Representation Theory and Duality associated with Non-commutative Special Functions

非交换变元的特殊函数理论是深入理解对偶对的一个框架，是经典不变理论的一种复兴。研究的目的是将特殊函数、不变量和表示三个理论相互联系起来，为非交换性下的数学世界提供新的视角。在我们研究的中心，我们有Capelli恒等式，不变微分算子的等式，它产生于泛包络代数的中心的表示。围绕Capelli恒等式，我们发现了从交换理论到非交换理论的转变过程中所引起的许多有趣现象。为了得到最终的Capelli型恒等式，我们将非对易矩阵元、转移概念的推广、符号方法和生成函数方法结合起来，采用了联合收割机的新思想，取得了很大的进展.虽然程序还没有完成，我们的结果将是非常有用的最终Capelli恒等式的理解。一个重要的例子是在处理欧拉的五边形数定理，我们理解了无限大矩阵的迹恒等式。其中，我们发现推广五边形数定理的一些恒等式确实是q-超几何级数的一类求和公式。这个观点将引导我们通过无限维空间把表示论和不变量理论联系起来。另一个重要的发现是代数，它是研究高阶Capelli恒等式作为非交换形式变量的一个非常有用的工具。这是M做的。伊东，这项研究的研究员。

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.