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

Understand

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.