2000/01/28 by Dmitri I. Panyushev
Mathematics · #math.AG #math.RT
published as Transformation Groups, 6 (2001), 371-396 · LaTeX 2.09, 30 pages
arxiv created 2000/01/28 · arxiv updated 2009/11/30
A well-known result of Kostant gives a description of the G-module structure for the exterior algebra of Lie algebra \frak g. We give a generalization of this result for the isotropy representations of symmetric spaces. If \frak g=\frak g0+\frak g1 is a Z2-grading of a simple Lie algebra, we explicitly describe a \frak g0-module Spin0(\frak g1) such that the exterior algebra of \frak g1 is the tensor square of this module times some power of 2. Although Spin0(\frak g1) is usually reducible, we show that a Casimir element for \frak g0 always acts scalarly on it. We also a give classification of all orthogonal representations of simple algebraic groups having an exterior algebra of skew-invariants.