Bessel Models and the Transfer of Siegel Cusp Forms of Degree 2

本文建议详细研究非阿基米德局域上群GSP(4)的Bessel模型，并将结果应用于L-函数的解析性质、Siegel模形式到GL(4)的非一般变换以及L-函数的特殊值。惠特克模型在自同构表示理论中的重要性是众所周知的。对于某些群体，如GSP(4)，贝塞尔模型可以作为那些没有惠特克模型的表示的替代品。虽然贝塞尔模型的一般事实是已知的，例如贝塞尔模型的唯一性，但某些重要的问题仍然没有得到回答，并阻碍了这些模型的广泛部署。未知因素包括给定表示和给定贝塞尔模型的测试向量的确定，以及显式贝塞尔函数的计算。这项研究将填补这些知识差距，允许贝塞尔模型的几个新应用。这项提议的活动将通过将研究生纳入拟议研究的参与者来促进教学、培训和学习。这项提案的活动将扩大任职人数不足群体的参与范围。国际和平协会和联合国际计划继续与俄克拉何马大学麦克奈尔学者计划以及俄克拉荷马大学更早的传统学者计划保持密切联系。这些计划的本科生要么是第一代低收入者，要么来自代表性不足的群体，要么来自俄克拉荷马城和塔尔萨地区传统上处于不利地位的高中。这项提议的活动将加强研究和教育的基础设施。作为俄克拉荷马大学自形形式小组的一部分，国际和平协会和共同国际组织已经并将继续参加一些涉及其他机构的研究人员和学生的科学活动。在这些活动中，包括与邻近大学的联合研讨会和具有很大区域吸引力的会议。其他活动将包括来自该国其他地区和海外的研究人员的短期和中期访问。最后，拟议的研究活动将导致在俄克拉荷马大学自形形式小组的框架内创建一个网站，为学生和研究人员提供自形形式领域的资源。除其他事项外，该网站将包含一份自我同构形式的全面活动清单。

A proposal is made for a detailed study of Bessel models for the group GSp(4) over a nonarchimedean local field, and the application of the results to analytic properties of L-functions, the non-generic transfer of Siegel modular forms to GL(4), and special values of L-functions. The importance of Whittaker models in the theory of Automorphic Representations is well known. For certain groups such as GSp(4) Bessel models can serve as a substitute for those representations that have no Whittaker model. While general facts for Bessel models are known, such as the uniqueness of Bessel models, certain important questions remain unanswered and prevent a widespread deployment of these models. Amongst the unknown facts are the determination of test vectors for a given representation and a given Bessel model, and the calculation of explicit Bessel functions. This research will close these knowledge gaps, allowing for several new applications of Bessel models.The activity of this proposal will promote teaching, training, and learning via the inclusion of graduate students as participants in the proposed research. The activity of this proposal will broaden the participation of under-represented groups. The PI and the Co-PI plan to continue their close contacts with the OU McNair Scholars Programs well as the Sooner Traditions Scholars program of the University of Oklahoma. These programs are comprised of undergraduate students which are either first-generation and low-income, or from underrepresented groups, or from traditionally disadvantaged high-schools in the Oklahoma City and Tulsa area. The activity of this proposal will enhance infrastructure for research and education. As part of the Automorphic Forms group at the University of Oklahoma, the PI and the Co-PI have participated, and will continue to participate, in a number of scientific activities involving researchers and students from other institutions. Amongst these activities are joint seminars with neighboring universities and conferences with a largely regional appeal. Other activities will include short- and medium-term visits from researchers from other parts of the country and also from overseas. Finally, the activity of the proposed research will lead, within the framework of the Automorphic Forms group of the University of Oklahoma, to the creation of a website with resources for students and researchers in the area of Automorphic Forms. Amongst other things, the web site will contain a comprehensive list of activities in Automorphic Forms.