# Actions of finite groups on compact symmetric spaces

I am interested in series of finite subgroups of the classical compact simple Lie groups which have big orbits on compact symmetric spaces and where the double coset space has some nice explicit description.

Most interseting example for me is the case of complex Grassmannians. So to formalize a little bit...

For any two natural numbers $$k, n$$ with $$k \leq n$$ I want to have a finite subgroup $$G_{n, k} \leq U(n)$$ such that

1. The generic orbit of $$G_{n,k}$$ on $$U(n) / U(k)\times U(n-k)$$ is big.
2. There is a nice model for $$G_{n,k} \backslash U(n)/ U(k)\times U(n-k)$$ where I would like to consider the double coset space as a metric space with distance function inherited from the symmetric space.

The much studied examples of discrete cocompact subgroups of noncompact symmetric spaces are messing up my Google / Mathscinet searches. I would be grateful for any pointers to relevant literature.

• What do you mean by a "big orbit" of a finite group? – Friedrich Knop Feb 26 at 16:10
• That it has many points. I am not sure what kind of growth can expect with respect to $k$ and $n$ (or for $n$ while keeping the $k$ fixed) so it's hard for me to formalize it. I'm really just fishing for some examples at this point. – Vít Tuček Feb 26 at 18:22
• The size of a generic orbit is pretty much equal to the group order. – Friedrich Knop Feb 26 at 21:23
• How does one see that? – Vít Tuček Feb 27 at 18:47
• When a group acts effectively on a manifold then the fixed point set of every $g\ne1$ is a proper submanifold. So if $G$ is finite then the set of points with trivial stabilizer is the complement of finitely many proper submanifolds. In particular it is dense and open. – Friedrich Knop Feb 28 at 7:26