On the Centralizer of the Sum of Commuting Nilpotent Elements

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V , where K is a field of characteristic p ≥ 0. If F = K(t) is the field of rational functions on the projective line P1/K , consider the K(t)-endomorphism A = X+ tY of V . If p = 0, or if A = 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V ). For all geometric points (a : b) of a suitable open subset of P, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX + bY . This answers a question of J. Pevtsova. Let G be a connected and reductive algebraic group defined over an arbitrary field K of characteristic p ≥ 0. Write g = Lie(G), and consider the extension field F = K(t) with t transcendental over K. For convenience, we fix an algebraically closed field k containing both K and t. If X,Y ∈ g(K) are nilpotent and [X,Y ] = 0, then A = X + tY ∈ g(F ) is again nilpotent. Write C for the centralizer of A in G, and write RuC for the unipotent radical of C. Under favorable restrictions on the characteristic, the groups C and RuC are defined over K(t). In this note, I want to answer – at least in part – a question put to me by Julia Pevtsova at the July 2004 meeting in Snowbird, Utah. With notation as before, this question may be stated as follows: Question 1. When is it true that X,Y ∈ LieRuC? To begin the investigation, the first section of the paper includes some elementary results concerning G-varieties in case the algebraic group G acts with a finite number of orbits. For the most part, the use of these results could be avoided in the present application, but there is perhaps some interest in recording them. After these preliminaries, I am mainly going to investigate Question 1 in case the K-group is G = GL(V ), where V is a finite dimensional k-vector space defined over K; this means there is a given K-subspace V (K) for which the inclusion induces an isomorphism V (K)⊗K k ≃ V . The second section contains well-known material on nilpotent orbits, mainly for the group GL(V ); this material is used in section three where we prove our main result – Date: April 12, 2005. 1991 Mathematics Subject Classification. 20G15.