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The Segal-Bargmann “coherent state” transform for compact Lie groups. (English) Zbl 0838.22004

The Segal-Bargmann transform is an integral transform $A$ that maps ${L}^{2}\left({ℝ}^{n}\right)$ isometrically onto $ℋ\left({ℂ}^{n}\right)\cap {L}^{2}\left({ℂ}^{n},\mu \right)$, where $\mu$ is the Gaussian measure on ${ℂ}^{n}$; $A$ can be expressed in the standard heat kernel on ${ℝ}^{n}$, ${\rho }_{t}\left(x\right)={\left(2\pi t\right)}^{-n/2}exp\left(-{\sum }_{i}{x}_{i}^{2}/2t\right)$, at $t=1$. Here $ℋ\left(\cdots \right)$ denotes the space of holomorphic functions. In the present paper a similar transform is investigated, with ${ℝ}^{n}$ replaced by a compact connected Lie group $K$, and ${ℂ}^{n}$ replaced by a complexification $G$ of $K$. Suitable heat kernels ${\rho }_{t}$ on $K$ and ${\mu }_{t}$ on $G$ are defined, and isometric isomorphisms

${A}_{t}:{L}^{2}\left(K\right)\to ℋ\left(G\right)\cap {L}^{2}\left(G,{\mu }_{t}\right),\phantom{\rule{4pt}{0ex}}{B}_{t}:{L}^{2}\left(K,{\rho }_{t}\right)\to ℋ\left(G\right)\cap {L}^{2}\left(G,{\mu }_{t}\right),$
${C}_{t}:{L}^{2}\left(K\right)\to ℋ\left(G\right)\cap {L}^{2}\left(G,{\nu }_{t}\right)$

are obtained, for $t>0$; here ${\nu }_{t}$ is the $K$-average of ${\mu }_{t}$. In section 10 some Paley-Wiener type theorems are proved, in section 11 transforms on quotient spaces are considered, and in an appendix the main results are described in the language of creation operators, annihilation operators, and coherent states. The paper is well organized, and the proofs are transparent, even where things become rather technical.

##### MSC:
 22E30 Analysis on real and complex Lie groups 81R30 Coherent states; squeezed states (quantum theory) 58J35 Heat and other parabolic equation methods for PDEs on manifolds