Commuting varieties are classical objects of study in Lie theory. Historically,
the first result was the proof by Motzkin-Taussky in 1955 (established independently
by Gerstenhaber) that the variety of pairs of commuting n x n matrices is irreducible.
Richardson extended this to an arbitrary reductive Lie algebra in characteristic zero.
More recently there has been interest in the subvariety of pairs of commuting nilpotent
elements. One highlight was Premet's proof that this nilpotent commuting variety of
a reductive Lie algebra is equidimensional, and is irreducible for \gl_n.
In positive characteristic, the nilpotent commuting variety is related to cohomology
of the second Frobenius kernel of the corresponding group by a result of
Suslin-Friedlander-Bendel.
Motivated by this connection, I will explore two variations on the theme. First of all,
I will summarize some recent results on the variety of r-tuples of commuting nilpotent
elements of \gl_n or \sp_{2n}. This is joint work with N. Ngo and K. Sivic. Secondly,
I will outline some results on the irreducible components of certain generalisations
of the nilpotent commuting variety, for which we confine the first coordinate to
a fixed nilpotent orbit closure. This is joint work with N. Ngo.