## *Fourier Transform on the Isotropic Cone for the Indefinite
Quadratic Form*.
Analysis Seminar. Aarhus University, Denmark, 23 August 2007.

In this talk, I will introduce an involutive unitary operator *F* on the Hilbert space *L*^{2}(*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 *L*^{2}(*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 *R*^{n} gives rise to the Schrödinger model of the Weil representation of the metaplectic group.

© Toshiyuki Kobayashi