Normality of closure of nilpotent conjugacy classes

In this work the author studies the geometry of the conjugacy classes in the space of matrices and in its subspaces of symmetric and skew-symmetric matrices under the actions of the general linear group, the orthogonal group and the symplectic group. An extensive and self-contained review of the current state of the art concerning the description of nilpotent conjugacy classes and their closures is provided. Many geometrical properties of the closure of a conjugacy class are interesting from a representation theoretic viewpoint; in particular, their normality. A complete discussion of the normality of the closures of the conjugacy classes for the stated actions takes a prominent role in this work. The main new result completes the picture and it states that a nilpotent symmetric conjugacy class for the orthogonal group has normal closure if and only if the associated partition has consecutive parts of length differing by at most one.