Module structure of certain induced representations of compact Lie groups

Abstract:

Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $\lambda$, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $\lambda$. This paper studies the G-module structure of the space $\Gamma$ of square integrable sections of the Borel-Weil line bundle. It is found that $\Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $\Gamma (n) \subset \Gamma (n + 1) \subset \Gamma$ and $\Gamma (n)$ is isomorphic, as G-module, to \[ V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast }),\] where $V(\mu )$ denotes the irreducible representation of highest weight $\mu$, ’+’ is the Cartan semigroup operation, and ’$^\ast$’ is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.