Sums of Adjoint Orbits

We show that the sum of two adjoint orbits in the Lie algebra of an exponential Lie group coincides with the Campbell-Baker-Hausdorff product of these two orbits. Introduction N. Wildberger and others have recently investigated the structure of the hypergroup of the adjoint orbits in relation with the class hypergroup of compact Lie groups. A generalization of the notion of this type of hypergroup to non-compact groups, for instance to nilpotent or exponential Lie groups, leads to the problem of determining a precise relation between the sum of adjoint orbits in the Lie algebra and the product of the corresponding conjugacy classes in the group (see [1], and [4]). In ([3]) Wildberger has shown that for nilpotent Lie groups G the exponential of the sum of two adjoint orbits Ω1 + Ω2 is equal to the product exp Ω1 ·exp Ω2 in G . In this paper we consider the same problem for exponential groups. Let us recall that by the definition of exponential Lie groups, the mappings exp: g→ G and log:G→ g are diffeomorphisms. We can transfer the group multiplication in G via exp to a group multiplication in the Lie algebra g and we shall denote it by the symbol ∗ . We obtain the so called Baker-Campbell-Hausdorff multiplication in g , which is given by U ∗ V = U + V + 1 2 [U, V ] + 1 12 [U, [U, V ]] + 1 12 [V, [V, U ]] + · · · for small U and V in g . Let X and Y be two elements of the Lie algebra g of the exponential group G . We denote by X = Ad(A)X the adjoint action of the element A of G on X , and by X = {X | A ∈ G} ISSN 0940–2268 / $2.50 C © Heldermann Verlag 106 Arnal and Ludwig the adjoint orbit of X . For h ∈ G , let C(h) = {g · h · g−1 | g ∈ G} be the conjugacy class of h . We show in this note that exp(X+Y ) is equal to C(expX)·C(expY ). Theorem A. Let G be an exponential Lie group with Lie algebra g . For any elements X and Y of g we have