Decay of eigenfunctions on semisimple symmetric spaces (Q1190826)

Let \(G\) be a connected semisimple real Lie group with finite center, \(\sigma\) an involution of \(G\), and \(H\) an open subgroup of the fixed point set of \(\sigma\). Let \(\theta\) be a Cartan involution of \(G\) commuting with \(\sigma\), \(K\) the corresponding maximal compact subgroup. Let \(f \in C^ \infty(G/H)\) be a left \(K\)-finite eigenfunction of \(Z( {\mathfrak g})\). The authors assume that the \(({\mathfrak g},K)\)-module \(V_ f\) generated by \(f\) is unitarizable and that its restriction to any noncompact factor of \(G\) does not contain the trivial representation. Then the authors prove that \(f\) vanishes at infinity on \(G/H\) and give rates of decay for \(f\), depending on the Langlands parameters of \(V_ f\).