Are the canonical actions on Schubert Cells Linearizable?
G. Schwarz constructed a (counter)example for an action of a simple algebraic group on an affine space that is not linearizable (i.e., it is not a representations).
Natural examples of affine spaces that are not readily vector spaces are Schubert cells. So it was tempting to look for reductive group actions on them and see if they can lead to more counter-examples.
For a parabolic subgroup $P$ of a linear algebraic group $G$,
(say $G$ semi-simple) we can take a Schubert cell $C\subset G/P$.
By definition $C$ is the orbit for a (maximal) solvable subgroup
In a 5-minute meeting with M. Brion I asked this question and he said `Yes, it would follow from the slice theorem'. Can any one elaborate on his brief answer?