Graded multiplicities in the exterior algebra

This paper deals with the graded multiplicities of the "smallest" irreducible representations of a simple Lie algebra in its exterior algebra. An explicit formula for the graded multiplicity or the adjoint representation in terms of the Weyl group, exponents was conjectured by A. Joseph: a proof of this conjecture, based on the properties of Macdonald polynomials, is given in the present payer. The same method allows us to calculate the multiplicity of the simple module with highest weight equal to the short dominant root. (C) 2001 Academic Press.