THE SMOOTHNESS of ORBITAL MEASURES on NONCOMPACT SYMMETRIC SPACES

Let be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any r=r (G/K)continuous orbital measures has its density function in L2(G) and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank G/K of. For the special case of the orbital measures vai, supported on the double cosets KaiK, where ai belongs to the dense set of regular elements, we prove the sharp result that va1∗va2 ϵ L2 except for the symmetric space of Cartan class when the convolution of three orbital measures is needed (even though va1∗va2 is absolutely continuous).