A Residue Calculus for Root Systems

Research paper by E. P. van den Ban, H. Schlichtkrull

Indexed on: 01 Aug '00Published on: 01 Aug '00Published in: Compositio Mathematica


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 ε Vℂ∣〈 λ, α 〉 = s, α ε Σ, s ε 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 ε 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.