See: arXiv:dg-ga/9406006 and arXiv:math.DG/9506223.
- Edit:
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,
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:
- Onishchik, Arkadi L. Topology of transitive transformation groups. (English) Zbl 0796.57001 Leipzig: Johann Ambrosius Barth. xv, 300 p. (1994).
