A Residue Calculus for Root Systems

Abstract

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