Sur la dimension de l'espace des orbites d'une grassmannienne sous l'action d'un groupe algébrique

Abstract

We are interested in the actions of an algebraic group G over the grassmannians of a finite dimensional K-vector space V (K algebraically closed) deduced from an action of G over V. We prove that the dimension of the orbit space of G(i,V) is smaller than that of G(j,V) if and only if the dimension of G(i,V) is smaller than that of G(j,V). We then get similar results for flag varieties. Different methods allow us to get results about the number of orbits when the ground field is finite.