Commuting varieties of r-tuples over Lie algebras

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p and let g be the Lie algebra of G. It is well known that for p large enough the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of elements in the nilpotent cone of g (Suslin et al., 1997, [27]). In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties Cr(gl2), Cr(sl2) and Cr(N) where N is the nilpotent cone of sl2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of r-tuples. Furthermore, in the case when g=sl2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of sl3, we are able to verify the aforementioned properties for Cr(u). Finally, applying our calculations on the commuting variety Cr(Osub¯) where Osub¯ is the closure of the subregular orbit in the nilpotent cone of sl3, we prove that the nilpotent commuting variety Cr(N) has singularities of codimension ⩾2.