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We are considering the commuting variety of the Lie algebra pglₙ over an algebraically closed field of characteristic p >0, namely the set of pairs \ (A, B) pglₙ pglₙ A, B=0 \. We prove that if n=pr, then there are precisely two irreducible components, of dimensions n²+r-1 and n²+n-2.
Vlad Roman (Fri,) studied this question.