  This is the first in a series of papers devoted to an analogue of the
metaplectic representation, namely, the minimal unitary representation of an
indefinite orthogonal group; this representation corresponds to the minimal
nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant.
  We begin by applying methods from conformal geometry of pseudo-Riemannian
manifolds to a general construction of an infinite-dimensional representation
of the conformal group on the solution space of the Yamabe equation. By
functoriality of the constructions, we obtain different models of the unitary
representation, as well as giving new proofs of unitarity and irreducibility.
  The results in this paper play a basic role in the subsequent papers, where
we give explicit branching formulae, and prove unitarization in the various
models.
