的成员.研究了G'的极大紧子群K'的幂零K '_C-轨道的θ提升，并给出了对称对（G，K）的提升幂零轨道的闭包的正则函数环的精确K_C-模结构.作为应用，我们证明了用这种方法得到的某些幂零K_C-轨道的闭包的球性和正规性。我们还给出了它们的度的积分公式。
We study a new lifting of automorphic representations using the theta representation $Θ$ on the $4$-fold cover of the symplectic group, $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. This lifting produces the first examples of CAP representations on higher-degree metaplectic covering groups. Central to our analysis is the identification of the maximal nilpotent orbit associated to $Θ$. We conjecture a natural extension of Arthur's parameterization of the discrete spectrum to $\overline{\mathrm{Sp}}_{2r}(\mathbb{A})$. Assuming this, we compute the effect of our lift on Arthur parameters and show th…
Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ of characteristic 0, and denote by $V$ and $\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $Max Hom(V,\tilde{V})$ of full rank elements in $Hom(V,\tilde{V})$, and the nilpotent orbit correspondence $\mathcal{O} \subset \mathfrak{g}$ and $Θ(\mathcal{O})\subset \tilde{\mathfrak{g}}$ induced by elements of $Max Hom(V,\tilde{V})$ via the moment maps. Let $(π,\mathscr{V})$ be a smooth irreducible representation of $G$. We show that there is a correspondence of the generalized Whit…
Godement--Jacquet L-function and homological theta lifting
本文结合同调与解析方法，研究非阿基米德局部域上 II 型对偶对的 theta 提升，完全确定了临界点处 L-函数全纯的不可约表示的大 theta 提升，计算了所有特征标的大 theta 提升，并给出了 Weil 表示射影性的充要条件。
Global theta lifting and automorphic periods associated to nilpotent orbits
本文利用矩映射导致的幂零轨道转移，在全局且完全一般的情形下建立对偶对成员间自守周期的 theta 对应关系，给出 Gomez–Zhu 局部结果的全局类比。
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. So, they realize the geometric Langlands functoriality for the corresp…
In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally we will use these results to study subquotients of some cohomologically induced modules.
Compatibility of the Theta correspondence with the Whittaker functors
本研究证明了三类经典李群对的整体几何theta提升函子与Whittaker规范化相容，即theta提升函子与目标群Whittaker函子的复合同构于源群的Whittaker函子，推进了几何表示论中Theta对应理论的发展。
In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.
Theta lifts to certain cohomological representations of indefinite orthogonal groups
本文利用 Borcherds 方法将全纯 Mp₂(ℝ) 尖点形式 theta 提升到不定正交群的上同调表示 Π 中，并通过 Fourier 展开计算与尖点支集分析确认所得自守形式的平方可积性。
In this paper we observe that isomorphism classes of certain metrized vector bundles over P^1-{0,infinity} can be parameterized by arithmetic quotients of loop groups. We construct an asymptotic version of theta functions, which are defined on these quotients. Then we prove the convergence and extend the theta functions to loop symplectic groups. We interpret them as sections of line bundles over an infinite dimensional torus, discuss the relations with loop Heisenberg groups, and give an asymptotic multiplication formula.
Geometric theta-lifting for the dual pair SO_{2m}, Sp_{2n}
本研究在特征大于2的代数闭域光滑射影曲线上，针对对偶对SO_{2m}, Sp_{2n}定义几何theta提升函子，刻画其与Hecke算子的关系，在两种特殊情形下验证几何Langlands函子性，并借助局部结果（Rallis定理几何类比）完成证明。
Geometric Waldspurger periods II
本文将几何Waldspurger周期计算推广到分歧覆叠情形，验证了相关Whittaker系数猜想，得到新的GL_2自守层，并提出更强的相关猜想。
Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta. Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducib…
Some local-global non-vanishing results for theta lifts from orthogonal groups
本文首先改进了B. Roberts，其特征在于从O（X）到Sp（n）的全局theta提升的非零性，即局部theta提升的非零性。特别地，我们将去掉加在他的定理上的所有阿基米德条件。其次，在Roberts之后，我们将把我们的定理应用于低秩相似群的θ提升。也就是说，我们在我们的改进设置中描述了从GO（4）到GSp（2）的全局theta提升的非零条件。我们还考虑了从GO（4）到GSp（1）的全局theta提升的非零条件，并显式计算了存在的提升。
In this paper, we associate a family of infinite-rank pro-Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac--Moody algebra. In the case that the element has a polynomial representative, we can prove our lattices are theta-finite in the sense of Bost, allowing us to attach to each of our lattices a well-defined theta-like function.
Theta Correspondence of Automorphic Characters
本文描述了将$\GO(3)(\A)$的自守特征标提升为$\SLT(\A)$。它通过将此提升的图像与从$\GO(1)(\A)$到$\SLT(\A)$的自守字符的提升进行匹配来实现。我们的匹配实际上给出了单个自守形式的匹配，而不仅仅是表示空间的匹配。设$\V$是$3-$维二次向量空间，$\U$是某个$1-$维二次空间.对于由角色$χ$的提升中的Schwartz函数$ϕ\in \Sc(\V(\A))$确定的自守形式$I_{\V}(χ,ϕ)$，我们匹配由角色$μ$的提升中的Schwartz函数$ϕ_{0}\in \Sc(\U(\A))$确定的自守形式$I_{\U}(μ,ϕ_{0})$。我们的工作表明，空间$\U$是明确确定的字符$χ$。特征符$μ$由空间$\V$明确确定，函数$ϕ_{0}$由涉及$ϕ$的轨道积分给出。
Theta series for quantum loop algebras and Yangians
该研究引入并研究了系数在量子loop代数张量平方中的Theta级数，证明其权分量的多项式性，给出O范畴中R矩阵的分解与多项式性结果，并将相关级数推广到Yangian中且通过结合子解释证明其多项式性。
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …
Geometric Waldspurger periods
本文在几何朗兰兹纲领框架下研究对偶约化对的θ提升，定义了对应导出范畴间的函子，证明了GO_2与GL_2对偶对的几何朗兰兹函子性，并应用于周期计算，给出GSp_4上自守层的三种猜想构造。
Lifting Deligne-Lusztig Reduction and Geometric Coxeter Type Elements
本文提出统一框架，通过提升与Deligne-Lusztig约化，给出判据证明仿射Deligne-Lusztig簇为经典簇与仿射空间的乘积，并引入几何Coxeter型元素类。
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