Dimension of the intersection of a pair of orthogonal groups

Abstract

Let g, h:V×V→ℂ be two non-degenerate symmetric bilinear forms on a finite-dimensional complex vector space V. Let G (resp. H) be the Lie group of isometries of g (resp. h). If the endomorphism L:V→V associated to g, h is diagonalizable, then the dimension of the intersection group G∩H is computed in terms of the dimensions of the eigenspaces of L.

Keywords:

• diagonalizable endomorphism
• isometry
• matrix exponential
• orthogonal group
• symmetric bilinear form

2000 AMS Subject Classifications :

• 15A57
• 15A63
• 22E10
• 22E60

Acknowledgements

R. Durán Díaz and J. Muñoz Masqué are supported by Ministerio de Educación y Ciencia (Spain) under grant MTM2005-00173, and L. Hernández Encinas and Seok-Zun Song are supported by Korean Science and Engineering Foundation (Korea) under grant F01-2007-000-10047-0.