Complementary Series and Exotic Sobolev Norms

Complementary series representations of the Lorentz groups can be realized on eigenspaces of the Laplacian on hyperbolic space. Here we give an explicit formula for the norm on the eigenspace, at least for the K-finite eigenfunctions and a portion of the complementary series. The norm is given by a variant of the usual Sobolev norm involving two derivatives in L2 with the "exotic" feature that the integrand assumes both positive and negative values, namely Q(ƒ, ƒ) = ∫ (〈∇2ƒ, ∇2ƒ〉 − a〈∇ƒ, ∇ƒ〉 + b |ƒ|2) dx, where the positive coefficients a and b are "tuned" to the particular complementary series representation. An equivalent formula is Q(ƒ, ƒ) = ∫ p(Δ)ƒ2dx, where p is a certain quadratic polynomial. Similar results are obtained for the groups SO(p, q) in terms of eigenspaces of the Laplacian on semi-Riemannian spaces of constant curvature.