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 V$\Bbb C$=V+iV, the singular locus of which is a locally finite union of hyperplanes of the form {λ ∈ V$\Bbb C$[mid ]〈 λ, α 〉 = s}, α ∈ Σ, s ∈ $\Bbb R$. 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 ∈ 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.