Representation Theory and Duality associated with Non-commutative Special Functions

非交换变量的特殊函数理论是我们深入理解对偶的一个框架，是对经典不变理论的一种复兴。本研究的目的是将三种关于特殊函数、不变量和表示的理论联系起来，给非交换性下的数学世界带来新的启示。在我们研究的中心，我们有Capelli恒等式，不变微分算子的等式，它来自于全域包络代数中心的表示。围绕Capelli恒等式，我们发现了许多有趣的现象，这些现象是由交换理论向非交换理论转变而引起的。即使在通常的交换理论背后，我们也经常发现占主导地位的非交换变量。为了得到最终的Capelli型恒等式，我们通过结合非交换矩阵元素的新思想、对迁移概念的推广、符号方法和生成函数的方法取得了很大的进展。虽然该程序尚未完成，但我们得到的结果将对理解最终的Capelli恒等式非常有用。一个重要的例子是在欧拉五边形数定理的处理中，我们理解了无限大小矩阵的迹恒等式。在那里，我们发现了一些推广五边形数定理的恒等式实际上是q-超几何级数的求和公式。这一观点将引导我们通过无限维空间将表征理论与不变理论联系起来。另一个重要的发现是代数，它对于作为非交换形式变量的高Capelli恒等式是一个非常有用的工具。这是由本研究的研究员伊藤先生完成的。

Theory of special functions of non-commutative variables is a framework for us to understand deeply the dual pairs, which is a sort of revival of the classical invariant theory. The aim of the research is to link each other the three theories on special functions, invariants, and representations, to throw a new light to the mathematical world under the non-commutativity. In the center of our study, we have the Capelli identities, the equalities of the invariant differential operators, which arise from the representations of the centers of the universal enveloping algebras. Around the Capelli identities, we found many interesting phenomena caused from the transition from commutative theories to non-commutativeones. And even behind the usual commutative theories, we often found the dominating non-commutative variables.For the goal to obtain the ultimate Capelli type identities, we made a big progress through the new ideas which combine the non-commutative matrix elements, the generalization of the notion of transfer, symbolic method, and the method of generating functions. Though the program has not been accomplished, the results we had will be very useful for the understanding of the ultimate Capelli identities.An important example is in the treatment of Euler's pentagonal number theorem, which we understand a trace identity of matrices of infinite size. In there, we discover the fact that some identities generalizing the pentagonal number theorem are indeed sort of summation formula for q-hypergeometric series. This point of view will lead us to a new link between the representation theory and invariant theory through the infinite-dimensional spaces.Another important discovery is the algebra, which is a very useful toolfor the higher Capelli identities as the non-commutative formal variables. This is done by M. Itoh, an investigator of this research.