Journal of Lie Theory Volume 16 (2006) 743–776 c  2006 Heldermann Verlag Orbital Convolution Theory for Semi-direct Products A. H. Dooley and N. J. Wildberger 1 Communicated by J. Ludwig Abstract. We extend previous results of the authors on orbital convolutions for compact groups, to compact times vector semidirect products. In particular, we define convolutions of noncompact coadjoint orbits and recover the character formulae and Plancherel formula of Lipsman. Mathematics Subject Index 2000: Primary: 43A80; Secondary: 22E15. Keywords and phrases: Lie group, semi-direct product, character formula, coad- joint orbit. 1. Introduction Previous work of the authors ([5], [7], [6]), in the setting of compact groups, introduced the wrapping map Φ. This map associates, to each Ad-invariant distribution µ of compact support on the Lie algebra g , a central distribution Φµ on the Lie group G , via the formula, for f ∈ C ∞ c (G), 〈Φµ,f 〉 = 〈µ,j · f ◦ exp〉, (1) where j is the square root of the Jacobian of exp : g → G . The remarkable thing about Φ is that it provides a convolution homomor- phism between the Euclidean convolution structure on g and the group convolution on G , that is Φ(µ ∗ g ν )=Φµ ∗ G Φν. (2) This mapping is a global version of the Duflo isomorphism — there are no condi- tions on the supports of µ and ν (they need not, for example, lie in a fundamental domain). As pointed out in [5], we may interpret the dual of Φ, a map from the Gelfand space of M G (G) to that of M G (g), in such a way as to obtain the Kirillov character formula for G . In a recent paper [1], Andler, Sahi and Torossian have extended the Duflo isomorphism to arbitrary Lie groups. Their results give a version of equation (2) which holds for germs of hyperfunctions with support at the identity. In fact, equation (2) can be viewed as a statement that, for compact Lie groups, the 1 We gratefully acknowledge the support of the Australian Research Council ISSN 0949–5932 / $2.50 c  Heldermann Verlag