Central subalgebras of the centralizer of a nilpotent element

Let GG be a connected, semisimple algebraic group over a field kk whose characteristic is very good for GG. In a canonical manner, one associates to a nilpotent element X∈Lie(G)X \in \mathrm {Lie}(G) a parabolic subgroup PP – in characteristic zero, PP may be described using an sl2\mathfrak {sl}_2-triple containing XX; in general, PP is the “instability parabolic” for XX as in geometric invariant theory. In this setting, we are concerned with the center Z(C)Z(C) of the centralizer CC of XX in GG. Choose a Levi factor LL of PP, and write dd for the dimension of the center Z(L)Z(L). Finally, assume that the nilpotent element XX is even. In this case, we can deform Lie(L)\mathrm {Lie}(L) to Lie(C)\mathrm {Lie}(C), and our deformation produces a dd-dimensional subalgebra of Lie(Z(C))\mathrm {Lie}(Z(C)). Since Z(C)Z(C) is a smooth group scheme, it follows that dim⁡Z(C)≥d=dim⁡Z(L)\dim Z(C) \ge d = \dim Z(L). In fact, Lawther and Testerman have proved that dim⁡Z(C)=dim⁡Z(L)\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.