Fourier-like transformation and a representation of the Lie algebra \({\mathfrak so}(n+1,2)\) (Q791218)

The geometric quantization, i.e. representation via Kirillov theory of the natural action of \(SO(n+1,2)\) on \(T^*S^ n\) is studied. The Lie algebra \({\mathfrak so}(n+1,2)\) is realized as a Poisson subalgebra of functions on \(T^*S^ n\). Since the action does not preserve any polarization on \(M=T^*S^ n-zero\) section, the standard tools of geometric quantization fail in this case. The author uses a pair of transversal polarizations: the vertical polarization and a partially complex one. To obtain sufficiently many sections, singular sections in the associated quantum bundle are considered. Using methods of \textit{K. Gawedzki} [Diss. Math. 128 (1976; Zbl 0343.53024)], a pairing between the two spaces of sections is constructed (a Fourier-like transform). Finally an irreducible representation of \(so(n+1,2)\) as an algebra of skew- Hermitian, pseudodifferential operators of order one on a Sobolev space over \(S^ n\) is obtained.