The image of the Lepowsky homomorphism for the group $F_4$

Abstract

Let $G_o$ be a semisimple Lie group, let $K_o$ be a maximal compact subgroup of $G_o$ and let $\mathfrak{k}\subset\mathfrak{g}$ denote the complexification of their Lie algebras. Let $G$ be the adjoint group of $\mathfrak{g}$ and let $K$ be the connected Lie subgroup of $G$ with Lie algebra $ad(\mathfrak{k})$. If $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$ then $U(\mathfrak{g})^K$ will denote the centralizer of $K$ in $U(\mathfrak{g})$. Also let $P:U(\mathfrak{g})\longrightarrow U(\mathfrak{k})\otimes U(\mathfrak{a})$ be the projection map corresponding to the direct sum $U(\mathfrak{g})=\bigl(U(\mathfrak{k})\otimes U(\mathfrak{a})\bigr)\oplus U(\mathfrak{g})\mathfrak{n}$ associated to an Iwasawa decomposition of $G_o$ adapted to $K_o$. In this paper we give a characterization of the image of $U(\mathfrak{g})^K$ under the injective antihomorphism $P:U(\mathfrak{g})^K\longrightarrow U(\mathfrak{k})^M\otimes U(\mathfrak{a})$, considered by Lepowsky, when $G_o$ is locally isomorphic to F$_4$.