Fixed point spaces in actions of exceptional algebraic groups

Abstract

Let G be a simple algebraic group of exceptional type acting transitively on an algebraic variety. We provide estimates for the dimensions of the subvarieties of fixed points of elements of G. These translate into estimates for the dimensions of intersections of conjugacy classes of G with closed subgroups.Introduction.

Fixed point ratios in actions of finite exceptional groups of Lie type

“…In Case (II), where H is a subgroup of maximal rank, we may as above take X = G σ , and so H = M σ with M reductive of maximal rank. For this case we use results from the paper [40], in which we considered an analogous question for the exceptional algebraic groups G. For these groups the quantity analogous to the fixed point ratio is…”

“…In [40], upper bounds are obtained for −f (x, G/M ). By passing from the algebraic groups G to the finite groups G σ , we are able in Section 4 to use these dimension bounds to obtain the bounds for fixed point ratios recorded in column (II) of Table 2.…”

“…There are not many possibilities under (i) or (ii), and they are dealt with fairly easily using the methods of [40]. The subgroups in (iii) are handled in Section 5 using some rather lengthy ad hoc arguments.…”

“…In the proof we shall make heavy use of the main result in [40], namely [40,Theorem 2]; for x = s, u this result provides strong lower bounds for the quantities…”

“…The next result is our main tool for passing from the dimension bounds of [40,Theorem 2] to the bounds for the finite groups G σ that we require. In the statement, we denote by f the order of the fundamental group of G (so f = 2, 3 for G = E 7 , E 6 respectively, and f = 1 otherwise).…”