Jesua Epequin
We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs over finite fields with odd characteristic. We define a bijection obtained from this correspondence, and enjoying the property of ``having minimal unipotent support''. Finally, we examine the interaction between the Howe correspondence and weak cuspidality.
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
Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a subset of irreducible characters of $G'$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^*)^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^*$. In this paper, we prove the commutativity (up to…
We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs ($\mathbf{U}_m(\mathbb{F}_q),\mathbf{U}_n(\mathbb{F}_q$)) and ($\mathbf{Sp}_{2m}(\mathbb{F}_q),\mathbf{O}^\pm_{2n}(\mathbb{F}_q)$), where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd). We use the Lusztig correspondence to reduce the study of the Howe correspondence between Harish-Chandra series to a correspondence between unipotent Harish-Chandra series. Finally, we provide two applications of these results.
In this article we prove a Howe correspondence for a family of representations of sl(2n), which was introduced by Benkart, Britten, and Lemire.
In this paper we give a complete description of the Howe correspondence of unipotent characters for a finite dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization under a mild restriction of the characteristic of the base field. That is, the conjecture by Aubert-Michel-Rouquier is confirmed.
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.
We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^ε_{2n}(\mathbb{F}_q))$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd) and $ε=\pm 1$. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of $π$ under the correpondence of a unipotent representation $π$ of $G$.
The main purpose of this article is to supplement the authors' results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these representations play a key role in understanding the basic structure of Howe correspondence (and its complications in the archimedean case), and their global counterparts play an equally essential part in the…
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…
A sub-relation of the $Θ$-correspondence called the \emph{$η$-correspondence} is defined by Gurevich-Howe for a finite reductive dual pair in stable range. In this paper we propose an extension of the correspondence to general finite reductive dual pairs. Then we determine the domain the correspondence and prove a formula on the difference of degrees in $q$ of two irreducible characters paired by the correspondence.
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 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.
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.
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.
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.
Howe duality over finite fields II: Explicit stable computation
本文基于Howe双生理论，通过G. Lusztig的参数化方法，明确描述了eta和zeta对应关系，并证明了稳定范围内的不可约表示出现的非零多重性。
Correspondences between affine Hecke algebras and applications
本文综述广义仿射 Hecke 代数的构造，并利用其与 p-adic 约化群两类 L-参数 Bernstein 级数的关联研究 Howe 对应。
This paper provides a construction of the unipotent representations for classical complex groups in terms of the Theta correspondence as introduced and studied by R. Howe. The K-type structure of unipotent representations is obtained as a consequence of the character formulas for unipotent representations of D. Vogan and the author. This provides a tight link between unipotent representations and the orbit philosophy. A parametrization of unipotent representations for the Spin groups is obtained.
Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences
本文通过代数簇的中阶 étale 上同调几何构造了有限酉群的 Heisenberg-Weil 表示，并利用 Frobenius 作用实现了任意有限域上 $(\mathit{Sp}_{2n},O_2^-)$ 的 Howe 对应，证明了幂单性在该对应下保持不变。
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 article, we give a new method for proving Howe correspondence in the case of dual pairs of type $({\rm GL}_n, {\rm GL}_m)$ over a non-Archimedean locally compact field $F$. The proof consists in combining a study on Kudla's filtration with the results of a previous article of the autor about the irreducibility of a parabolically induced representation. The proof is valid for $F$ of any characteristic and allows us to make the correspondence explicit in terms of Langlands parameters.
Given a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank. If the pair is in the stable range and if the representation is unitary, then the resulting distribution is the character of the representation obtained via Howe's correspondence. This construction was transferred to the p-adic case and a conjecture was formulated. In this note we verify a weaker version of this conjecture…
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 $G'$.
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.
Hecke algebras for tame genuine principal series and local Shimura correspondence
该研究扩展局部志村对应，利用 Bushnell–Kutzko 类型理论与线性代数群方法刻画真正主级数分量的 Hecke 代数并构造类型，同时以反例说明其与主内窥镜群对应代数未必同构。
If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go sl}_{2}$. Therefore the associative algebra they generate is a quotient of the universal enveloping algebra ${\cal U}({\go sl}_{2})$. This fact is in some sense the foundation of the metaplectic representation. The present paper is devoted to the study of the case where $Q(x)$ is replaced by $Δ_{0}(x)$, where $Δ_{0}(x)$ is the relative invariant of a prehomogeneous…
Dirac cohomology and $Θ$-correspondence for complex dual pairs
本文研究复约化对偶对下 Dirac 上同调在 Howe 的 $\Theta$-对应中的行为，刻画了具有非零 Dirac 上同调的 Harish-Chandra 模，其 $\Theta$-提升仍保持非零，并显式计算了提升后的 Dirac 上同调。
We give a proof of the Howe duality conjecture in local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.
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。