On Abstract Homomorphisms of Chevalley Groups with Nonreductive Image, I

Abstract

homomorphisms of chevalley groups 375 defined over the field of constants k0 = x ∈ k δ x = 0 , one can consider a homomorphism ηδ G k → G K K ηδ g = g δ g where the semi-direct product is formed using the action of G on its Lie algebra via the adjoint representation, δ g = g−1 δ g , and δ g is obtained by applying δ to every matrix entry of g; moreover, if δ is nontrivial, ηδ has a Zariski dense, hence nonreductive (as is the unipotent radical of G ), image (for details cf. Section 2). In [T], Tits formulated a general conjecture that, under sufficiently general hypotheses on G and k and without assuming G′ to be reductive, for any abstract homomorphism φ G k → G′ k′ there should exist a commutative finitedimensional k′-algebra A and a ring homomorphism α k → A such that φ can be written as φ = ψ ◦ rA/k′ ◦ α, where α G k → AG A is induced by α (AG is the group obtained by the change of scalars), rA/k′ AG A → RA/k′ AG k′ is the canonical isomorphism (RA/k is the functor of restriction of scalars), and ψ is a rational k′-morphism of RA/k′ AG to G′. In the same paper Tits proved this conjecture for k = k′ = and also announced its truth for G a simple simply connected split k-group if k is not a nonperfect field of characteristic two. However, this result still leaves open the question about an explicit description of abstract homomorphisms as one would like to know precisely which algebras A and which rational homomorphisms ψ can actually arise. Subsequently, abstract homomorphisms with nonreductive images were not analyzed (to the best of our knowledge). The goal of this paper is twofold. First, we show that if G is an absolutely simple simply connected split (in other words, Chevalley) group over a field k of characteristic zero, then any homomorphism of G k such that the Zariski closure of its image has a commutative unipotent radical can be obtained from Borel–Tits’ construction. This result does not depend on Tits’ result [T] and gives an explicit description of such homomorphisms. Second, we describe a generalization of Borle–Tits’ construction which allows one to construct abstract homomorphisms for which the unipotent radical of the Zariski closure of the image has arbitrarily large dimension and nilpotency class. For abstract homomorphisms whose image has a commutative unipotent radical we prove the following. Theorem 3. Let G be a simple simply connected Chevalley group over a field k of characteristic zero. Furthermore, let be a connected algebraic group over an extension K of k and μ G k → K be an abstract homomorphism 376 lifschitz and rapinchuk with Zariski dense in image. Assume that: (1) the unipotent radical V = Ru is commutative, and (2) if G′ = /V , then the composition G k → K → G′ K of μ with the canonical morphism → G′ extends to a rational K-homomorphism λ G → G′.