On a theorem of Schmid

Abstract

We establish for which parabolic subgroups $P$ of a simply connected and semisimple algebraic group $G$ with unipotent radical $U$ and Levi factor $H$ the two rings $\mathbb{k}[G/H{]}^U$ and $\mathbb{k}[U^{-}]$ are isomorphic as $H$ algebras. We show the relation of this problem with a Theorem of Schmid and we compare the multiplications in the rings $\mathbb{k}[U^{-}]$ and $\mathbb{k}[G/H]$.