Mixed Commuting Varieties over simple Lie algebras

Let $\mathfrak{g}$ be a simple Lie algebra defined over an algebraically closed field $k$ of characteristic $p$. Fix an integer $r>1$ and suppose that $V_1,\ldots,V_r$ are irreducible closed subvarieties of $\mathfrak{g}$. Let $C(V_1,\ldots,V_r)$ be the closed variety of all the pairwise commuting elements in $V_1\times\cdots\times V_r$. This paper studies the dimension and irreducibility of such varieties with various $V_i$ in a Lie algebra $\mathfrak{g}$. In particular, we complete the problem for the case when $V_i$'s are either $\overline{\mathcal{O}_{\text{sub}}}$ the closure of the subregular orbit or $\mathcal{N}$ the nilpotent cone of any rank two Lie algebra $\mathfrak{g}$. A result on the dimension of these mixed commuting varieties is generalized for higher ranks. Finally, we apply our calculations to study properties of support varieties for a simple module over the $r$-th Frobenius kernels of $G$.