A Gelfand Model for Weyl Groups of Type D2n

A Gelfand model for a finite group G is a complex representation of G , which is isomorphic to the direct sum of all irreducible representations of G . When G is isomorphic to a subgroup of G L n ( ℂ ) , where ℂ is the field of complex numbers, it has been proved that each G -module over ℂ is isomorphic to a G -submodule in the polynomial ring ℂ [x 1 , … , x n] , and taking the space of zeros of certain G -invariant operators in the Weyl algebra, a finite-dimensional G -space Gin ℂ [x 1 , … , x n] can be obtained, which contains all the simple G -modules over ℂ . This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of typeD 2 n,E 7, orE 8. This paper presents a model of Gelfand for a Weyl group of typeD 2 nwhose construction is based on the same principles as the polynomial model.