A Formula for the Multiplicity of a Weight.

1. Introduction. 1. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra of g. Let ir\ be an irreducible representation of g, with highest weight X, on a finite dimensional vector space V\. A well known theorem of E. Cartan asserts that the highest weight, X, of w\ occurs with multi-plicity one. It has been a question of long standing to determine, more generally , the multiplicity of an arbitrary weight of irx. Weyl's formula (1.12) for the character of tt\ is an expression for the function xx(x) =tr exp ^x(x), x (Ef), on I) in terms of X and quantities independent of the representation. In the same spirit the author has always understood the multiplicity question to mean the following: Let I be the set of all integral linear forms on fi. Let m\ be the function in / which assigns to each integral linear form vE I the mul-tiplicity m\(v) of its occurrence as a weight of w\. Find a formula for the multiplicity function m\ in terms of X and quantities independent of the