Mixed Commuting Varieties over simple Lie algebras

TL;DR

This paper investigates the structure, dimension, and irreducibility of mixed commuting varieties over simple Lie algebras, extending known results and applying findings to support varieties of modules over Frobenius kernels.

Contribution

It completes the classification of mixed commuting varieties for rank two Lie algebras and generalizes dimension results to higher ranks, with applications to representation theory.

Findings

Determined irreducibility and dimension for specific subvarieties in rank two Lie algebras.

Extended dimension bounds of mixed commuting varieties to higher rank Lie algebras.

Applied results to properties of support varieties for modules over Frobenius kernels.

Abstract

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 $G$.