I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of the dominant weights of root systems of type $B_n$ or $C_n$. So far I haven't found an accessible reference that draws complete analogy with the case of $U(n)$ and $S_n$. I am also interested in the analogue of symmetric functions in the case of $H_n$. Any pointer to literature would be greatly appreciated.

The $O(n)$ version of SchurWeyl duality involves Brauer algebras, the structure of which was not worked out completely until the 1980s by Hans Wenzl (Ann. of Math. (2) 128 (1988), no. 1, 173–193.) So perhaps you are looking for $H_n$ as a subgroup of the Brauer algebras? Googling "Brauer algebra" and "Hyperoctahedral group" gives: this recent preprint for example. 

