Algebraic automorphisms of algebraic groups with stable maximal tori

Sarah J. Gottlieb

Vol. 72 (1977), No. 2, 461–470

DOI: 10.2140/pjm.1977.72.461

Abstract

Let T₁ and T₂ be maximal tori of a connected linear algebraic group G ⊆ GL(n,κ), and suppose some (algebraic group) automorphism σ of G stabilizes both T₁ and T₂. Suppose further that σ also stabilizes two Borel subgroups, B₁ and B₂, of G. This paper is about the following natural questions:

1. Are T₁ and T₂ conjugate by a σ-fixed point of G?
2. Are B₁ and B₂ conjugate by a σ-fixed point of G?
3. If T_i ⊆ B_i, (i = 1,2), are the T_i and B_i respectively conjugate by a single σ-fixed point of G?
4. Are at least T₁ and T₂ described in (3) above conjugate by a σ-fixed point of G?

Mathematical Subject Classification 2000

Primary: 20G99

Secondary: 14L99

Milestones

Received: 12 January 1977

Revised: 23 March 1977

Published: 1 October 1977