A Gel'fand model for a Weyl group of type $B_{n}$

J. O. Araujo

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 [S]). Gel'fand models for the symmetric group and the linear group over a finite field can be found in [A] and [K]. Using the same ideas as in [A], in this work we describe a Gel'fand model for a Weyl group of type $B_n$. When $K$ is a field of characteristic zero and $\mathfrak G$ is a Weyl group of type $B_n$, we give a finite dimensional $K$-subspace $\mathcal N$ of the polynomial ring $K[x_1,\ldots,x_n]$. If $K$ is the field of complex numbers, then $\mathcal N$ provides a Gel'fand model for $\mathfrak G$. \noindent The space $\mathcal N$ can be defined in a more general way (see [AA]), obtained as the zeros of certain differential operators (symmetrical operators) in the Weyl algebra. However, in the case of a group $G$ of type $D_n$ ($n$ even), $\mathcal N$ is not a Gel'fand model for $G$.

[A] Aguado, J. L.; Araujo, J. O.: A Gel'fand Model for the Symmetric Group. Communications in Algebra, {\bf 29}(4) (2001), 1841--1851.

[AA] Araujo, J. O.; Aguado, J. L.: Representations of Finite Groups on Polynomial Rings. Actas, V Congreso de Matemática Dr. Antonio A. R. Monteiro, 35-40, Bahía Blanca 1999.

[K] Klyachko, A. A.: Models for the complex representations of the groups $G(n,q)$. Math. of the USSR - Sbornik {\bf 48} (1984), 365--380.

[S] Soto-Andrade, J.: Geometrical Gel'fand Models, Tensor Quotients and Weil Representations. Proceedings of Symposia in Pure Mathematics {\bf 47}, part. 2 (1987), 306--316.

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