Does there exist a description of the odd dimensional spheres as homogeneous spaces of the symplectic group. For $S^7$ it seems to me that we should have $S^7 \simeq Sp(3)/Sp(2)$, but I can't make a proof in this easy case or in the general picture.
2026-03-25 20:35:20.1774470920
Spheres as Symplectic Homogeneous Spaces
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The symplectic group $Sp(n)$ acts unitarily on $\mathbb{H}^n\cong \mathbb{R}^{4n}$, and the induced action on $S^{4n-1}$ is transitive with isotropy group $Sp(n-1)$. Then $$ Sp(n)/Sp(n-1)\cong S^{4n-1}. $$ In particular, $S^7\cong Sp(2)/Sp(1)$.