Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

In view of the many applications of the theory of symmetric functions to representation theory it seems desirable to have a theory of symmetric functions in the spirit of Macdonald's book Mac] for Weyl groups other than the symmetric group. Our approach to this problem is to nd an algebraic structure which motivates each statement in the classical symmetric function theory. If this algebraic notion occurs across the board then this should indicate what the proper generalization is for other types. Note that from this point of view there may be several useful generalizations of a given concept depending on what symmetric function properties are desirable. The goal of this paper is to ooer a suggestion for the analogue of the basis of homogeneous symmetric functions for Weyl group symmetric functions. In this case the deenition is motivated by the theory of centralizer algebras. The idea motivating the generalization is that it is really the Frobenius image of the homogeneous symmetric function that is the useful object. It is clear from the double centralizer theory that an analogue of the Frobenius characteristic map is a feature of the double centralizer mechanism, see R]. With this point of view one nds an analogue of the \Jacobi-Trudi" formula in the work of Verma V], Zelevinsky Z], Akin A] and Goodman-Wallach GW]. In this paper I simply ooer a mechanism by which to transfer their results to Weyl group symmetric functions. I would like to thank D.-N. Verma for so patiently explaining the many many things about Lie algebras and their representations which he considered useful for me to know. In particular, he showed me the representation theoretic result, Theorem (4.6) in this paper, which motivates the deenition of the homogeneous symmetric functions. I would like to thank N. Wallach for further useful discussions with me on this topic.