Chen-Bo Zhu
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We describe two algorithms for finding representatives of the nilpotent orbits of a theta-group. The algorithms have been implemented in the computer algebra system GAP (inside the package SLA). We comment on their performance. We apply the algorithms to study the nilpotent orbits of theta-groups, where theta is an N-regular automorphism of a simple Lie algebra of exceptional type.
Quasi-admissible, raisable nilpotent orbits, and theta representations
本文研究覆盖群幂零轨道的拟容许性与可提升性，确定了特定覆盖次数下分裂幂零轨道的性质，显式计算了theta表示的波前集，并求得了其Harish-Chandra特征展开的首项系数。
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…
We consider a reductive dual pair (G, G') in the stable range with G' the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K'_C-orbits, where K' is a maximal compact subgroup of G' and we describe the precise K_C-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an application, we prove sphericality and normality of the closure of certain nilpotent K_C-orbits obtained in this way. We also give integral formulas for their degrees.
In this paper, we prove that theta correspondence preserves unitarity under certain restrictions.
In this paper, we will use the local theta correspondences for the dual pair (Sp(W),O(V)) to investigate some branching law problems.
In his article "Transcending Classical Invariant Theory" (J.A.M.S., 1989, Vol 2), Roger Howe established a correspondence between representations of a dual pair of reductive groups. This correspondence is known as Howe's correspondence or local theta correspondence. In my paper "Unitary Representations and Theta Correspondence for Classical Group of Type I"(to appear in J. Func. Anal.), we proved that theta correspondence over the real field preserves unitarity under certain restrictions, generalizing some results of Jian-Shu Li (Invent. Math., Vol 97, 1989). In this paper, we study the compos…
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.
We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all type I irreducible dual pairs and all local fields of characteristic zero.
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.
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。
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.
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。
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.
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.
We give the number of nilpotent orbits in the Lie algebras of orthogonal groups under the adjoint action of the groups over $\tF_{2^n}$. Let $G$ be an adjoint algebraic group of type $B,C$ or $D$ defined over an algebraically closed field of characteristic 2. We construct the Springer correspondence for the nilpotent variety in the Lie algebra of $G$.
In this note, we consider perverse sheaves on the nilpotent cone. We prove orthogonality relations for the equivariant category of sheaves on the nilpotent cone in a method similar to Lusztig's for character sheaves. We also consider cleanness for cuspidal perverse sheaves and the generalized Lusztig--Shoji algorithm.
Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.
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".
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.
Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaves and the derived category of differential graded modules over a dg-ring related to the Weyl group.
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 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.
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。
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.
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。
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.
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.
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。
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.
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。
Let $G$ be a simply connected algebraic group of type $B,C$ or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the dual vector space of the Lie algebra of $G$. In particular, we classify the nilpotent orbits in the duals of symplectic and orthogonal Lie algebras over algebraically closed or finite fields of characteristic 2.
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。
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).
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.
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.
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.
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.
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 …
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。
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。
Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of $E_7$ to the dual pair $G_2 \times{}$Sp(6) gives the non-endoscopic Langlands lift of irreducible representations of $G_2$ to Sp(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 $\tilde{G}'$.
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…
This is a set of notes on automorphic forms and theta correspondence, based on my lectures at the 2022 Arizona Winter School.
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.
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…
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 $\GSp_4$.
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.
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.
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.
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.
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 $G'$.
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。
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。
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。
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。
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.
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 $-1$.
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.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。