Reducibility of commuting varieties of elements of simple Lie algebra

Abstract

In this paper, we prove that the variety $C_m(L)$ of commuting $m$-tuples of elements of simple Lie algebra $L$ is often reducible. Explicitely, we prove it is reducible for all simple Lie algebra $L$ not isomorphic to $\mathfrak{sl}_2$ and $\mathfrak{sl})_3$, and all $m \geq 4$. We also prove it is reducible for $C_3(L)$ for $L$ of types $B_k,C_k,E_7,E_8,F_4,G_2$, $k \geq 2$, as well as for $D_l$ for $l \geq 10$. We do this by proving Theorem on Adding Diagonals, that says that if we can find a simple Lie subalgebra $L'$ whose Dynkin diagram is a subdiagram of the Dynkin diagram of $L$, then under mild conditions, from the fact that $C_m(L')$ is reducible, it follows that $C_m(L)$ is also reducible.