Central subalgebras of the centralizer of a nilpotent element

Abstract:

Let $G$ be a connected, semisimple algebraic group over a field $k$ whose characteristic is very good for $G$. In a canonical manner, one associates to a nilpotent element $X \in \mathrm {Lie}(G)$ a parabolic subgroup $P$ – in characteristic zero, $P$ may be described using an $\mathfrak {sl}_2$-triple containing $X$; in general, $P$ is the “instability parabolic” for $X$ as in geometric invariant theory.

In this setting, we are concerned with the center $Z(C)$ of the centralizer $C$ of $X$ in $G$. Choose a Levi factor $L$ of $P$, and write $d$ for the dimension of the center $Z(L)$. Finally, assume that the nilpotent element $X$ is even. In this case, we can deform $\mathrm {Lie}(L)$ to $\mathrm {Lie}(C)$, and our deformation produces a $d$-dimensional subalgebra of $\mathrm {Lie}(Z(C))$. Since $Z(C)$ is a smooth group scheme, it follows that $\dim Z(C) \ge d = \dim Z(L)$.

In fact, Lawther and Testerman have proved that $\dim Z(C) = \dim Z(L)$. Despite only yielding a partial result, the interest in the method found in the present work is that it avoids the extensive case-checking carried out by Lawther and Testerman.