Abstract
This chapter is devoted to a detailed development of the theory of square integrable group representations, including the resulting orthogonality relations. Then we study a particular class of semidirect product groups, namely, groups of the form \(G = {\mathbb{R}}^{n} \rtimes H\), where H is an n-dimensional closed subgroup of GL\((n, \mathbb{R})\). Several concrete examples are presented. Finally we generalize the theory to representations that are only square integrable on a homogeneous space. This allows one to treat CS of the Gilmore-Perelomov type and, in particular, CS of the Galilei group.
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Notes
- 1.
This is an unconventional notation for spherical harmonics, and it differs from the one used in Chap. 7.
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Ali, S.T., Antoine, JP., Gazeau, JP. (2014). Coherent States from Square Integrable Representations. In: Coherent States, Wavelets, and Their Generalizations. Theoretical and Mathematical Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8535-3_8
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