A classical theorem connects Lie group cohomology to the theory of differential-forms invariant under the action of a finite Weyl group. We will examine the space of derivation differential-forms invariant under Coxeter and complex reflection groups more generally. We will present new structure theorems describing these invariants and show the existence of a set of free generators for any Coxeter group. These theorems connect with a 1997 conjecture of Reeder and results in Lie theory by Bazlov, Broer, Joseph, and Stembridge and more recent work of DeConcini, Papi, and Procesi.
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