Chevalley restriction theorem for exterior algebras Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem says that there is isomorphism of algebras of invariants $\mathbb{C}[\mathfrak{g}]^{G}\simeq \mathbb{C}[\mathfrak{h}]^{W}$ given by restriction.
Here $\mathbb{C}[\mathfrak{g}]$ and $\mathbb{C}[\mathfrak{h}]$ denote the algebras of polynomial functions on correspondent algebras viewed as vector spaces. In other words, $\mathbb{C}[\mathfrak{g}]=Sym(\mathfrak{g}^*)$ is a symmetric algebra.
My question is: is there a similar result for algebra of $G$-invariants of the exterior algebra $\bigwedge(\mathfrak{g}^*)$? 
Thank you very much!
 A: See: arXiv:dg-ga/9406006 and arXiv:math.DG/9506223.


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See lemma 3.3 in the first paper: $C^\infty$ and polynomial (by Solomon, cited in the paper)
$W$-invariant differential forms on $\mathfrak h$ correspond to horizontal $G$-invariant forms on $\mathfrak g$. 
I just noted: You are asking for $G$-invariants of $\bigwedge \mathfrak g^\star$, i.e., constant differential forms.
These describe the de Rham cohomology of a compact form of $G$, by the theorem of Chevalley and Eilenberg, and can be described as the set of primitive elements. See books, for example, 


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*MR0400275 (53 #4110) Reviewed 
Greub, Werner; Halperin, Stephen; Vanstone, Ray
Connections, curvature, and cohomology. 
Volume III: Cohomology of principal bundles and homogeneous spaces. Pure and Applied Mathematics, Vol. 47-III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. xxi+593 pp. 

*MR1379333 (97j:57057) Reviewed 
Onishchik, A. L.(RS-YAR)
\cyr Topologiya tranzitivnykh grupp preobrazovaniĭ. (Russian. English, Russian summary) [Topology of transitive transformation groups] Fizmatlit ``Nauka'', Moscow, 1995. 384 pp. ISBN: 5-02-014724-9 
There is a translation into English which cannot be found in MathRev:


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*Onishchik, Arkadi L.
Topology of transitive transformation groups. (English) Zbl 0796.57001
Leipzig: Johann Ambrosius Barth. xv, 300 p.  (1994).

