Abstract involutions of algebraic groups and of Kac–Moody groups

Abstract

Abstract. Based on the second author's thesis [33], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [32] and [49] for algebraic groups and in [39] for certain Kac-Moody groups, we analyze the filtration studied in [24] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [7].

The Combinatorics of Automorphisms and Opposition in Generalised Polygons

“…For each possibility we compute y = 2592 mod 3240 (using Lemma 8.13), and so y = 2592. The cases (f 2 , f 3 ) = (20,13), (24,18) give X 4 (p 0 ) < 0. Thus the only surviving cases are (f 1 , f 2 , f 3 ) = (4, 12, 3), (4,16,8) with x = 0 and y = 2592.…”

On flips of unitary buildings I: Classification of flips

“…Flips are studied in a more general context in [9] and [8] where their properties are explored, however the authors do not make a closer study of flips of the building under consideration here.…”

Infinite generation of non-cocompact lattices on right-angled buildings

“…(c) By Section 7.3 of Gramlich, Horn and Mühlherr [7], the fixed set G  of certain involutions  of ƒ is a lattice in Aut.X C /, which is sometimes cocompact and sometimes non-cocompact. Moreover, by [7,Remark 7.13], there exists  such that G  is not finitely generated.…”