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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.)

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You can describe both constructions directly in terms of the weight lattice $P$. The first one (up to taking a dual) is the $W$-invariants of the symmetric algebra $\text{Sym}(P \otimes \mathbb{C})$; the second one is the $W$-invariants of the group algebra $\mathbb{C}[P]$, or maybe more evocatively, $\mathbb{C}[e^P]$, which both correctly describes the relationship between addition in $P$ and multiplication in the group algebra, and correctly suggests how to relate these two constructions even before taking Weyl group invariants.

Namely, the symmetric algebra admits a completion to a formal power series ring

$$\widehat{\text{Sym}}(P \otimes \mathbb{C}) \cong \prod_{n=0}^{\infty} \text{Sym}^n(P \otimes \mathbb{C})$$

which can be thought of as completion with respect to the ideal of elements of positive degree, and now there's a natural $W$-equivariant map

$$\mathbb{C}[e^P] \ni e^p \mapsto 1 + p + \frac{p^2}{2!} + \dots \in \widehat{\text{Sym}}(P \otimes \mathbb{C})$$

which is an isomorphism provided that we also complete the LHS with respect to a suitable ideal.

All of this works with $\mathbb{C}$ replaced by $\mathbb{Q}$, and here there's the following natural connection to algebraic topology. Let $G$ be a compact connected (semisimple?) Lie group with maximal torus $T$ and Weyl group $G$. The splitting principle gives a natural isomorphism

$$H^{\bullet}(BG, \mathbb{Q}) \cong H^{\bullet}(BT, \mathbb{Q})^W$$

which we can rewrite as an isomorphism in equivariant cohomology

$$H^{\bullet}_G(\text{pt}, \mathbb{Q}) \cong H^{\bullet}_T(\text{pt}, \mathbb{Q})^W.$$

Now, $H^{\bullet}_T(\text{pt}, \mathbb{Q})$ can canonically be identified with the symmetric algebra of $P \otimes \mathbb{Q}$ ($W$-equivariantly), so $H^{\bullet}_G(\text{pt}, \mathbb{Q})$ is linear invariants, again up to taking a dual. We also have a natural isomorphism

$$K(BG, \mathbb{Q}) \cong K(BT, \mathbb{Q})^W$$

which refines to a natural isomorphism in equivariant K-theory

$$K_G(\text{pt}, \mathbb{Q}) \cong K_T(\text{pt}, \mathbb{Q})^W.$$

Now, $K_T(\text{pt}, \mathbb{Q})$ can canonically be identified with the rational group algebra of $P$ (again, $W$-equivariantly), so $K_G(\text{pt}, \mathbb{Q})$ is multiplicative invariants, and $K(BG, \mathbb{Q})$ is a suitable completion of it by the Atiyah-Segal theorem (or direct computation with the RHS).

The relationship between these two constructions is given by the Chern character map

$$K_G(\text{pt}, \mathbb{Q}) \xrightarrow{\text{ch}} H^{\bullet}_G(\text{pt}, \mathbb{Q})$$

which is an isomorphism after completing the LHS appropriately, or equivalently after replacing it with $K(BG, \mathbb{Q})$. This map is precisely the map above.

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Qiaochu Yuan satisfactorily answered my question, but I feel I should add an answer to mention the following nice paper which I just stumbled upon:

Sanghoon Baek, Erhard Neher & Kirill Zainoulline, "Basic polynomial invariants, fundamental representations and the Chern class map", Doc. Math. 17 (2012), 135–150 (arXiv:1106.4332)

Which investigates the relation between additive and multiplicative invariants of the Weyl group in the more general case of integer coefficients. Qiaochu Yuan's answer is very much related to their remark 6.2.

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