Weyl orbit sum in a group algebra

Use the finite reduced crystallographic root system and weight lattice $P$ of Finite Weyl root system, lattice and chamber conventions, with the proved finite $W$ action and lattice preservation of Finite Weyl positive roots and simple reflections. The complex group algebra $\mathbb{C}[P]$ consists of finitely supported sums ${\sum }_{\mu \in P}{c}_{\mu }{e}^{\mu }$, with formal basis indexed by $P$ and multiplication $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$ extended distributively. Finite supports make each product finite, and the abelian group law of $P$ proves associativity and commutativity; $e^0$ is the unit. The action $w(e^{\mu })=e^{w\mu }$ extends to algebra automorphisms because $w$ is additive and permutes $P$.

For $\lambda \in P$ define the orbit sum $m_{\lambda }={\sum }_{\mu \in W\lambda }{e}^{\mu }$, summing each distinct orbit element once. The orbit is finite and nonempty, and $W$ permutes it, so $m_{\lambda }\in \mathbb{C}[P{]}^W$. Stabilizers do not multiply coefficients. In particular $m_0=e^0=1$. For rank zero, $P=\{0\}$ and this is the one-dimensional algebra $\mathbb{C}$. No choice of representatives for a family of orbits is part of the definition.