Structure of commutative hypergroups

超群是局部紧空间，其上的有界测度卷积类似于局部紧群上的卷积。自1975年左右，超群理论就有了一种公理的方法。超群的例子是由Gelfand对衍生的双陪集空间，更具体的例子是非负整数或非负实数，它们的卷积分别由特殊函数的正交多项式定义。超群结构有着广泛的应用，从二阶微分方程组理论(Sturm-Liouville问题)到概率论(随机游动的性质)。有关该理论的全面论述，请参阅申请人的专著[1]和[2]。这里将要描述的研究计划的中心主题是超群的扩展，即从已知的超群中产生新的超群的雄心。通过这种方式，人们希望得出超群的结构理论。在方法论上，首先要取得成功的一个想法是将群的上同调理论推广到超群，特别是推广G.W.Mackey的上循环理论。在申请人与其来自奈良教育大学的同事S.Kawakami一起发表的早期论文中，已经成功地处理了扩展问题(参考清单一和二)。对于Pontryagin超基团这一特殊类别，取得了令人惊讶的结果(参考清单二[HK1])。向交换超群的上同调理论迈进的第一步是成功的(参考文献II[HK 24])。本申请所涉及的项目涉及交换上推群的非本原定理。我们猜想，这个定理至少可以建立在由群在超群上的作用所定义的半直积上，也可以建立在具有超正稳定性超群的一般超群上。在半直积的情况下，与S.Kawakami共同撰写的一篇论文几乎完成[3]。为了丰富有趣的例子，我们打算研究离散Mautner群的特征标超群，即对偶[4]。显然，这样的研究与非信仰局部紧群的诱导表示和对偶理论是狭义相关的。

Hypergroups are locally compact spaces on which the bounded measures convolve similar to those on a locally compact group. There exists an axiomatic approach to the theory of hypergroups since around 1975. Examples of hyper groups are double coset spaces derived from Gelfand pairs, more concrete examples are the nonnegative integers or the nonnegative real numbers whose convolutions are defined by orthogonal polynomials of by special functions respectively. Hyper group structures enjoy widespread applications ranging from the theory of differential equations of second order (Sturm-Liouville problems) to the theory of probability (properties of random walks). For comprehensive expositions on the theory see the applicant’s monographs [1] and [2]. The central topic of the research program to be described here is the extension of hyper groups, i.e. the ambition to produce new hyper groups from already known ones. This way one hopes to arrive at a structural theory of hyper groups. Methodically a first idea to succeed is to extend the cohomology theory for groups to hyper groups, in particular to generalize G.W. Mackey’s theory of cocycles. In earlier papers published by the applicant together with his colleague S. Kawakami from the Nara University of Education the extension problem has been successfully dealt with (Reference lists I and II). For the special class of Pontryagin hyper groups surprising results have been obtained ([HK1] of Reference list II). A first step in the direction to a cohomology theory for commutative hyper groups has been successful ([HK 24] of Reference List II). The project which the present application refers to concerns the imprimitivity theorem for commutative hypger groups. There is the conjecture that this theorem can be established at least for semidirect products which are defined by an action of a group on a hyper group, also for general hyper groups with supernormal stability hyper groups. In the case of semi direct products a paper jointly written with S. Kawakami is almost complete [3]. In order to enrich the repertoire of interesting examples it is intended to study the character Hyper group, i.e. the dual, of the discrete Mautner group [4]. Clearly, such a study is narrowly related to the theory of induced representation and duality of nonbelief locally compact groups.