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We know $U(m)$ is one the subgroups of $SO(2m)$ acting transitively on the sphere $S^{2m-1}$ (one of the groups in the Borel's list). What is the explicit formula of this embedding (or it's action)?

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Turn a complex number $a+bi$ into $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ and do it for each entry. –  user27126 Jan 26 '13 at 9:59
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$U(m)$ acts naturally on the unit sphere in $\mathbf C^m$ (vectors of norm $1$ for the standard Hermitian form), which under the identification of $\mathbf C$ with $\mathbf R^2$ becomes the unit sphere $\mathbf S^{2m-1}$. So the embdedding is just restriction of scalars from $\mathbf C$-linear operators to $\mathbf R$-linear operators.

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