Galleries for Root Subsystems

We consider the operations of projection and lifting of Weyl chambers to and from a root subsystems of a finite roots system. Extending these operations to labeled galleries, we produce pairs of such galleries that satisfy some common wall crossing properties. These pairs give rise to certain morphisms in the category of Bott-Samelson varieties earlier considered by the author. We prove here that all these morphisms define embeddings of Bott-Samelson varieties (considered in the original interpretation based on compact Lie groups due to Raoul Bott and Hans Samelson) skew invariant with respect to the compact torus. We prove that those embeddings that come from projection and lifting preserve two natural orders on the set of the points fixed by the compact torus. We also consider the application of these embeddings to equivariant cohomology. The operations of projection and lifting can also be applied separately to each segment of a gallery. We describe conditions that allow us to glue together the galleries obtained this way.