I am looking for generalizations of the following construction.
Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of $\theta$-fixed points. Then $G/H$ is known as a symmetric variety. The map $\tau: G \rightarrow G$ given by $\tau(g) = g \theta(g)^{-1}$ descends to an embedding $\overline{\tau} : G / H \hookrightarrow G$.
I would like to consider a more general setting. Let $G$ be a connected, reductive group and $H$ a spherical subgroup of $G$ (meaning that some Borel subgroup of $G$ has a dense orbit in $G/H$ under the usual left multiplication action). Let $G'$ be another reductive group. I am interested in examples of the following:
- a morphism of algebraic groups $\alpha : G \rightarrow G' \times G'$; let $\alpha_1, \alpha_2 : G \rightarrow G'$ be the corresponding morphisms obtained by composing with the two projections;
- a locally closed $\alpha$-equivariant embedding $f : G / H \hookrightarrow G'$, where $\alpha$-equivariance means that $f(g \cdot xH) = \alpha_1(g) f(xH) \alpha_2(g)^{-1}$.
The embedding of a symmetric variety $G/H$ into $G$ fits into this setup by taking $G' = G$ and $\alpha(g) = (g, \theta(g))$. I would like to ask if anyone knows of any other examples of this general construction in the literature, or results that would rule out such constructions in certain situations. My expectation is that, if such a construction exists for a wider class of spherical varieties, then the group $G'$ will likely be considerably larger than $G$.
Edit: The construction for symmetric varieties is used in work of R.W. Richardson and T.A. Springer (The Bruhat order on symmetric varieties and Combinatorics and geometry of K-orbits on the flag manifold). The generalized construction outlined above would possibly aid in a study of Bruhat order on a wider class of spherical varieties.
