Thanks Jeff. If I understand correctly, in general the Richardson-Springer map from $K\backslash G/B$ to the set of twisted involutions in the Weyl group is neither injective nor surjective. Can you please explain how one can characterize set for the orbits? For $(\mathrm{GL}_n,\mathrm{O}_n)$ one can prove bijectivity by an explicit construction, but I don't know what happens in geneal. It seems to me that existence of $K$-orbits corresponding to the "clans" of Oshima-Matsuki can be proved by a similar technique, but it is not clear to me why this method is supposed to produce ALL orbits.

Thanks for the references. However, I am not sure how those papers are related to the Matsuki-Oshima parameterisation by "clans". For example, my understanding is that the work of Springer (and Richardson-Springer) provides a correspondence between orbits and twisted involutions but this correspondence might not always be a bijection. If you could explain where one can find the connection between these works, or where the more concrete method of Matsuki-Oshima is done in more detail, that would be great.

Thanks for describing the construction of $\overline{H}$. I understand that it has a coproduct, but what is its antipode?

Thanks for your comment. At least for modules whose highest weights are in the root lattice the factor $1/2$ should go away. I am wondering if one can make 1 and 2 work in this special setting.

Thanks for the answer. Actually what I am looking for is a formula for $\mathrm{Prob}(d,a)$ that can be used for estimates as $d\to\infty$ and $a\to 0$. In fact I am interested in an integral over the variable $a$.