The image of the lepowsky homomorphism for the group F-20 4

Abstract

Let Go be a semisimple Lie group, let Ko be a maximal compact subgroup of Go and let k ?g denote the complexification of their Lie algebras. Let G be the adjoint group of g and let K be the connected Lie subgroup of G with Lie algebra ad(k). If U(g) is the universal enveloping algebra of g, then U(g)K will denote the centralizer of K in U(g). Also let P :U(g) →U(k) ⊗ U(a) be the projection map corresponding to the direct sum U(g)=(U(k) ⊗ U(a)) ⊗ U(g)n associated to an Iwasawa decomposition of Go adapted to Ko. In this paper, we give a characterization of the image of U(g)K under the injective antihomorphism P :U(g)K →U(k)M ⊗ U(a), considered by Lepowsky in [12], when Go is isomorphic to the rank 1 real form F -204 of the exceptional Lie group F4. © 2012 The Author(s).