# 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

buildings.

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

English

2024/04/29