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

About this Title

Toshiyuki Kobayashi, Graduate School of Mathematical Sciences, IMPU, the University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan and Gen Mano, Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 213, Number 1000
ISBNs: 978-0-8218-4757-2 (print); 978-1-4704-0617-2 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00592-7
Published electronically: February 4, 2011
Keywords: Minimal representation, Schrödinger model, generalization of the Fourier transform, Weil representation, indefinite orthogonal group, unitary representation, isotropic cone, Bessel functions, Meijer’s $G$-functions
MSC: Primary 22E30; Secondary 22E46, 43A80

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Table of Contents

Chapters

• 1. Introduction
• 2. Two models of the minimal representation of $O(p,q)$
• 3. $K$-finite eigenvectors in the Schrödinger model $L^2(C)$
• 4. Radial part of the inversion
• 5. Main theorem
• 6. Bessel distributions
• 7. Appendix: special functions

Abstract

We introduce a generalization of the Fourier transform, denoted by $\mathcal {F}_C$, on the isotropic cone $C$ associated to an indefinite quadratic form of signature $(n_1,n_2)$ on $\mathbb {R}^n$ ($n=n_1+n_2$: 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 $\mathcal {F}_{\mathbb {R}^n}$ on $L^2(\mathbb {R}^n)$. 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 $\mathcal {F}_C$ expands functions on $C$ into joint eigenfunctions of fundamental differential operators which are mutually commuting, self-adjoint, and of second order. We decompose $\mathcal {F}_C$ into the singular Radon transform and the Mellin–Barnes integral, find its distribution kernel, and establish the inversion and the Plancherel formula. The transform $\mathcal {F}_C$ reduces to the Hankel transform if $G$ is $O(n,2)$ or $O(3,3) \approx SL(4,\mathbb {R})$.

The unitary operator $\mathcal {F}_C$ together with multiplications and translations coming from the conformal transformation group $CO(n_1,n_2)\ltimes \mathbb {R}^{n_1+n_2}$ 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–Ørsted, and others. Among them, our model gives the global formula of the whole group action on the simple Hilbert space $L^2(C)$, and generalizes the classic Schrödinger model $L^2(\mathbb R^n)$ of the Weil representation. Here, $\mathcal {F}_C$ plays a similar role to $\mathcal {F}_{\mathbb {R}^n}$.

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.