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

TL;DR

This paper characterizes the image of the Lepowsky homomorphism for the universal enveloping algebra of a Lie algebra associated with the group F4, providing insights into its structure and representation theory.

Contribution

It offers a new characterization of the Lepowsky homomorphism's image specifically for the group F4, extending previous understanding to this exceptional Lie group.

Findings

Explicit description of the image of the Lepowsky homomorphism for F4

Advances understanding of invariant differential operators in this context

Provides tools for further representation theory analysis

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$_4$.