On linearity of some automorphism groups of projective varieties

We are going to prove the following recent result of M. Brion. Let $G$ be a linear algebraic group acting on a normal projective variety $X$ by automorphisms. Then the subgroup of all automorphisms, fixing the field of $G$-invariant rational functions pointwise, is linear algebraic as well. Also we will consider some applications of this theorem to structure of automorphism groups.