  We introduce the `Fourier transform' F_C on the isotropic cone C associated
to an indefinite quadratic form of signature (n_1,n_2) on R^n (n=n_1+n_2:
even). This transform is in some sense the unique and natural unitary operator
on L^2(C), as is the case with the Euclidean Fourier transform.
  Inspired by recent developments of algebraic representation theory of
reductive groups, we shed new light on classical analysis on the one hand, and
give the global formulas for the L^2-model of the minimal representation of the
simple Lie group G=O(n_1+1,n_2+1) on the other hand.
  The transform F_C expands functions on C into joint eigenfunctions of the n
commuting, self-adjoint, second order differential operators. We decompose F_C
into the singular Radon transform and the Mellin--Barnes integral, find its
distribution kernel, and establish the inversion and the Plancherel formula.
F_C reduces to the Hankel transform if G is O(n,2) or O(3,3).
  The unitary operator F_C together with the simple action of the conformal
transformation group generates the minimal representation of the indefinite
orthogonal group G. Various different models of the same representation have
been constructed by Kazhdan, Kostant, Binegar-Zierau, Gross-Wallach, Zhu-Huang,
Torasso, Brylinski, and Kobayashi-Orsted, and others. Among them, our model
built on L^2(C) generalizes the classic Schrodinger model of the Weil
representation.
  Yet another motif is special functions. Large group symmetries in the minimal
representation yield functional equations of various special functions. We find
explicit K-finite vectors on L^2(C), and give a new proof of the Plancherel
formula for Meijer's G-transforms.
