There is a nice theory of discrete reflection groups and discrete generalized reflection groups. What about compact Lie groups (subgroups of U(n) ) that are generated by reflections ( or generalized reflections)? the unitary and orthogonal groups are examples. Have such groups been studied/ classified? References?
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1$\begingroup$ Two trivial remarks: 1. If the subgroup G is simple as an abstract group, this is just the question on whether G contains any (complex) reflections at all. 2. One then asks which irreducible representations $G\to U(n)$ sends any involution to a reflection. The latter question probably has the answer "hardly any", but making this precise would require doing some computations. $\endgroup$– MishaCommented May 14, 2013 at 3:26
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$\begingroup$ Indeed, if we compose $G \to U(V) \to U(V \otimes {\mathbb C}^2)$ then $G$ is not such a group. Hi Harm, welcome to MO! $\endgroup$– Allen KnutsonCommented May 14, 2013 at 3:53
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1$\begingroup$ I don't have a useful answer to your question, but I vaguely recall at least one conference talk in which $O(n)$ was compared, with quite some success, to a finite reflection group, so this question is not entirely out of left field. $\endgroup$– Theo Johnson-FreydCommented May 14, 2013 at 4:16
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$\begingroup$ There are no simple reflection Lie groups. If $G$ is a simple reflection Lie group, then it is generated by commutators and commutators lie in $SU(n)$. So G is contained in $SU(n)$ but $SU(n)$ does not contain any (generalized) reflections. But perhaps one may consider the case for which $G/[G,G]$ is finite. $\endgroup$– Harm DerksenCommented May 14, 2013 at 15:12
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