Given a finite-dimensional crystallographic root system, we can construct an associated Kac-Moody group, with a corresponding flag variety and Littlewood-Richardson coefficients. Between a pair of distinct generators in the Coxeter system, either $ss'$ is of infinite order, $(ss')^2=1$, $(ss')^3=1$, $(ss')^4=1$, or $(ss')^6=1$.
Using Kostant and Kumar's construction of the equivariant cohomology ring of the generalized flag variety, the restriction to crystallographic root systems is unnecessary. We can construct an equivariant Schubert calculus ring with non-integral Littlewood-Richardson coefficients just fine without an associated flag variety.
My question is: is there any geometric interpretation of these non-integral Schubert calculus rings? My conjecture is that their structure constants are positive, but it would obviously be quite hard to prove that combinatorially (which doesn't mean I'm not trying of course, but success is not expected). I have verified that they are indeed positive for $H_4$, and I have a Littlewood-Richardson rule for $I_2(n)$ for all $n$.