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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.

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A reference or two would help fill in background. Does the field matter? Does the reductive case involve essentially more than the semisimple case? – Jim Humphreys Apr 16 '13 at 23:09
I am generally considering an algebraically closed field, and I am fine with characteristic zero assumptions if necessary. I don't think reductive adds anything important, but I tried to cast as wide a net as possible while fishing for examples. I'll add an edit to say a little about motivation. – Michael Joyce Apr 17 '13 at 2:43
Concerning the field, the default assumption is that the characteristic not be 2. I'm not sure what's been done by now, but a simple search of MathSciNet shows at least 20 relevant items in the subject area 20G including Springer's last papers and a mini-workshop at Oberwolfach in 2008. By now there are quite a few directions of research, involving algebraic geometry and partly inspired by classical symmetric spaces of Lie groups, so it's worth searching the literature. (However, the type of examples you want may not be there yet.) – Jim Humphreys Apr 17 '13 at 13:25

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