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Generalizations of the Bargmann transform. (English) Zbl 0916.22006
Doebner, H.-D. (ed.) et al., Lie theory and its applications in physics. Proceedings of the international workshop, Clausthal, Germany, August 14–17, 1995. Singapore: World Scientific. 3-14 (1996).
Consider the Heisenberg group ${ℂ}^{n}×ℝ$. It has a family of distinguished representations on infinite dimensional Hilbert spaces. There are two standard models for the representations, the one is the Fock space of holomorphic functions on ${ℂ}^{n}$, the other is the Schrödinger model ${L}^{2}\left({ℝ}^{2}\right)$. The Bargmann transform then gives the intertwining operator between the two. In this descriptive paper the authors explain that the Bargmann transform can be obtained by the so called restriction principle, namely the restriction of a holomorphic function on ${ℂ}^{n}$ to its real form ${ℝ}^{n}$. The same idea is applied to Fock spaces on the complexification of a compact Lie group and to Bergman spaces on a Hermitian symmetric space with a certain Riemannian symmetric space as its real form, thus obtaining the Bargmann transform in these settings.

##### MSC:
 22E30 Analysis on real and complex Lie groups 43A85 Analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations