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Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: \langle X,Y \rangle=\operatorname{tr}(X^TY) $).

The metric obtained $g$ is left-$GL_n$ invariant and right-$O_n$ invariant. ($O_n$ is the orthogonal group).

Question: Are there isometries of $(GL_n(\mathbb{R}),g)$ which are not compositions of the above mentioned isometries? (i.e I am asking whether these symmetries generate $\text{Iso}(GL_n(\mathbb{R}),g)$ as a group).

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    $\begingroup$ I don't have an answer, but you may find this paper of Ravi Shankar of interest: www2.math.ou.edu/~shankar/papers/paper_thesis.pdf In it, he computes the full isometry group of several homogeneous spaces. Unfortunately, he only deals with compact things, so I'm not sure how much of what he does carries over to your case. $\endgroup$ – Jason DeVito Mar 11 '17 at 17:32

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