Orbits in real loci of spherical varieties

The talk is based on a joint work in progress with S. Cupit-Foutou. Given a spherical variety $X$ for a complex reductive group $G$ defined over real numbers, we address the problem of describing orbits of the real Lie group $G(R) $ in the real locus $X(R)$. (There may be several real orbits even if $X$ is $G$-homogeneous.) We concentrate on two cases: (1) $X$ is a symmetric space; (2) $G$ is split over $R$ and $X$ is $G$-homogeneous. The answer is similar in both cases: the $G(R)$-orbits are classified by the orbits of a finite reflection group $W_X$ (the “little Weyl group”) acting in a fancy way on the set of orbits of $T(R)$ in $Z(R)$, where $T$ is a maximal torus in $G$ and $Z$ is a “Brion–Luna–Vust slice” in $X$. The latter orbit set can be described combinatorially. We use different tools: Galois cohomology in (1) and Knop's theory of polarized cotangent bundle in (2), and we expect that the second approach can be extended to the non-split case.