Geometric invariants of representations and the Whittaker models

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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.