The inversion formula and holomorphic extension of the minimal representation of the conformal group

Details The inversion formula and holomorphic extension of the minimal representation of the conformal group The inversion formula and holomorphic extension of the minimal representation of the conformal group

2006-06-30

arxiv.org/abs/math/0607007v1

Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: In Honour of Roger E. Howe, Singapore Univer

Mathematics

Representation Theory

Scientific paper

The minimal representation $\pi$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on $\mathbb R^m$ with respect to the measure $|x|^{-1} dx_1... dx_m$. This article gives an explicit integral formula for the holomorphic extension of $\pi$ to a holomorphic semigroup of $O(m+3, \mathbb C)$ by means of the Bessel function. Taking its `boundary value', we also find the integral kernel of the `inversion operator' corresponding to the inversion element on the Minkowski space $\mathbb R^{m,1}$.

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