Geometric invariants of representations and the Whittaker models

在研究期间，我们得到了许多关于交换微分算子对称性的结果，这些结果与本研究的主要主题--Whittaker模型和表示不变量有关。完全可积系统Calogero-Moser-Sutherland模型(以下简称CMS模型)与根系和Weyl群密切相关。该模型的哈密顿算符在Weyl群的作用下是不变的，其势函数用相应的根系表示。请注意，它的势函数沿Weyl腔壁具有反平方奇点。我们研究了哈密顿算子的交换子，它的势函数沿着通过原点的一些超平面具有反平方奇性。结果表明，势函数和交换子的Weyl群对称性是由这种奇点和耦合常数的一般性质自然产生的。此外，我们还得到了以下结果：(1)如果这种对称性是经典类型的，则势函数一定是已知的。(2)在二阶情形下，势函数一定满足某种线性关系。(3)在二阶情形下，(4)当且仅当两个耦合常数为1时，A_2型CMS模型才可能发生形变。(5)即使没有耦合常数为1时，B_2型CMS模型也可能发生形变。(6)构造了B_2型CMS模型的新形变。

During the research period, we obtain many results about the symmetry of commuting differential operators, which is relevant to the main subjects of this research, namely Whittaker models and invariants of representations.The completely integrable system called Calogero-Moser-Sutherland model (which we abbreviate as CMS model in the following of this abstract) is closely related to root systems and Weyl groups. The Hamiltonian operator of this model is invariant under the action of a Weyl group and the potential function of it is expressed by means of the corresponding root system. Note that the potential function of it possesses inverse square singularities along the walls of a Weyl chamber. We study the commutants of a Hamiltonian operator whose potential function possesses inverse square singularities along some hyperplanes passing through the origin. It is shown that the Weyl group symmetry of the potential function and the commutants naturally results from such singularities and the generic nature of the coupling constants. Moreover, we have obtained the following results :(1)If this symmetry is of the classical type, the potential function must be one of the known ones.(2)In the rank two cases, the potential function must satisfy some linear relations.(3)In the rank two cases, the order of the commutant is at least the number of singular lines.(4)Deformation of A_2 type CMS model is possible if and only if two of the coupling constants are 1.(5)Deformation of B_2 type CMS model is possible even if no coupling constant is 1.(6)A new deformation of the B_2 type CMS model is constructed.Besides the above investigation, we study an elementary method of constructing F_4 type invariant polynomials.