Harmonic analysis on solvable Lie groups and discrete subgroups

可解李群和离散子群的调和分析

• 批准号: 05640237
• 负责人: FUJIWARA Hidenori
• 金额: $ 0.77万
• 依托单位: Kinki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1993
• 资助国家: 日本
• 起止时间: 1993 至 1994
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-05640237/
• 关键词: Exponential Solvable Lie Group; Monomial Representation; Intertwining Operator; Plancherel Formula; Symmetric Space; Farey Fraction; Riemann Hypothesis; Zeta Function

Head investigator Fujiwara studied monomial representations of exponential solvable Lie groups and obtained the following results.1. On intertwining operators between equivalent monomial irreducible representations (joint work with D.Arnal and J.Ludwig of Metz university in France).(1) Explicit description of the operator for monomial representations induced from polarizations of Vergne.(2) Construction of an operator verifying the composition formula with Maslov index.(3) Local expression at the unit element of the operator in general case.2. As an application of the Penney's Plancherel formula, we gave another proof of the commutativity of the algebra of invariant differential operators associated with monomial representations of finite multiplicities. It is more direct than the original proof due to Corwin and Greenleaf.3. Under some additional condition on the support of distributions, we proved the Frobenius reciprocity in the case of nilpotent symmetric spaces with character, namely that the dimension of the space of bi-semiinvariant distributions is equal to the multiplicity in the irreducible decomposition.Investigator Kanemitsu studied the Riemann hypothesis and got the following.1. We can get a condition equivalent to the Riemann hypothesis by the evaluation of Farey series on a short interval.2. Expression in a closed form of series involving Hurwitz-Lerch zeta function.3. Calculation of series involving Hurwitz zeta function.

首席研究员Fujiwara研究了指数可解李群的单项式表示，得到了以下结果：等价单项式不可约表示之间的缠结算子（与法国梅斯大学的D.Arnal和J.Ludwig合著）。(1)对由Vergne偏振引起的单项式表示的算子进行了显式描述。(2)构造一个算子来验证含有马斯洛夫指数的成分公式。(3)一般情况下，运算符单位元素处的局部表达式。作为Penney's Plancherel公式的一个应用，我们给出了与有限多重的单项式表示相关的不变微分算子代数的交换性的另一个证明。由于Corwin和greenleaf的研究，它比原来的证明更直接。在分布支持的一些附加条件下，证明了具有特征的幂零对称空间的Frobenius互易性，即双半不变分布空间的维数等于不可约分解中的重数。研究者Kanemitsu研究了黎曼假设，得到了以下结论：通过对短区间上的Farey级数求值，可以得到一个等价于Riemann假设的条件。包含Hurwitz-Lerch ζ函数的级数的封闭形式的表达式。计算包含Hurwitz zeta函数的级数。

项目成果

Didier Arnal, Hidenori Fujiwara et Jean Ludwig: "Operateurs d'entrelacement pour les groupes de Lie exponentiel" C.R.Acad.Sci.Paris, Serie I. 319. 549-551 (1994)

Didier Arnal、Hidenori Fujiwara 和 Jean Ludwig：“Operateurs dentrelacement pour les groupes de Lie exponentiel”C.R.Acad.Sci.Paris，Serie I. 319. 549-551 (1994)

Didier ARNAL, Hidenori FUJIWARA, Jean LUDWIG: "Operateurs dentrelacement pour les groupes de Lie exponentiels" C.R.Acad.Sci.Paris,Serie I. 319. 549-551 (1994)

Didier ARNAL、Hidenori FUJIWARA、Jean LUDWIG：“Operateurs dentrelacement pour les groupes de Lie exponentiels”C.R.Acad.Sci.Paris，Serie I. 319. 549-551 (1994)

Didiev ARNAL,Hidenori FUJIWARA et Jean LUDWIG: "Operateurs dentrelacement pour les groupes de Lie exponentiels" C.R.Acad.Sci.Paris,Sarie I. 319. 549-551 (1994)

Didiev ARNAL、Hidenori FUJIWARA 和 Jean LUDWIG：“Operateurs dentrelacement pour les groupes de Lie exponentiels”C.R.Acad.Sci.Paris，Sarie I. 319. 549-551 (1994)