The exterior algebra and `Spin' of an orthogonal g-module

Abstract

A well-known result of Kostant gives a description of the G-module structure for the exterior algebra of Lie algebra $\mathfrak{g}$. We give a generalization of this result for the isotropy representations of symmetric spaces. If $\mathfrak{g}={\mathfrak{g}}_0+{\mathfrak{g}}_1$ is a Z_2-grading of a simple Lie algebra, we explicitly describe a ${\mathfrak{g}}_0$-module $Spi{n}_0({\mathfrak{g}}_1)$ such that the exterior algebra of ${\mathfrak{g}}_1$ is the tensor square of this module times some power of 2. Although $Spi{n}_0({\mathfrak{g}}_1)$ is usually reducible, we show that a Casimir element for ${\mathfrak{g}}_0$ 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.

LaTeX 2.09, 30 pages