  We develop analogues for Grauert tubes of real analytic Riemannian manifolds
(M,g) of some basic notions of pluri-potential theory, such as the Siciak
extremal function. The basic idea is to use analytic continuations of
eigenfunctions in place of polynomials or sections of powers of positive line
bundles for pluripotential theory. The analytically continued Poisson-wave
kernel plays the role of Bergman kernel. The main results are Weyl laws in the
complex domain, distribution of complex zeros of eigenfunctions on locally
symmetric spaces, and estimates of triple products of eigenfunctions.
