Abstract
Let
$G/K$
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
$L^{2}(G)$
and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of
$G/K$
. For the special case of the orbital measures,
$\nu _{a_{i}}$
, supported on the double cosets
$Ka_{i}K$
, where
$a_{i}$
belongs to the dense set of regular elements, we prove the sharp result that
$\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$
except for the symmetric space of Cartan class
$AI$
when the convolution of three orbital measures is needed (even though
$\nu _{a_{1}}\ast \nu _{a_{2}}$
is absolutely continuous).
Regular elements in complete uniquely complemented lattices
