On nilpotent commuting varieties and cohomology of Frobenius kernels

Abstract The paper studies the dimensions of irreducible components of commuting varieties of (restricted) nilpotent r-tuples in a classical Lie algebra g defined over an algebraically closed field k. As applications, we obtain some new results on the structure of the (even) cohomology ring of Frobenius kernels G r for each r ≥ 1 , where G is the simply connected, simple algebraic group such that Lie ( G ) = g . Explicit calculations for rank two groups are also presented.