Unipotent Conjugacy in General Linear Groups

Hence, it is easy to count the orbits of GL n q under the conjugation action of U n q but seems hard to do the same for the group U n q itself! We fix some further notation. Let V n q be the vector space of column vectors, a module for GL n q . Recall that a flag is a totally ordered set of n− 1 nonzero proper subspaces of V n q . For g in GL n q let f g be the number of flags fixed by g, so it is zero unless g is conjugate with an element of B n q the group of upper triangular matrices. For a unipotent element u of GL n q let u be the partition of n given by the Jordan form of u.