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Let F be a local field. In the case of F being the real field, Pierre Cartier constructed Heisenberg-Weil representations of a Heisenberg group in families using non-self-dual lattices. This result was later reformulated by Jae-Hyun Yang in another paper. We extend this family of representations to a representation of a Jacobi subgroup by incorporating a rational Metaplectic group. In the case of F being a non-archimedean local field of odd residual characteristic or a finite unramified extension of the field of 2-adic numbers, we obtain similar results for a Jacobi group by incorporating a Metaplectic group., Comment: 48 pages, comments welcome

A common challenge in Bicep Curls rehabilitation is muscle compensation, where patients adopt alternative movement patterns when the primary muscle group cannot act due to injury or fatigue, significantly decreasing the effectiveness of rehabilitation efforts. The problem is exacerbated by the growing trend toward transitioning from in-clinic to home-based rehabilitation, where constant monitoring and correction by physiotherapists are limited. Developing wearable sensors capable of detecting muscle compensation becomes crucial to address this challenge. This study aims to gain insights into the optimal deployment of wearable sensors through a comprehensive study of muscle compensation in Bicep Curls. We collect upper limb joint kinematics and surface electromyography signals (sEMG) from eight muscles in 12 healthy subjects during standard and fatigue stages. Two muscle synergies are derived from sEMG signals and are analyzed comprehensively along with joint kinematics. Our findings reveal a shift in the relative contribution of forearm muscles to shoulder muscles, accompanied by a significant increase in activation amplitude for both synergies. Additionally, more pronounced movement was observed at the shoulder joint during fatigue. These results suggest focusing on the shoulder muscle activities and joint motions when deploying wearable sensors to effectively detect compensatory movements., Comment: 11 pages, 7 figures, accepted by T-MRB

Let F be the usual real field. Let W be a symplectic vector space over F. It is known that there are two different Weil representations of a Meteplectic covering group $\widetilde{Sp}(W)$. By some twisted actions, we reorganize them into a representation of $\widetilde{Sp}^{\pm}(W)$, a covering group over a subgroup $Sp^{\pm}(W)$ of $GSp(W)$. Based on the works of MVW, Kudla, and Howe on reductive dual pairs in $Sp(W)$, we explore the analogous dual pairs in $Sp^{\pm}(W)$ . Finally, following Lion-Vergne's classical book on Weil representations and theta series, we investigate some simple theta series in $Sp^{\pm}(W)$ where $W$ has dimension two., Comment: 67 pages, comments welcome

Let F be a non-archimedean local field of odd residual characteristic. Let W be a symplectic vector space over F. It is known that there are different Weil representations of a Meteplectic covering group Mp(W). By some twisted actions, we reorganize them as a representation of $PGMp^{\pm}(W)$, which is a covering group related to the projective similitude symplectic group., Comment: 35 pages, a reference added, some errors modified, comments welcome

It is well known(cf. Weil, G\'erardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related projective symplectic similitude group. We also discuss the even field case by following Genestier-Lysenko and Gurevich-Hadani's works on geometric Weil representations in characteristic two. As a result, we approach some of their results from the lattice model, which is inspired by MVW, Prasad and Takeda's works., Comment: 30 pages, some mistakes corrected,comments welcome!

In 1978, Munn proved that a bounded complex representation of an inverse semigroup is semiunitary and completely reducible. We consider the converse question in the finite case. We provide some sufficient conditions from a finite regular semigroup to a finite inverse semigroup., Comment: 16 pages, comments welcome!

(I) We study Clifford-Mackey-Rieffel's theory for finite monoid; (II) We prove some results of Theta Representations of finite inverse monoids., Comment: 60 pages, Comments welcomes

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., Comment: 23 pages. Improve some proofs and correct some mistakes, and add some examples. Comments welcome

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., Comment: A revised version. Comments welcome

If $(G_1, G_2)$ is a dual reductive pair of type I in $Sp(W)$, it is known that the degree $8$ metaplectic cover of $Sp(W)$ splits over $G_1G_2$, with one obvious exception. In this paper we replace $G_1G_2$ by a larger subgroup obtained via similitude groups, and show that the degree $8$ metaplectic cover splits, with the same obvious exception., Comment: 17 pages, a revised version

Let Sp_V(F) be the group of isometries of a symplectic vector space V over a finite field F of odd cardinality. The group Sp_V(F) possesses distinguished representations--- the Weil representations. We know that they are compatible with base change in the sense of Shintani for a finite extension F'/F. The result is also true for the group of similitudes of V.

In this paper, we give a way to define the Howe correspondences for the similitudes groups over p-adic fields by following the work of Brooks Roberts on the groups (GSp, GO)., Comment: 38 pages, no figure; v3: Many mistakes corrected. One reference added, the proof of Corollary 5.2(2) changed, minor correction in section 10

Let F be a finite field of odd cardinality, and let G= GL2(F). The group G \times G \times G acts on F^2 \otimes F^2 \otimes F^2 via symplectic similitudes, and has a natural Weil representation. Answering a question rasised by V. Drinfeld, we decompose that representation into irreducibles. We also decompose the analogous representation of GL2(A), where A is a cubic algebra over F., Comment: 29 pages. Comments welcome