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March 2007 Integral formula of the unitary inversion operator for the minimal representation of $O(p, q)$
Toshiyuki Kobayashi, Gen Mano
Proc. Japan Acad. Ser. A Math. Sci. 83(3): 27-31 (March 2007). DOI: 10.3792/pjaa.83.27

Abstract

The indefinite orthogonal group $G = O(p,q)$ has a distinguished infinite dimensional unitary representation $\pi$, called the \textit{minimal representation} for $p+q$ even and greater than $6$. The \textit{Schr\"odinger model} realizes $\pi$ on a very simple Hilbert space, namely, $L^2(C)$ consisting of square integrable functions on a Lagrangean submanifold $C$ of the minimal nilpotent coadjoint orbit, whereas the $G$-action on $L^2(C)$ has not been well-understood. This paper gives an explicit formula of the unitary operator $\pi(w_0)$ on $L^2(C)$ for the `conformal inversion' $w_0$ as an integro-differential operator, whose kernel function is given by a Bessel distribution. Our main theorem generalizes the classic Schr\"odinger model on $L^2(\mathbf R^n)$ of the Weil representation, and leads us to an explicit formula of the action of the whole group $O(p,q)$ on $L^2(C)$. As its corollaries, we also find a representation theoretic proof of the inversion formula and the Plancherel formula for Meijer's $G$-transforms.

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Toshiyuki Kobayashi. Gen Mano. "Integral formula of the unitary inversion operator for the minimal representation of $O(p, q)$." Proc. Japan Acad. Ser. A Math. Sci. 83 (3) 27 - 31, March 2007. https://doi.org/10.3792/pjaa.83.27

Information

Published: March 2007
First available in Project Euclid: 9 April 2007

zbMATH: 1230.22007
MathSciNet: MR2317306
Digital Object Identifier: 10.3792/pjaa.83.27

Subjects:
Primary: 22E30, 22E46, 43A80

Rights: Copyright © 2007 The Japan Academy

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Vol.83 • No. 3 • March 2007
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