arXiv subjects
Publications and source records attributed to Hung Yean Loke.
Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.
We show that Howe's big quotient is obtained via the tensoring over appropriate algebra.
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 dual pairs in the stable range over a p-adic field.
For 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. The purpose of this article is to transfer this construction to the p-adic case. We provide the precise definition of the integral kernel operator and formulate a conjecture.
Exceptional groups of type $E_6$ contain dual pairs where one member is $\mathrm{Spin}(8)$, and the other is $T\rtimes \mathbb Z/2\mathbb Z$, where $T$ is a two-dimensional torus and the non-trivial element in $\mathbb Z/2\mathbb Z$ acts on $T$ by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.
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.
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".
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.
We show that every exceptional Lie algebra over a number field can be obtained by Tits' construction from an octonion algebra O and a cubic Jordan algebra J. In particular, the exceptional Lie algebra contains a dual pair which is the direct sum of the derivation algebras of O and J. We determine rational forms of this dual pair.
Let $(G,G')$ be a type I irreducible reductive dual pair in $\mathrm{Sp}(W_{\mathbb{R}})$. We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ and $\mathfrak{g}' = \mathfrak{k}' \oplus \mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\widetilde{K}}$ and ${\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\widetilde{\mathrm{Sp}}(W_\mathbb{R})$ of ${\mathrm{Sp}}(W_\mathbb{R})$. Let $ρ$ be a genuine irreducible $(\mathfrak{g},{\widetilde{K}})$-module. Our first main result is that if $ρ$ is unitarizable, then except for one special case, the full local theta lift $ρ' = Θ(ρ)$ is equal to the local theta lift $θ(ρ)$. Thus excluding the special case, the full theta lift $ρ'$ is an irreducible and unitarizable $(\mathfrak{g}',{\widetilde{K}}')$-module. Our second main result is that the associated variety and the associated cycle of $ρ'$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation $ρ^*$ respectively. Finally we obtain some interesting $(\mathfrak{g},{\widetilde{K}})$-modules whose ${\widetilde{K}}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_\mathbb{C}$-orbits in $\mathfrak{p}^*$.
In this paper we first construct natural filtrations on the full theta lifts for any real reductive dual pairs. We will use these filtrations to calculate the associated cycles and therefore the associated varieties of Harish-Chandra modules of the indefinite orthogonal groups which are theta lifts of unitary lowest weight modules of the metaplectic double covers of the real symplectic groups. We will show that some of these representations are special unipotent and satisfy a K-type formula in a conjecture of Vogan.
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.
In this paper we develop representation theory of the two fold central extension of SL(2) over the field of 2-adic numbers. As a consequence, we obtain a local Shimura correspondence between the two fold central extension of SL(2) and the linear group PGL(2), which we show to be compatible with the global Shimura correspondence at p=2.