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

Abstract

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 FG (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.