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### What do you call a scaled orthogonal map?

Wikipedia suggests "conformal orthogonal group" for the group of all such maps; see the articles https://en.wikipedia.org/wiki/Conformal_group https://en.wikipedia.org/wiki/Orthogonal_group#...
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"Orthogonal similitude" would be consistent with a very-common use of "symplectic similitude" (in automorphic forms and repn theory) for $g\in GL_n$ such that \$g^\top J g=\nu(g)\...