In this talk, I will introduce an involutive unitary operator F on the Hilbert space L2(C), where C is the isotropic cone for the indefinite quadratic form of signature (p,q). This operator F gives the expansion of functions on C into joint eigenfunctions of commuting family of self-adjoint operators of second order.By using this operator F , we also establish global formulas for the L2(C)-model of the minimal representation of the indefinite orthgonal group O(p+1,q+1), which parallels to the known fact that the ordinary Fourier transform on Rn gives rise to the Schrödinger model of the Weil representation of the metaplectic group.
© Toshiyuki Kobayashi