Root group data (RGD) systems of affine type for significant subgroups of isotropic reductive groups over k[t,1/t]

Zhang, Yuan, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Abramenko, Peter, AS-Mathematics (MATH), University of Virginia

Given a Chevally group G, G = G(k[t,1/t]) can be seen as the k[t,1/t]-points of a linear algebraic
group, and as a Kac-Moody group over k of affine type (in Tits' sense) as well. In [27], Tits provided
an RGD system of not necessarily spherical type for Kac-Moody groups of \split" type over fields
(in Tits' sense, which covers the case of G we just described) which give rise to twin BN pairs and
hence twin buildings. This has many applications, chief among which the study of action on twin

However, for isotropic reductive k-group G, G = G(k[t,1/t]) is no longer a Kac-Moody group over k
in Tits' sense. But Tits mentioned without concrete proof that (see [26, sec: 3.2] and [26, sec: 3.3])
his RGD axioms will also apply to this more general situation when G is almost simple and simply
connected. Neither the proof nor the construction justifying this claim is present in the literature
as of now. This brings us to the main goal of this thesis: Provide a concrete construction of RGD
system in this general reductive case for some significant subgroups (the elementary subgroup
G(k[t,1/t])+, and a larger subgroup G(k[t,1/t])+CG (S)(k)) of G. Note, in case that G is simply
connected, and any normal semisimple subgroup of G has k-rank at least 2, we in fact can see
G(k[t,1/t]) = G(k)G(k[t,1/t])+ = G(k[t,1/t])+CG (S)(k) (see [21, cor:6.2]).

PHD (Doctor of Philosophy)
Group theory, Algebraic groups, Building theory, Combinatorics
Issued Date: