Fusion Product of Co-adjoint Orbits

In this paper, we introduce the fusion product of a generic pair of co-adjoint orbits. This construction provides the geometric dual object to the product in Verlinde fusion algebra. The latter is a quantum deformation of the standard tensor product, the fusion product constructed here is the corresponding deformation of the Cartesian product. Let G be a connected and simply-connected compact simple Lie group, P+ be the set of dominant integral weights, θ be the highest root. Suppose λ ∈ P+, and denote by [λ] the irreducible G-module defined by λ. Let (·|·) be the invariant bilinear form on g, g, normalized so that (θ|θ) = 2. For k ∈ Z+, define P k + = {λ ∈ P+| (θ|λ) ≤ k}.