Let $\mathcal{W}$ be a Weyl group, acting variously on a root system $\Phi$, the real vector space $V = \langle\Phi\rangle_{\mathbb{R}}$ in which they live, or the associated weight lattice $P \subset V$ (vectors on which the coroots $\Phi^\vee\subset V^\vee$ take integral values). There are two obvious "rings of invariants" we can associate to this situation:

The

**linear invariants**: here $\mathcal{W}$ is seen as acting on $V$ as a group of $\mathbb{R}$-linear transformations, so it also acts on the set $\mathbb{C}[V] := \mathrm{Sym}(V^\vee_{\mathbb{C}})$ of polynomials on $V$, and the ring $\mathbb{C}[V]^{\mathcal{W}}$ is the set of invariants for this action.The

**multiplicative**(or**exponential**) invariants: here, $\mathcal{W}$ is seen as acting on $P$ as a group of $\mathbb{Z}$-linear transformations, and this time we consider $P$ as the set of monomials on the group algebra (an algebra of Laurent series), $\mathbb{C} P$, and we consider the ring $(\mathbb{C}P)^{\mathcal{W}}$ of invariants. (Sometimes this is written with the same bracket notation, but I think this is really*too*confusing.)

Both occur in the context of Lie algebras. If $V_{\mathbb{C}} = \mathfrak{h}^\vee$ is the dual of the Cartan algebra of a semisimple Lie algebra $\mathfrak{g}$, then $\mathbb{C}[V]^{\mathcal{W}} = \mathrm{Sym}(\mathfrak{h})^{\mathcal{W}}$ is the source of the Harish-Chandra isomorphism, whereas $(\mathbb{C}P)^{\mathcal{W}}$ is the ring in which formal characters of finite dimensional representations of $\mathfrak{g}$ live.

Furthermore, in the situation considered above, both turn out to be, in fact, isomorphic to polynomial rings: for the ring of linear invariants, this is the Shephard-Todd-Chevalley theorem because $\mathcal{W}$ is generated by reflections, whereas for multiplicative invariants the fact is attributed to Bourbaki (Bourbaki, LIE, VI, §3, nº4, théorème 1; cf. Lorenz, *Multiplicative Invariant Theory* (Springer 2005, EMS 135), theorem 3.6.1). So, in fact, the two are isomorphic. But:

**Question:** are they isomorphic in a "nice" way? Or is there some other easily described relation between the two?

(I realize that my question might be a little bit vague, but really, what I'm trying to get is a feel of why the two occur, why they look so similar, and, if possible, how not to get the two mixed up. Any indication on where both might occur side by side in a book or course would be welcome.)