Dedicated to Jiro Sekiguchi on the occasion of his sixtieth birthday.Abstract. Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair.We take parabolic subgroups P of G and Q of K respectively and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it.In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.

We consider a generalization of the Mahler measure of a multivariable polynomial P as the integral of log k |P | in the unit torus, as opposed to the classical definition with the integral of log |P |. A zeta Mahler measure, involving the integral of |P | s , is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet L-functions, and polylogarithms.

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