Harmonic analysis on solvable Lie groups and discrete subgroups

首席研究员藤原研究了指数可解李群的单项表示，并获得了以下结果。1.关于等价单项不可约表示之间的交织算子（与法国梅斯大学的D.Arnal和J.Ludwig的联合工作）。(1)由Vergne极化引起的单项表示算子的显式描述。(2)用Maslov指标证明复合公式的算子构造。(3)一般情况下，运算符的单位元素处的局部表达式.作为Penney的Plancherel公式的应用，我们给出了与有限重数的单项表示相联系的不变微分算子代数的交换性的另一个证明.它比Corwin和Greenleaf的原始证明更直接。在分布支撑的一些附加条件下，我们证明了具有特征的幂零对称空间的Frobenius互反性，即双半不变分布空间的维数等于不可约分解的重数. Kanemitsu研究了Riemann假设，得到如下结论.通过对Farey级数在短区间上的求值，可以得到与Riemann假设等价的条件.包含Hurwitz-Lerch zeta函数的级数的封闭形式的表达式.包含Hurwitz zeta函数的级数的计算。

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.