Optimal SL(2)-homomorphisms

Let G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf has associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X in Lie(G) lies in the image of the differential of a homomorphism SL(2) --> G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL(2) is optimal for X in Kempf's sense. We show here that any two SL(2)-homomorphisms which are optimal for X are conjugate under the connected centralizer of X. This implies, for example, that there is a unique conjugacy class of principal homomorphisms for G. We show that the image of an optimal SL(2)-homomorphism is a completely reducible subgroup of G; this is a notion defined recently by J-P. Serre. Finally, if G is defined over the (arbitrary) subfield K of k, and if X in Lie(G)(K) is a K-rational nilpotent element whose p-th power is 0, we show that there is an optimal homomorphism for X which is defined over K.Comment: AMS-LaTeX, 26 pages. To appear in Comment. Math. Helv. The most substantial modification found in the revision is a proof of the G(K)-conjugacy of any 2 optimal SL(2)-homomorphisms for X in Lie(G)(K) which are defined over K; see Prop/Def 21 and Theorem 4

Sub-principal homomorphisms in positive characteristic

Let G be a reductive group over an algebraically closed field of characteristic p, and let u in G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism phi:SL_2/k --> G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of phi generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, phi is obtained by reduction modulo P from a homomorphism of group schemes over a valuation ring in a number field. This permits us to show moreover that the weight spaces of a maximal torus of phi(SL_2/k) on Lie(G) are ``the same as in characteristic 0''; the existence of a phi with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000).Comment: 20 pages, AMS LaTeX. This version fixes some minor glitches, and includes a more detailed section 5.3. To appear in Math. Zeitschrif

Faithful representations of SL(2) over truncated Witt vectors

Let G be the six dimensional linear algebraic k-group SL_2(W_2), where W_2 is the ring of Witt vectors of length two over the algebraically closed field k of characteristic p>2. Then the minimal dimension of a faithful rational k-representation of G is p+3.Comment: New title. New section 9 contains a finite group version of the main result. 11 pages, AMS LaTeX. To appear in the Journal of Algebr

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, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 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 the projective line, 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.Comment: 12 pages. To appear in the Friedlander birthday volume of J. Pure and Applied Algebr

Completely reducible Lie subalgebras

Let G be a connected and reductive group over the algebraically closed field K. J-P. Serre has introduced the notion of a G-completely reducible subgroup H of G. In this note, we give a notion of G-complete reducibility -- G-cr for short -- for Lie subalgebras of Lie(G), and we show that if the closed subgroup H < G is G-cr, then Lie(H) is G-cr as well.Comment: 7 pages; AMS LaTeX. To appear in *Transformation Groups

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in Lie(G)(F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).Comment: 32 pages, AMSLaTeX. To appear: Math. Annalen. This version has a new title; it also contains various corrections of typographic errors and such. More significantly, it contains "cleaner" statements of convergence for unipotent (as opposed to nilpotent) orbital integrals; see sections 8.5-8.

Abelian Unipotent Subgroups of Reductive Groups

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebra of G. Second, if G is semisimple and p is sufficiently large, we show that G always has a faithful representation (r,V) with the property that the exponential of dr(X) lies in r(G) for each p-nilpotent X in Lie(G). This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the Frobenius kernels G_d, d > 1. The previous authors already observed that the natural representation of a classical group has the above property (with no restriction on p). Our methods apply to any Chevalley group and hence give the result also for quasisimple groups with ``exceptional type'' root systems. The methods give explicit sufficient conditions on p; for an adjoint semisimple G with Coxeter number h, the condition p > 2h -2 is always good enough.Comment: 27 pages; AMS LaTeX. This version fixes an error in section 7 (the fix makes the main result of section 9 true only under a condition on the prime). Moreover, it contains a number of changes in expositio