On $${\mathbb{F}_q}$$ -Rational Structure of Nilpotent Orbits in the Witt Algebra.

Let $${\mathfrak{g}=W_1}$$ be the p-dimensional Witt algebra over an algebraically closed field $${k=\overline{\mathbb{F}}_q}$$ , where p > 3 is a prime and q is a power of p. Let G be the automorphism group of $${\mathfrak{g}}$$ . The Frobenius morphism F (resp. $${F_\mathfrak{g}}$$ ) can be defined naturally on G (resp. $${\mathfrak{g}}$$ ). In this paper, we determine the $${F_\mathfrak{g}}$$ -stable G-orbits in $${\mathfrak{g}}$$ . Furthermore, the number of $${\mathbb{F}_q}$$ -rational points in each $${F_\mathfrak{g}}$$ -stable orbit is precisely given. Consequently, we obtain the number of $${\mathbb{F}_q}$$ -rational points in the nilpotent variety. [ABSTRACT FROM AUTHOR]