The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$

Abstract

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=\lambda I, \lambda \in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.