Wee Teck Gan, Gordan Savin
We study three exceptional theta correspondences for p-adic groups, where one member of the dual pair is the exceptional group G2. We prove the Howe duality conjecture for these dual pairs and a dichotomy theorem, and determine explicitly the theta lifts of all non-cuspidal representations.
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Dichotomy for generic supercuspidal representations of $G_2$
局部朗兰兹图意味着，对于$p$ -adic域上的$G_2$的每一个一般超尖点不可约表示，人们可以将$PGSp_6$或$PGL_3$的一般超尖点不可约表示联系起来.我们证明了这种结构上的二分法，证明了$G_2$的某些表示和$PGSp_6$和$PGL_3$的其他表示之间的精确对应。这种对应关系源于$E_6$和$E_7$中的theta对应关系、Shalika泛函的分析和自旋L函数。我们的主要结果减少了广义超尖形不可约表示的$G_2$的几何Langlands参数化为一个关于$PGSp_6$的参数化的猜想。
We give a proof of the Howe duality conjecture for the (almost) equal rank dual pairs in full generality. For arbitrary dual pairs, we prove the irreducibility of the (small) theta lifts for all tempered representations. Our proof works for any nonarchimedean local field of characteristic not 2 and in arbitrary residual characteristic.
Parameters and Theta lifts
本文通过朗兰兹-沃根参数明确齐性群和正交群的参数对应关系，并重新表述了霍华德对应关系的相关研究。
For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patters…
Similitude exceptional theta correspondences
该研究构建发展了例外θ对应的相似型版本，证明其Howe对偶定理由等距情形推导而来，并将跷跷板恒等式等基本工具推广到相似型框架，拓展了例外θ对应理论的研究范畴。
The big theta
该研究在数学表示论领域，证明了Howe大商能够通过在合适的代数上做张量积来得到，丰富了相关代数结构研究。
Theta Correspondence and Quantum Induction
在他的文章“超越经典不变理论”（J.A.M.S.，1989年，罗杰·豪（Roger Howe）建立了一个对偶对约化群的表示之间的对应关系。这种对应关系被称为豪氏对应或局部θ对应。在我的论文“I型典型群的酉表示和θ对应”（将出现在J. Func.肛门），证明了实域上的θ对应在一定的限制条件下保持酉性，推广了李建树（Invent.数学，1997年，1989年）。本文研究了θ对应的合成。在一定的条件下，我们证明了θ对应的合成产生酉表示。我们称这个过程为量子感应。量子归纳法在许多方面类似于抛物线归纳法。本文的部分结果已在ICM-2002上公布。
We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors.
An "automatic continuity" question has naturally occurred since Roger Howe established the local theta correspondence over $\mathbb R$: does the algebraic version of local theta correspondence over $\mathbb R$ agrees with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor.
In this paper, we propose two maximal one-to-one sub-relations $\underlineθ, \overlineθ$ of the Howe correspondence $Θ$ for a finite reductive dual pair consisting of a symplectic group and an orthogonal group. Moreover, we show that both the correspondences $\underlineθ$ and $\overlineθ$ are extensions beyond the stable range of the $η$-correspondence defined by Gurevich-Howe.
In this paper, we completely describe the Howe correspondence for the dual pairs from the title over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.
This note shows a property of degree-parity preservation for $K$-types under Howe's theta correspondence. As its application, we deduce the preservation of parity of all $K$-types occurring in an arbitrary irreducible $(\mathfrak{g},K)$-module of any Lie group in reductive dual pairs.
It is known that irreducible cuspidal characters satisfy the preservation principle in the Howe correspondences of finite reductive dual pairs. In this article, we generalize the preservation principle to any irreducible characters of classical groups.
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。
Boundary behavior of special cohomology classes arising from the Weil representation
在我们的前一篇文章[math.NT/0408050]中，我们建立了附在$(p,q)$型实正交群上的局部对称空间$X$的向量值全纯Siegel模形式与局部系数的上同调之间的对应关系.这种对应关系是使用与显式构造的“特殊”Schwartz形式相关联的θ函数来实现的。此外，θ函数导致在具有系数的$X$中生成一系列特定的“特殊循环”。在本文中，我们研究了这些θ函数在非紧情况下的边界行为，并证明了θ函数扩展到了$X$的Borel-Sere紧化$\bar{X}$。然而，对于签名$(p,p)$的$\Q$ -split情形，我们必须构造并考虑一个稍大的紧化，即“大”Borel-Serre紧化。对$\bar{X}$的每个面的限制再次是如在[math.NT/0408050]中的θ级数，现在对于较小的正交群和较大的系数系统。作为应用，我们建立了当传递到$X$的一个适当的有限覆盖时特殊（余）圈的上同调非零性。特别地，所讨论的（co）同调群不为零。
It is known that the $Θ$-correspondence for a finite reductive dual pair is not one-to-one in general. In this paper, we propose two maximal one-to-one sub-relations $\underlineθ,\overlineθ$ of the $Θ$-correspondences for a finite dual pair of two unitary groups. These two correspondences are extensions of the analogue of the $η$-correspondence proposed by Gurevich-Howe.
We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.
During the last two decades, great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs. However, uniform formulas remain elusive for real dual pairs of type I. The purpose of this paper is twofold: to formulate an explicit version of induction principle for dual pairs $(O(p,q),Sp(2n,\mathbb{R}))$ with $p + q$ even, and to apply it to obtain a complete and explicit description of the local theta correspondence when $p + q = 4$. Our approach is very elementary by analysis on the infinitesimal characters and K-types under the correspondence.
We give an elementary introduction to Classical Invariant Theory and its modern extension "Transcending Classical Invariant Theory", commonly known as the theory of local theta correspondence. We explain the two fundamental assertions of the theory: the Howe duality conjecture and the Kudla-Rallis conservation relation conjecture. We give a status report on the problem of explicitly describing local theta correspondence in terms of Langlands-Vogan parameters. We conclude with a discussion on a certain problem of automatic continuity, which manifests unity of the theory in algebraic and smooth …
Lectures on local theta correspondence
本文介绍了局部theta对应关系的基本理论、守恒关系、不变量性质及其在单位ary表示论中的应用。
We prove Howe duality for the theta correspondence arising from the $p$-adic dual pair $G_2 \times (\text{PU}_3 \rtimes \mathbb{Z}/2\mathbb{Z})$ inside the adjoint quasi-split group of type $E_6$.
In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible representation $Π$ of $\tilde{G}$ appearing in Howe correspondence, we compute explicitly the restriction of the character $Θ_{Π'}$ of the associated representation $Π'$ of $\tilde{G}'$ on the set of regular points on the compact Cartan subgroup $\tilde{H}'$ of $\tilde{G}'$.
For an irreducible dual pair $(G, G') \in Sp(W)$ with one member compact and two representations $Π\leftrightarrow Π'$ appearing in the Howe duality, we give an expression of the character $Θ_{Π'}$ of $Π'$ via the character of $Π$. We make computations for the dual pair $(G = U(n, \mathbb{C}), G' = U(p, q, \mathbb{C}))$, which are explicit in low dimensions. For $(G = U(1, \mathbb{C}), G' = U(1, 1, \mathbb{C}))$, we verify directly a result of H. Hecht saying that the character has the same value on both Cartan subgroups of $G'$.
Howe duality for the dual pair $SL_2(\mathbb R) \times F_{4,1}$: a ping-pong of $K$-types
本文通过见字如面恒等式证明了例外theta对应中的Howe对偶，研究了相应表示的K型关系。
We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic.
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