  Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with
maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a
generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric
variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the
classification of smooth Fano generic closures given by Voskresenski\u{\i} and
Klyachko. As an application, we apply the well known correspondence between
Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit
which reflexive polytopes correspond to generic closures -- this list includes
the reflexive root polytopes.

Algebraic Geometry;Combinatorics;Group Theory;Representation Theory