Determinant Functions and the Geometry of the Flag Manifold for SU(p,q)

Let G 0 = SU(p, q) with q p, K 0 = S(U(p)U(q)) a maximal compact subgroup, and let G, K be their complexifications.Finally, let B be a Borel subgroup of G.We define a number of algebraic functions on G/B G/K and use them to describe the closures of codimension one K orbits of the flag manifold G/B .We show how the underlying geometry of the flag manifold interacts with these functions.In particular, we shall use these functions to construct a Stein extension of the Riemannian symmetric space G 0 /K 0 , whose connected component turns out to be the space of linear cycles in most cases.