Representation Theory and Duality associated with Non-commutative Special Functions

非对易变量的特殊函数理论是我们深入理解对偶偶的一个框架，它是经典不变量理论的一种复兴。研究的目的是将特殊函数、不变量和表示这三个理论联系起来，为非对易下的数学界带来新的曙光。在我们研究的中心，我们有Capelli恒等式，不变微分算子的等式，它们产生于泛包络代数的中心的表示。围绕Capelli恒等式，我们发现了从对易理论向非对易理论过渡所产生的许多有趣的现象。为了得到最终的Capelli型恒等式，我们通过将非交换矩阵元、转移概念的推广、符号方法和生成函数方法相结合的新思想，取得了很大的进展。虽然程序还没有完成，但我们所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处理欧拉五角数定理时，我们理解了无穷大矩阵的迹恒等式。在这里，我们发现一些推广的五边形数定理的恒等式确实是一种q-超几何级数的求和公式。这一观点将使我们在无限维空间中找到表示理论和不变量理论之间的新联系。另一个重要的发现是代数，它是研究作为非对易形式变量的更高Capelli恒等式的非常有用的工具。这项研究是由M.Itoh完成的，他是一名研究人员。

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.