The nilpotent commuting variety of the Witt algebra

Let gg be the Witt algebra over an algebraically closed field k   of characteristic p>3p>3. Let N={x∈g|x[p]=0}N={x∈g|x[p]=0} be the nilpotent variety of gg, and C(N):={(x,y)∈N×N|[x,y]=0}C(N):={(x,y)∈N×N|[x,y]=0} the nilpotent commuting variety of gg. As an analogue of Premet's result in the case of classical Lie algebras (Premet (2003) [6]), we show that the variety C(N)C(N) is reducible and equidimensional. Irreducible components of C(N)C(N) and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.