Rational conjugacy classes of Frobenius stable tori (in a finite group of Lie type) are in bijection with Frobenius-conjugacy classes of the corresponding Weyl group.When the group is the Symplectic group Sp$(2n)$ then the Frobenius action on the Weyl group is trivial and classes of rational tori are in bijection with conjugacy classes of the type C Weyl group $W\_n$. These conjugacy classes are parametrized by bipartitions of n.


The groups of Frobenius-stable points, of tori belonging conjugacy class of rational tori, are all isomorphic. Is there a way of constructing these groups from the corresponding bipartition of n? I tried this for Sp(4) but I cannot make much sense of what I got. I also did this computation for GL(n), for this group it is possible (and quite simple). 


I hope someone can give me a hint. Thank you in advance. Hans.



Conjugacy classes of rational tori in Symplectic group
See Proposition 3.1 in <https://link.springer.com/article/10.1007%2Fs10469-007-0009-z>, for example.


*Buturlakin, A. A.; Grechkoseeva, M. A.*, [**The cyclic structure of maximal tori of the finite classical groups.**](http://dx.doi.org/10.1007/s10469-007-0009-z), Algebra Logika 46, No. 2, 129-156 (2007); translation in Algebra Logic 46, No. 2, 73-89 (2007). [ZBL1155.20047](https://zbmath.org/?q=an:1155.20047), [MR2356522](https://mathscinet.ams.org/mathscinet-getitem?mr=2356522).


