Meromorphic Basis of H-Invariant Distribution Vectors for the Generalized Principal Series of Reductive Symmetrical Spaces: Functional Equation

Let G be the group of real points of a reductive algebraic group defined over R, σ an involution of G, and θ a Cartan involution of G commuting with σ. Let H be an open subgroup of the group of fixed points for σ. One builds a meromorphic basis for the space of H-fixed distributions vectors of induced representations from a σθ-stable parabolic subgroup P of G. For this, we use a method which extends the domain of application of Bruhat′s thesis (in particular, to the irreducibility problem for generalized principal series). The meromorphy is obtained by means of a functional equation that we establish and which generalizes the equation obtained by E. van den Ban in the case when P is minimal σθ-stable.