Strong Transitivity and Weyl Transitivity of Group Actions on Affine Buildings

Let H be a group acting on a building ∆. We analyze three transitivity properties that this action could have, namely strong, Weyl and weak transitivity. We present and analyze a collection of groups H and buildings ∆ for which the action is not weakly transitive but may nonetheless be Weyl transitive. In these examples, the failure to be weakly transitive is in some sense precisely determined, and in some cases is shown to be extremely severe.

The first situation we consider is Chevalley groups. Let K be a local field and G = g(K) a Chevalley group. Let (B,N) be the standard spherical BN-pair and W = N/B ∩ N the Weyl group. We precisely characterize which elements w of W admit only finite-order representatives in N. In particular for such a w of order m, all representatives of w in N have the same order, and that order is either m or 2m. Using this we can find a variety of subgroups H of G, in particular if H is dense and torsionfree, such that H acts Weyl transitively but not weakly transitively on the affine building arising from G.

Next we consider the case of division algebras, where the failure to be weakly transitive can be more precisely characterized and shown to be very extreme. Let D be a finite-dimensional F-division algebra of degree d > 2, and let H be either D× or SL1(D). For any splitting field K, H admits an action on the buildings associated to G = GLd(K) or G = SLd(K). It is easy to show that this action is not weakly transitive, and in the present context we can show that it even fails “dramatically” to be weakly transitive. If F is a global field we can construct examples where the action of H on the affine building of G is nonetheless Weyl transitive. In the global case, for “most” D we can even show that SL1(D) acts on the fundamental affine apartment only by translations - the most extreme possible situation.