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THE INVERSION FORMULA AND HOLOMORPHIC EXTENSION OF THE MINIMAL REPRESENTATION OF THE CONFORMAL GROUP
- TOSHIYUKI KOBAYASHI and
- GEN MANO
Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory01 Nov 2007
Preview Abstract
The minimal representation π of the indefinite orthogonal group O(m+1, 2) is realized on the Hilbert space of square integrable functions on ℝ^m with respect to the measure |x|⁻¹dx₁ ⋯ dx_m. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m+3, 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 ℝ ^{m, 1}.

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