Central Multiplier Theorems for Compact Lie Groups

The purpose of this note is to describe how central multiplier theorems for compact Lie groups can be reduced to corresponding results on a maximal torus. We shall show that every multiplier theorem for multiple Fourier series gives rise to a corresponding theorem for such groups and, also, for expansions in terms of special functions. We use the notation and terminology of N. J. Weiss [4]. Let G denote a simply connected semisimple Lie group, g its Lie algebra and ï) a maximal abelian subalgebra; P the set of positive roots in I)*, the dual of t) (with respect to some order), and ( , ) is the inner product on f)* induced by the Killing form. With X=(Xl9 • • • , Xx) e Z l we associate the weight A=2l=i^"<> where TTÎ are the fundamental weights adapted to the simple roots. The characters %x of G are then indexed by those X with nonnegative integer coefficients. The degree dx of the corresponding representation is then given by