Analysis on the minimal representation of O(p,q) -- I. Realization via conformal geometry

Details Analysis on the minimal representation of O(p,q) -- I. Realization via conformal geometry Analysis on the minimal representation of O(p,q) -- I. Realization via conformal geometry

2001-11-07

arxiv.org/abs/math/0111083v1

Adv. Math., 180, (2003), pp.486-512

Mathematics

Representation Theory

Scientific paper

10.1016/S0001-8708(03)00012-4

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.

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