K-orbit closures and Barbasch–Evens–Magyar varieties

We define the Barbasch-Evens-Magyar varieties. show they are isomorphic to smooth varieties defined in [D.~Barbasch-S.~Evens '94] that map generically finitely symmetric orbit closures, thereby giving resolutions of singularities certain cases. Our definition parallels [P.~Magyar '98]'s construction Bott-Samelson [H.~C.~Hansen '73, M.~Demazure '74]. From this alternative viewpoint, one deduces a graphical description type $A$, stratification into closed subvarieties same kind, and determination torus-fixed points. Moreover, we explain how these manifolds inherit natural symplectic structure with Hamiltonian torus action. then express moment polytope terms variety.