A Gelfand Model for Weyl Groups of TypeD2n

摘要

A Gelfand model for a finite groupGis a complex representation ofG, which is isomorphic to the direct sum of all irreducible representations ofG. WhenGis isomorphic to a subgroup ofGLn(ℂ), whereℂis the field of complex numbers, it has been proved that eachG-module overℂis isomorphic to aG-submodule in the polynomial ringℂ[x1,…,xn], and taking the space of zeros of certainG-invariant operators in the Weyl algebra, a finite-dimensionalG-space𝒩Ginℂ[x1,…,xn]can be obtained, which contains all the simpleG-modules overℂ. This type of representation has been named polynomial model. It has been proved that whenGis a Coxeter group, the polynomial model is a Gelfand model forGif, and only if,Ghas not an irreducible factor of typeD2n,E7, orE8. This paper presents a model of Gelfand for a Weyl group of typeD2nwhose construction is based on the same principles as the polynomial model.