  A scalar field in four-dimensional deSitter spacetime (dS_4) has quasinormal
modes which are singular on the past horizon of the south pole and decay
exponentially towards the future. These are found to lie in two complex
highest-weight representations of the dS_4 isometry group SO(4,1). The
Klein-Gordon norm cannot be used for quantization of these modes because it
diverges. However a modified `R-norm', which involves reflection across the
equator of a spatial S^3 slice, is nonsingular. The quasinormal modes are shown
to provide a complete orthogonal basis with respect to the R-norm. Adopting the
associated R-adjoint effectively transforms SO(4,1) to the symmetry group
SO(3,2) of a 2+1-dimensional CFT. It is further shown that the conventional
Euclidean vacuum may be defined as the state annihilated by half of the
quasinormal modes, and the Euclidean Green function obtained from a simple mode
sum. Quasinormal quantization contrasts with some conventional approaches in
that it maintains manifest dS-invariance throughout. The results are expected
to generalize to other dimensions and spins.
