Exceptional Theta-correspondences I

Explicit Construction of Local Langlangds Correspondence of GL(2,F) Using Theta Correspondence

The Langlands correspondence of GL(2,F) over a non-Archimedean local field F of characteristic 0 has been well studied. The construction uses the theta correspondence. In this paper, we are going to describe explicitly how this construction is done. We will then provide new proofs to several properties of the correspondence. Including that the correspondence does not depend on the additive character introduced by the Weil representation.

局部theta对应与广义Shalika模型

Local theta correspondence and generalized Shalika models

本文研究等秩II型对局部theta对应下广义Shalika模型的行为，推广了Gan关于Shalika模型与线性模型关系的结论。

Theta correspondence for symplectic-orthogonal and metaplectic-orthogonal p-adic dual pairs

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.

A note on the local theta correspondence for unitary similitude dual pairs

Following Roberts' work in the case of orthogonal-symplectic similitude dual pairs, we study the local theta correspondence for unitary similitude dual pairs over a $p$-adic field.

Howe duality and dichotomy for exceptional theta correspondences

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.

Local Theta Correspondences between Supercuspidal Representations

By the works of Yu, Kim and Hakim-Murnaghan, we have a parameterization and construction of all supercuspidal representations of a reductive $p$-adic group in terms of supercuspidal data, when $p$ is sufficiently large. In this paper, we will define a correspondence of supercuspidal data via moment maps and theta correspondences over finite fields. Then we will show that local theta correspondences between supercuspidal representations are completely described by this notion. In Appendix B, we give a short proof of a result of Pan on "depth preservation".

The explicit theta correspondence for reductive dual pairs $(Sp(p,q),O^{*}(4))$

For every irreducible Harish-Chandra module of $O^{*}(4)$, we determine its theta lift to $Sp(p,q)$ in terms of the Langlands parameter, for all non-negative integers $p$ and $q$. Our strategy is to determine the desired theta lifts by their infinitesimal characters and lowest $K$-types.

Parity preservation of $K$-types under theta correspondence

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.

On Theta and Eta Correspondences for Finite Unitary Dual Pairs

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.

$\ell$-模局部theta对应

The $\ell$-modular local theta correspondence

本文在模表示理论语境下，证明了非阿基米德局部域上 $p$-进域的辛-正交与酉-酉对偶对在系数域特征足够大时存在双射对应，即模局部theta对应。

On Theta and Eta Correspondences for Finite Symplectic/Orthogonal Dual Pairs

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.

Local theta correspondence and nilpotent invariants

We consider two types of nilpotent invariants associated to smooth representations, namely generalized Whittaker models, and associated characters (in the case of a real reductive group). We survey some recent results on the behavior of these nilpotent invariants under local theta correspondence, and highlight the special role of a certain double fiberation of moment maps.

Theta对应与Springer对应

Theta correspondence and Springer correspondence

本文通过Springer对应推导出偶正交群与辛群之间，以及一般线性群在有限域上的unipotent主系列表示的theta对应公式，并研究了相关Hecke范畴的模块结构。

Remarks on the theta correspondence over finite fields

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.

Generalized Preservation Principle in Finite Theta Correspondence

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.

阿基米德 theta 对应的重数一性质

Multiplicity One Property for Archimedean Theta Correspondence

本文证明阿基米德约化对偶对中，小 theta 提升在完整 theta 提升中的 Jordan-Hölder 重数为 1，并给出基于最低次数 K-型的 theta 对应新描述。

Local theta correspondences between epipelagic supercuspidal representations

In this paper we study the local theta correspondences between epipelagic supercupsidal representations of a type I classical dual pair $(G,G')$ over $p$-adic fields. We show that, besides an exceptional case, an epipelagic supercupsidal representation $π$ of $\widetilde{G}$ lifts to an epipelagic supercupsidal representation $π'$ of $\widetilde{G}'$ if and only if the epipelagic data of $π$ and $π'$ are related by the moment maps.

Duality for spherical representations in exceptional theta correspondences

We study the exceptional theta correspondence for real groups obtained by restricting the minimal representation of the split exceptional group of the type E_n, to a split dual pair where one member is the exceptional group of the type G_2. We prove that the correspondence gives a bijection between spherical representations if n=6,7, and a slightly weaker statement if n=8.

局部theta对应关系讲义

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。

Theta对应与Arthur包：关于Adams猜想

Theta correspondence and Arthur packets: on the Adams conjecture

本文研究symplectic--even orthogonal双对的Adams猜想，通过精确描述成立条件，验证了局部theta对应与Arthur包的关系。

Theta correspondence and the Prasad conjecture for SL(2)

We use relations between the base change representations and theta lifts, to give a new proof to the local period problems of SL(2) over a nonarchimedean quadratic field extension E/F. Then we will verify the Prasad conjecture for SL(2). With a similar strategy, we obtain a certain result for the Prasad conjecture for Sp(4).

Tensor products of singular representations and an extension of the theta-correspondence

In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space G'/H'. Our procedure establishes a correspondence between certain unitary representations of G and those of G'. This extends the usual theta--correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E_7 and F_4.

Finite Theta Correspondence of Almost Characters

The theory of almost characters which is closely related to character sheaves is proposed by Lusztig to study the representation theory of finite reductive groups. In this article we show that the decomposition of the Weil character for finite reductive dual pairs $(\Sp_{2n},\rmO^\pm_{2n'})$ or $(\Sp_{2n},\SO_{2n'+1})$ with respect to the almost characters is exactly the same as the decomposition with respect to the irreducible characters. As a consequence, the finite theta correspondence on almost characters is established.

On the local theta representation

We study the algebraic framework in which one can define, in the manner of the theta correspondence, a correspondence between representations of two locally profinite groups $H_1$, $H_2$. In particular, we examine when and how such a correspondence can be extended to bigger groups $G_1$, $G_2$ containing $H_1$, $H_2$ respectively as normal subgroups. As an application, we discuss the theta correspondence for a reductive dual pair of the similitude groups in the non-archimedean case.

Base change and theta correspondences for supercuspidal representations of SL(2)

Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of SL(2,F). The argument involves the theory of types and the lattice model of the Weil representation.

Local theta correspondence: the basic theory

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 …

有限域上的θ对应关系

The theta correspondence over finite fields

本文探讨有限域上θ对应关系的主要性质，包括其与Harish-Chandra和Lusztig系列以及表示的Jordan分解的兼容性，以及其完整显式描述。

关于Weil表示的有理性及局部theta对应关系

On the rationality of the Weil Representation and the local theta correspondence

本文证明Weil表示可系数于数域，并通过显式降维论证确定其降维至的数域，同时展示theta对应关系在完美域上成立当且仅当在代数闭包上成立，从而证明经典局部theta对应关系的有理性。

相似型例外θ对应

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。

Characters of some unitary highest weight representations via the theta correspondence

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 $\tilde{G}'$.

Relative character identities and theta correspondence

In this paper, we will determine the factorization of global period attached to the spherical variety $X:=\mathrm{SO}(n-1)\backslash \mathrm{SO}(n)$, which is a special case of the Sakellaridis-Venkatesh conjecture. The main idea is to build a connection between the periods of $X$ and the periods of the Whittaker case $N,ψ\backslash \mathrm{SL}_2$ (n even) or $N,ψ\backslash \mathrm{Mp}_2$ (n odd) using the tool of theta correspondence. In the local setting, we determine the Plancherel formula of $L^2(X)$, the relative character identities and give an explicite integral formula of transfer betw…

Automorphic Forms and the Theta Correspondence

This is a set of notes on automorphic forms and theta correspondence, based on my lectures at the 2022 Arizona Winter School.

Theta lifts and distinction for regular supercuspidal representations

This article has a twofold purpose. First, by recent works of Kaletha and Loke-Ma, we give an explicit description of the local theta correspondence between regular supercuspidal representations in the equal rank symplectic-orthogonal case. Second, based on this description, we show that the local theta correspondence preserves distinction with respect to unramified Galois involutions.

Theta correspondences for GSp(4)

We explicitly determine the theta correspondences for $\GSp_4$ and orthogonal similitude groups associated to various quadratic spaces of rank $4$ and $6$. The results are needed in our proof of the local Langlands correspondence for $\GSp_4$.

Local Theta correspondence of Tempered Representations and Langlands parameters

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 Correspondence and Arthur packets

In spirit of Gan-Ichino's work on the Arthur's multiplicity formula for metaplectic groups, we have established the Arthur's multiplicity formula for even orthogonal or unitary groups with Witt index less than or equal to one. In that multiplicity formula, some local packets defined using the stable range theta lifts are involved. In this paper, we prove that at non-Archimedean places, the definition of the local packets involved in that multiplicity formula is independent of the choice of the dual-pairs used in their construction. Moreover, at those places where the groups are quasi-split, we…

Explicit induction principle and symplectic-orthogonal theta lifts

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.

The Gross-Prasad conjecture and local theta correspondence

We establish the Fourier-Jacobi case of the local Gross-Prasad conjecture for unitary groups, by using local theta correspondence to relate the Fourier-Jacobi case with the Bessel case established by Beuzart-Plessis. To achieve this, we prove two conjectures of D. Prasad on the precise description of the local theta correspondence for (almost) equal rank unitary dual pairs in terms of the local Langlands correspondence.

引上性与theta对应关系

Functoriality and the theta correspondence

本文研究经典p-adic群局部theta对应关系的引上性，通过适应Adams猜想至ABV包，为猜想提供证据，尤其在一般线性群情况。

局部theta对应与Galois周期

Local theta correspondence and Galois periods

本文研究了局部theta对应下偶正交群和辛群Galois周期的性质，比较了乘数并建立了共轭与相对特征关系。

源自例外群的无重数Weil表示

Multiplicity free Weil representations arising from exceptional groups

该研究借助例外theta对应，证明p进群的部分Weil表示具有无重数性质，同时确定了这些表示的不可约商，推进了相关表示论研究。

作为 C* - 对应关系的西塔对应

Theta correspondence as a C*-correspondence

综述将西塔对应描述为 C* - 对应关系的近期工作，阐明了在西塔对应中涉及的一个群为紧致群的特殊情况下的主要机制。

Transfer of characters in the theta correspondence with one compact member

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 $G'$.

Springer correspondence for exceptional Lie algebras and their duals in small characteristic

We describe the Springer correspondence explicitly for exceptional Lie algebras of type $G_2$ and $F_4$ and their duals in bad characteristics, i.e. in characteristics 2 and 3.

Theta correspondence for almost unramified representations of unitary groups

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$.

Local Langlands Correspondence for Even Orthogonal Groups via Theta Lifts

Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.

Explicit Constructions of Automorphic Forms: Theta Correspondence and Automorphic Descent

This is an expanded set of notes based on two lectures given by the author at the 2022 IHES summer school on the Langlands program, on explicit techniques for the construction of automorphic representations.

刚性内扭曲的局部theta对应与朗兰兹参数

Local theta correspondences and Langlands parameters for rigid inner twists

本文提出用刚性内扭曲的局部朗兰兹对应描述局部theta对应（含四元数对偶对对应）的猜想，并验证其在特定情况下成立。

II型中的\(\ell -\)模局部西塔对应与部分置换

The $\ell$-modular local theta correspondence in type II and partial permutations

研究在非阿基米德域\(F\)上II型的\(\overline{\mathbb{F}}_\ell\)-模西塔对应中出现的重数，通过对称群对部分置换的作用及相关表示，将问题归结为对称群模表示理论中的分支问题。

The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case

We prove the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case under some assumptions. This is the last case of the local Gan-Gross-Prasad conjectures. We also prove two of Prasad's conjectures on the local theta correspondence in the almost equal rank case.

Theta Correspondence for U(1,1) and U(2)

In this paper, we parametrize certain irreducible supercuspidal representations of U(1,1) and U(2) via explicit induction data. The parametrization depends on traceless elements of negative valuation in a quadratic extension of base field. We use the lattice model of the Weil representation to determine which traceless elements are involved in the theta correspondence for reductive dual pair U(1,1) and U(2).

On the Howe duality conjecture in classical theta correspondence

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.

Local Langlands Correspondence for Unitary Groups via Theta Lifts

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which was established in an early paper of Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this provides an alternative approach to the works of Kaletha-Minguez-Shin-White and Moeglin-Renard.

Generalized Shalika model on $\mathrm{SO}_{4n}(F),$ symplectic linear model on $\mathrm{Sp}_{4n}(F)$ and theta correspondence

We show that if an irreducible admissible representation of $\mathrm{SO}_{4n}(F)$ has a generalized Shalika model, then its small theta lift to $\mathrm{Sp}_{4n}(F)$ has the symplectic linear model, thus answering a question posed by D. Jiang. Here $F$ is a non-archimedean field of characteristic zero.