On Metaplectic Forms of Classical Groups

论经典群的元合形式

• 批准号: 09640050
• 负责人: SUZUKI Toshiaki
• 金额: $ 1.22万
• 依托单位: University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640050/
• 关键词: Metaplectic group; Automorphic forms; Eisenstein series; Distinguished representations; メタプレテック群; スーパーカスピダル表現; ヒルベルト記号; メタプレクテック形式; 志村対応

(1) We established a relation between Dirichlet series obtained as Fourier coefficients of metaplectic Eisenstein series and Rankin-Selberg convolutions of metaplectic forms.(2) Unramified distinguished representations were determined and their Whittaker functins were explicitely given in some case. It was shown that the Rankin-Selberg convolutions of metaplectic forms against ditinguished metaplectic forms have Euler product. We propose a conjecture that auto-morphic distinguished representations on a n-fold metaplectic group of GL(nr) are parametrized by automorphic cuspidal representations of GL(r).(3) We give a explicite formula for the dimension of Whittaker functionals on supercuspidal representations of metaplectic groups over a local field. Hence distinguished supercuspidal representations of metaplectic groups were determined.(4) We proved the existence of a maximal compact subring of a local field with respect to 1-lilbert symbol. Hence it follows the existence of an open compact subgroup of a metaplectic group over a local field, which has very good properties.

(1)本文建立了作为亚纯Eisenstein级数的Fourier系数的Dirichlet级数与亚纯形式的Rankin-Selberg卷积之间的关系。(2)确定了非分歧的可分辨表示，并在某些情况下明确给出了它们的Whittaker函数.证明了元代数形式对双元代数形式的Rankin-Selberg卷积具有Euler积。我们提出了一个猜想：GL（nr）的n重亚群上的自同构可区别表示是由GL（r）的自同构尖点表示参数化的. (3)给出了局部域上亚群超尖点表示上Whittaker泛函维数的一个显式公式。从而确定了亚群的超尖点表示。(4)证明了局部域上关于1-lilbert符号的极大紧子环的存在性。因此，它遵循的存在性，一个开放的紧子群的亚群在一个局部域上，这有很好的性质。

项目成果

Masashi KOSUDA: "The homfly invariant of closed tangles" Ryukyu Math.J.10. 1-22 (1997)

Masashi KOSUDA：“闭合缠结的 homfly 不变量”琉球 Math.J.10。

Toshiaki SUZUKI: "Distinguished representations of metaplectic groups" American J.of Math. 120. 723-755 (1998)

Toshiaki SUZUKI：“元波群的杰出表示”美国数学杂志。

鈴木利明: "Metaplectic Eisenstein series and the Bump-Hoffrtein cenjecture" Duke Math.J. 90(3). 577-630 (1997)

Toshiaki Suzuki：“Metaplectic Eisenstein 级数和 Bump-Hoffrtein 猜想”Duke Math.J. 90(3) (1997)。

鈴木利明: "Metaplectic Eisenstein series and Bump-Hoffstein conjecture" Dulre Math.J. 90(14). 1-54 (1997)

Toshiaki Suzuki：“Metaplectic Eisenstein 级数和 Bump-Hoffstein 猜想”Dulre Math.J. 90(14) (1997)。