A class of primary representations associated with symmetric pairs and restricted root systems

Gang Han

Vol. 225 (2006), No. 1, 33–51

DOI: 10.2140/pjm.2006.225.33

Abstract

Let ν r → so(p) be a representation of a complex reductive Lie algebra r on a complex vector space p. Assume that ν is the complexified differential of an orthogonal representation of a compact Lie group R. Then the exterior algebra ∧ p becomes an r-module by extending ν. Let Spin ν r → End S be the composition of ν with the spin representation Spin : so(p) → End S. We completely classify the representations ν for which the corresponding Spin ν representation is primary, give a description of the r-module structure of ∧ p, and present a decomposition of the Clifford algebra over p. It turns out that, if the Spin ν representation is primary, ν must be an isotropy representation of some symmetric pair. Our work generalizes Kostant’s well-known results that dealt with the special case when ν is the adjoint representation of a semisimple Lie algebra. In the proof we introduce the “restricted” root system of a real semisimple Lie algebra, which is of independent interest.

Keywords

primary representation, spin representation, symmetric pair, restricted root system

Mathematical Subject Classification 2000

Primary: 17B10

Milestones

Received: 15 October 2004

Revised: 23 August 2005

Accepted: 24 August 2005

Published: 1 May 2006