Let \(\mathfrak{g}\) denote a classical Lie algebra (type \(A\), \(B\), \(C\), or \(D\)) over an algebraically closed field \(k\) of characteristic \(p > 0\). That is, \(\mathfrak{g}\) is either \(\mathfrak{sl}_{n}\), \(\mathfrak{so}_{2n + 1}\), \(\mathfrak{sp}_{2n}\), or \(\mathfrak{so}_{2n}\). Further it is assumed that \(p \neq 2\) in the latter three cases. Associated to \(\mathfrak{g}\) (which is a restricted Lie algebra) is the nilpotent cone \({\mathcal N}\) and the restricted nullcone \({\mathcal N}_1 := \{x \in {\mathfrak g}~ |~ x^{[p]} = 0\} \subset {\mathcal N}\). Note that \({\mathcal N}_1 = {\mathcal N}\) if \(p\) is at least the Coxeter number. Let \(V \subset {\mathfrak g}\) denote a closed subvariety. Given a positive integer \(r\), associated to \(V\) is the commuting variety \(C_r(V) \subset V^r\) consisting of all pairwise commuting \(r\)-tuples of elements in \(V\). Of particular interest in the work under review are the commuting varieties \(C_r({\mathcal N})\) and \(C_r({\mathcal N}_1)\).
In [Invent. Math. 154, No. 3, 653–683 (2003; Zbl 1068.17006)], A. Premet (in greater generality than considered here) gave a precise description of the variety \(C_2({\mathcal N})\), from which one concludes that it is equidimensional. For larger \(r\), much less is known. The author showed in his doctoral thesis that \(C_r({\mathcal N})\) is irreducible (hence equidimensional) for all \(r\) for \(\mathfrak{sl}_2\) or \(\mathfrak{sl}_3\). On the other hand, the author with K. Šivic [Linear Algebra Appl. 452, 237–262 (2014; Zbl 1291.15045)] showed for \(\mathfrak{sl}_n\) that \(C_r({\mathcal N})\) is reducible for \(r, n\) sufficiently large. In the work under review, the author shows for sufficiently large \(r\) (with \({\mathfrak g}\) as above) that the variety \(C_r({\mathcal N})\) is in fact not equidimensional. Explicit bounds on \(r\) are given for each type (approximately \(r > 3\)).
This conclusion is achieved by making a number of explicit computations of dimensions. First, using regular elements, the author identifies an irreducible component of \(C_r({\mathcal N})\) and computes its dimension. Then, using certain matrices that square to zero, the author constructs another closed subvariety of \(C_r({\mathcal N})\) whose dimension is also computable. For sufficiently large \(r\), this latter variety has larger dimension than the first, thus giving the desired conclusion. Further, the latter subvariety is always contained within \(C_r({\mathcal N}_1)\) and so can be used to give a lower bound on the dimension of \(C_r({\mathcal N}_1)\). From this, the author explicitly computes the dimension of \(C_r({\mathcal N}_1)\) in type \(A\) for \(p = 2\) (i.e., \(\mathfrak{sl}_n\) for arbitrary \(n\)) and for rank 2 Lie algebras for \(p > 2\) (i.e., \(\mathfrak{sl}_3\) or \(\mathfrak{sp}_4\)). Throughout this discussion, computations are also made with \(\mathfrak{g}\) replaced by its subalgebra of strictly upper triangular matrices.
Let \(G\) be a simple, simply connected algebraic group with \(\text{Lie}(G) = \mathfrak{g}\) and \(G_r\) denote the \(r\)th Frobenius kernel of \(G\). The spectrum of the even-dimensional part of the cohomology ring \(\text{H}^{\bullet}(G_r,k)\) can be identified with the commuting variety \(C_r({\mathcal N}_1)\). Using in part the preceding work on dimensions of commuting varieties, the author also obtains new information on \(\text{H}^{\bullet}(G_r,k)\). For example, while \(\text{H}^{\bullet}(G_r,k)\) is shown to be Cohen-Macaulay when \(G = \mathrm{SL}_2\), for larger rank groups and sufficiently large \(r\), it cannot be (based on the lack of equidimensionality of the commuting variety). The author also gives bounds on Krull dimensions of cohomology rings and the complexity of certain modules. Computations are also made for the cohomology of Frobenius kernels of a Borel subgroup of \(G\) (and the unipotent radical of a Borel).