A Residue Calculus for Root Systems

Let V be a finite-dimensional real vector space on which a root system is given. Consider a meromorphic function on V =V+iV, the singular locus of which is a locally finite union of hyperplanes of the form V , = s, , s . Assume is of suitable decay in the imaginary directions, so that integrals of the form _+iV , d make sense for generic V. A residue calculus is developed that allows shifting . This residue calculus can be used to obtain Plancherel and Paley–Wiener theorems on semisimple symmetric spaces.