All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a **semisimple** complex group. I am interested in whether the same results also apply to arbitrary **reductive** complex $G$? In particular, that one can make decomposition of $G/P$ into Schubert cells and that these cells are isomorphic to affine spaces.

Maybe it is true that the flag variety $G/P$ for an arbitrary reductive $G$ and parabolic $P$ is isomorphic to the flag variety $G'/P'$ for a semisimple $G'$ and a parabolic $P'\leq G,$ thus one can use the knowledge on the Schubert cells on $G'/P'$ in order to get the same for $G/P$?

isn'ttrue for reductive ones. (With obvious exceptions, like "the centre is finite" and "the group is semisimple".) $\endgroup$