Central subalgebras of the centralizer of a nilpotent element

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 is an element of Lie(G) a parabolic subgroup P - in characteristic zero, P may be described using an 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 Lie(L) to Lie(C), and our deformation produces a d-dimensional subalgebra of Lie(Z(C)). Since Z(C) is a smooth group scheme, it follows that dim Z(C) >= 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.