The image of the Lepowsky homomorphism for the split rank one symplectic group

Abstract

Let G o be a semisimple Lie group and let K o denote a maximal compact subgroup of G o . Let U(g) be the complex universal enveloping algebra of G o and let U(g) K denote the centralizer of K o in U(g). Also let P :be the projection map corresponding to the direct sum U(g) = (U (k) ⊗ U(a)) ⊕ U(g)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(g) K under the injective antihomomorphism P : U(g) K −→ U(k) ⊗ U(a), considered by Lepowsky in [J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1973) 1-44], when G o = Sp(n, 1).

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

“…The proof of this last theorem is given in Section 8, following the ideas developed in the symplectic case. In fact, most of the results proved in Section 6 of [3] hold in this case with appropriate changes.…”

“…In [3] we proved that P (U (g) K ) = B when G 0 = Sp(n, 1), and more recently, we showed that B Wρ = B when G 0 = SO(n,1) or SU(n,1) (see [4]). Hence these results established that P (U (g) K ) = B for every classical real rank one semisimple Lie group with finite center.…”

“…The proof of Theorem 1.1 follows the same ideas used to prove the analogue theorem for the symplectic group Sp(n, 1), however we had to overcome some difficulties to establish the transversality results needed (Section 6), and the a priori estimates of the Kostant degrees (Section 7). In Section 6 we give a new and simplified version of the corresponding transversality results obtained in the symplectic case (see Section 4 of [3]). This new version is sufficient because of the introduction of a simplifying hypothesis called the degree property, which is done in Section 7.…”

“…At this point it is convenient to state the following result. Its proof is given in Proposition 2.6 of [3], using the techniques and the notation of Section 3 of [12].…”

“…is the multiplicity of λ, then c = 1 when 2λ is not a restricted root and m(λ) is even, or when m(λ) is odd, and c = 3 2 when 2λ is a restricted root and m(λ) is even.…”