A Gelfand model for a Weyl group of type Bn

Abstract. A Gel’fand model for a finite group G is a complex representation of G which is isomorphic to the direct sum of all the irreducible representation of G (see [9]). Gel’fand models for the symmetric group and the linear group over a finite field can be found in [2] and [8]. Using the same ideas as in [2], in this work we describe a Gel’fand model for a Weyl group of type Bn. When K is a field of characteristic zero and G is a Weyl group of type Bn, we give a finite dimensional K-subspace N of the polynomial ring K [x1,..., xn]. If K is the field of complex numbers, then N provides a Gel’fand model for G. The space N can be defined in a more general way (see [3]), obtained as the zeros of certain differential operators (symmetrical operators) in the Weyl algebra. However, in the case of a group G of type Dn (n even), N is not a Gel’fand model for G. 1. Symmetrical operators and the space N Let K be a field of characteristic zero. Fix a natural number n. We will denote by A the polynomial ring K [x1,..., xn] and by W the Weyl algebra of K-linear differential operators K 〈x1,..., xn, ∂1,..., ∂n 〉 generated by the multiplication operators xi and the differential operators ∂i = ∂xi where i = 1,..., n. The angular brackets are used to indicate that the generators do not commute, indeed, ∂ixi = 1 + xi∂i for each i = 1,..., n. We will make use of some basic properties of the algebra W, which are proved in [4]. Let In = {1, 2,..., n} and M be the set of functions α: In → N0, where N0 denotes the set of non-negative integers. Such a function is called a multiindex, and we put αi = α (i) and α = (α1,..., αn). 0138-4821/93 $ 2.50 c © 2003 Heldermann Verlag 360 J. O. Araujo: A Gel’fand Model for a Weyl Group of Type Bn For two multiindexes α, β in M we will use the following notations: |α | = α1 + · · ·+ αn, α! = n∏ i=1 αi!