An analogue of Steinberg theory for symmetric pairs

The double flag variety $G/B\times G/B$ of a reductive group $G$ has a finite number of $G$-orbits parametrized by the Weyl group $W$. Relying on moment maps, one gets a map from the Weyl group to the set of nilpotent orbits of $\mathrm{Lie}(G)$, often called the Steinberg map. In type $\mathsf A$, this map can be computed explicitly in terms of classical combinatorial algorithms, namely the Robinson–Schensted correspondence. In this talk, we consider a double flag variety $X=G/P\times K/Q$ associated to a symmetric pair $(G,K)$. Following Steinberg's approach, we define two maps to the sets of nilpotent $K$-orbits of $\mathrm{Lie}(K)$ and of its Cartan complement, respectively. We focus on type $\mathsf{A}\mathrm{III}$, the orbits of $X$ are then parametrized by a set of pairs of partial permutations. We compute the two maps by relying on a combinatorial procedure that extends the classical Robinson–Schensted correspondence.
The talk is based on a joint work with Kyo Nishiyama; see arXiv:2103.08460, IMRN 2022, no. 1, 1–62.