The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p, q)

The indefinite orthogonal group G = O(p, q) has a distinguished infinite dimensional irreducible unitary representation π for p+ q even and greater than 4, which is the “smallest” in the sense that the Gelfand–Kirillov dimension of π attains its (positive) minimum value p + q − 3 among the unitary dual of G. Moreover, π is the minimal representation if p+ q > 6. The Schrödinger model realizes π on the Hilbert space L(C) consisting of square integrable functions on a Lagrangean submanifold C of the minimal nilpotent coadjoint orbit. Among various concrete models of π, the Hilbert structure (e.g. inner product) of the Schrödinger model is so simple, whereas the G-action on L(C) has not been wellunderstood except for a specific maximal parabolic subgroup. The subject of this paper is the analysis of the Schrodinger model of the minimal representation. We establish the “global formula” for the Schrödinger model with an explicit description of the action of the whole group G. For this, we describe the unitary operator π(w0) ∗Partially supported by Grant-in-Aid for Scientific Research (B) (18340037), Japan Society for the Promotion of Science. 2000 Mathematics Subject Classification. Primary 22E30; Secondary 22E46, 43A80