Geodesic
Double cosets of stabilizers of totally isotropic subspaces in a~special unitary group~I
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 30, Tome 452 (2016), pp. 86-107

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Let $D$ be a division algebra with a fixed involution and let $V$ be the corresponding unitary space over $D$ with $T$-condition (see [2]). For a pair of totally isotropic subspaces $u,v\leq V$ we consider the double cosets $P_u\gamma P_v$ of their stabilizers $P_u,P_v$ in $\Gamma=\mathrm{SU}(V)$. We give a description of cosets $P_u\gamma P_v$ in the terms of the intersection distance $d_\mathrm{in}(u,\gamma(v))$ and the Witt index of $u+\gamma(v)$. 
@article{ZNSL_2016_452_a4,
     author = {N. Gordeev and U. Rehmann},
     title = {Double cosets of stabilizers of totally isotropic subspaces in a~special unitary {group~I}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {86--107},
     publisher = {mathdoc},
     volume = {452},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_452_a4/}
}
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N. Gordeev; U. Rehmann. Double cosets of stabilizers of totally isotropic subspaces in a~special unitary group~I. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 30, Tome 452 (2016), pp. 86-107. http://geodesic.mathdoc.fr/item/ZNSL_2016_452_a4/