Hardy's uncertainty principle on semisimple groups

A theorem of Hardy states that, if f is a function on R such that \f(x)\ le ss than or equal to C e(-alpha\x\2) for all x in R and \(f) over cap(xi)\ l ess than or equal to C e(-beta\xi\2) for all xi in R, where alpha > 0, beta > 0, and alpha beta > 1/4, then f = 0. Sitaram and Sundari generalised thi s theorem to semisimple groups with one conjugacy class of Cartan subgroups and to the K-invariant case for general semisimple groups. We extend the t heorem to all semisimple groups.