Triviality of certain automorphisms of semi-simple groups over local fields

Abstract

The object of this article is to prove the following: Theorem A. Let n be a positive integer. For i ~ n let Gi be a connected semisimple algebraic group with trivial center, and defined, simple and isotropie over a local field (i,e. a non-discrete locally compact topological field) k i. Let Gi be the group of ki-rational points of G i with the natural locally compact Hausdorff topology. Let G = [I~= 1 Gi, and let F be a closed subgroup of G such that the homogeneous space G/F carries a .finite G-invariant Borel measure. Let q~ be a continuous automorphism of G such that the restriction of cp to F is the identity automorphism of F. Then (p is trivial. We recall that in [7] we have proved the following strong rigidity theorem. Theorem B. (see [7, Theorem 8.7]) Let n, n' be positive integers. For i<=n (resp. j <= n'), let Hi (resp. H~) be a connected, adjoint, semi-simple algebraic group defined, simple and isotropic over a non-archimedean local field k i (resp. k)), and let Hi (resp. Hj) be the group of k i (resp. k))-rational points of H i (resp. Hj) with the natural locally compact Hausdorff topology. Let H = l-I~= t Hi, H '= [if= ~ Hi, and let ni (resp. n}) be the natural projection of H on Hi (resp. of H' on Hi). Let A (resp. A') be a discrete subgroup of H (resp. H') such that H/A (resp. H/A') is compact. Let O: A ~ A ' be an isomorphism. Then n=n'. Now assume that for i (resp.j) <=n such that ki-rankH i = 1 (resp. k)-rankH}= 1), the closure of ni(A ) (resp. n)(A')) contains H + (resp. H~+). Then there is an isomorphism O: H ~ H ' , of topological groups, such that 0 is the restriction of 0 to A. (For unexplained notation see Preliminaries below.)