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How to write a Haar measure on $\operatorname{SO}(n)$ and $\operatorname{SU}(n)$ given Haar measure on $\operatorname{GL}(n)\,$?

I know that $\operatorname{GL}(n)$ has the Haar measure $(\det(A)^{-1} dA)$.

I try to write from these a Haar measure for the compact groups $\operatorname{SO}(n)$ and $\operatorname{SU}(n)$.

Samuel Adrian Antz
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Francisca
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    This post might be helpful. – Jake28 Feb 12 '22 at 14:33
  • something withouth spheric coordinates? – Francisca Feb 12 '22 at 15:00
  • If you want a density for the Haar measure, you will have to use some coordinates. – Jake28 Feb 12 '22 at 15:42
  • using classical coordinates? – Francisca Feb 12 '22 at 17:45
  • I don't really understand your question. Note that $SO(n)$ is a submanifold of $\mathbb{R}^{n^2}$ and not an open set like $GL(n)$. In particular, $SO(n)$ has Lebesgue measure zero (as a subset of $\mathbb{R}^{n^2}$) and therefore the Haar measure on $SO(n)$ cannot have a density with respect to the Lebesgue measure on $\mathbb{R}^{n^2}$. – Jake28 Feb 12 '22 at 18:02
  • I believe there is a typo in your formula for the Haar measure on the general linear group; see https://math.stackexchange.com/q/3815324/169085. – Alp Uzman Feb 12 '22 at 18:29
  • Further, a method of getting the density for the Haar measure on the special linear group would be through the Iwasawa decomposition. See Howe & Tan's book "Non-Abelian Harmonic Analysis", p.206, Prop.1.1.11 for the $\operatorname{SL}(2,\mathbb{R})$ case and Wallach's book "Harmonic Analysis on Homogeneous Spaces", p.177, Prop.7.6.4 for the general (connected) semisimple group case. – Alp Uzman Feb 12 '22 at 18:32

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