Counting Number Of Rational Points Of Nilpotent Orbits Of Reductive Lie Algebras In Finite Fields

In this thesis, we will try to count the numers of rational points of nilpotent orbits of reductive Lie algebras under Probenius maps in finite fields. The theory of nilpotent orbits plays an important role in the reprentation of modular Lie algebras and Lie groups . The topic itself has a great attraction to Mathamatians because of its own interest as a kind of geometry of Lie theory. We will concenntrate our arguments on the following problems: 1. we will give a direct formular for gl(n, k) to calculate the number of rational points of nilpotent orbits in the finite fields. 2. we generlize the Kawanaka's formular to the closure of nilpotent orbits. 3. we give some discrimination for orbits with the same numer of rational points. 4. we study the properties of the nilpotent orbits of gl(n,k).In this thesis, we give these problems a solution.