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

Abstract

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.