Fixed points on flag manifolds

Henry H. Glover and William Duncan Homer, II

Vol. 101 (1982), No. 2, 303–306

DOI: 10.2140/pjm.1982.101.303

Abstract

When K is R, C or H, let U_K(n) denote the group of n × n orthogonal, unitary, or symplectic matrices, respectively. If G is a closed connected subgroup of U_K(n) of maximal rank, then it is conjugate to a subgroup of the form U_K(n₁) × U_K(n₂) ×⋯ × U_K(n_k). A simple condition on the integers n_i is shown to be necessary for U_K(n)∕G to have the fixed point property (that every self map has a fixed point). It is conjectured that this condition is also sufficient, and a proof is given for some cases.

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 14M17, 54H25

Milestones

Received: 22 May 1981

Revised: 29 July 1981

Published: 1 August 1982