For k≤m, the pushforward of Lebesgue measure under the Gram map is (2^{-k} ∏_{j=1}^k Vol(S^{m-j})) |det G_k|^{(m-k-1)/2} dG, giving the Hilbert measure on the orbit space.
Michor, Basic differential forms for actions of Lie groups , Proc
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Hilbert measures on orbit spaces of coregular $\operatorname{O}_m$-modules
For k≤m, the pushforward of Lebesgue measure under the Gram map is (2^{-k} ∏_{j=1}^k Vol(S^{m-j})) |det G_k|^{(m-k-1)/2} dG, giving the Hilbert measure on the orbit space.