A Poisson Formula for Semi-Simple Lie Groups

for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it may be shown that h(g) is annihilated by a certain class of differential operators on G. The Poisson formula (1) may be used to express h(g), and we find that here it takes on a particularly simple form. Namely, if we denote by m the normalized Lebesgue measure on {j z I = 1}, and by gm, the transform of this measure by the group element g E G, then it can be seen that (1) becomes