In this talk we are going to discuss some fine details about the linear (fractional) self maps and in particular, the actions of the automorphism groups of a kind of domains on complex Grassmannians which generalized the classical domains of type I. Among other things, we will look at the behavior of the fixed-point sets of linear self maps, including one which generalizeds the following well known result of the complex unit ball:   any linear fractional map with more than two fixed points on the boundary must have fixed points in the interior. This is a joint work with Ng and Seo.