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Hermite functions on compact Lie groups. I. (English) Zbl 0836.43016
The formula for Hermite polynomials on ${R}^{d}$ is well known as well as the existence of a linear isometry of $\overline{S}$ onto ${L}^{2}\left({R}^{d},{p}_{1}\right)$; $S=S\left({C}^{d}\right)$ denotes the vector space of symmetric tensors over ${C}^{d}$, $\overline{S}$ is the completion of $S$ in a precise inner product, and ${p}_{1}\left(x\right)$ is the standard Gaussian on ${R}^{d}$. By L. Gross the above problem was extended to the case where ${R}^{d}$ is replaced by a compact Lie group $G$. The author exhibits explicitly, in a new and more convenient manner, the isometry proved by L. Gross. – Mention must be made of the remarks 2.7, 3.2 and 3.4.

##### MSC:
 43A77 Analysis on general compact groups 33C45 Orthogonal polynomials and functions of hypergeometric type 22E30 Analysis on real and complex Lie groups