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 $\frak g$. We give a generalization of this result for the isotropy representations of symmetric spaces. If $\frak g={\frak g}_0+{\frak g_1}$ is a Z_2-grading of a simple Lie algebra, we explicitly describe a ${\frak g}_0$-module $Spin_0({\frak g}_1)$ such that the exterior algebra of ${\frak g}_1$ is the tensor square of this module times some power of 2. Although $Spin_0({\frak g}_1)$ is usually reducible, we show that a Casimir element for ${\frak g}_0$ 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.