The Schrodinger Model for the Minimal Representation of the Indefinite Orthogonal Group $O(p,q)$
- ISBN-10
- 0821847570
- ISBN-13
- 9780821847572
- Category
- Representations of Lie groups
- Pages
- 132
- Language
- English
- Published
- 2011
- Publisher
- American Mathematical Soc.
- Authors
- Toshiyuki Kobayashi, Gen Mano
Description
The authors 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, the authors 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.
