[Math] Why are there no triple affine Hecke algebras

Question

This question arised after I recently stumbled upon the paper "Triple groups and Cherednik algebras". Doubly affine Hecke algebras are sort of a natural object to consider after finite and affine Hecke algebras. This makes one wonder, why are there no "triple affine Hecke algebras"? Or, if such a construction exists, why are they not useful? (The theory of doubly affine Hecke algebras has proved to have deep consequences and relations with many fields of mathematics, see this previous question.)

Answer 1

It would be great if they existed, even if only for the symmetric group. One possible application would be to algebraic combinatorics: Mark Haiman has collected data suggesting that the ring $R/R^{S_n}_+$ has dimension $2^n (n+1)^{n-2}$, where $$R=\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n,z_1,\dots,z_n],$$ the symmetric group acts by simultaneously permuting the three sets of variables, and the notation $R^{S_n}_+$ means the ideal generated by positive degree symmetric polynomials.

Iain Gordon showed that it is possible to use the representation theory of the rational Cherednik algebra (the rational object in the rational/trigonometric/elliptic trichotomy whose elliptic object is the DAHA) to establish the correct lower bound on this dimension in the case of two sets of variables (this dimension turns out to be $(n+1)^{n-1}$, a theorem proved by Haiman using a suggestion of Procesi and the geometry of the Hilbert scheme of points in the plane). So one might hope to use the representation theory of a TAHA to prove the correct lower bound, at least.

The reason the double affine Hecke algebra exists at all is a little subtle, and has to do with @Theo Johnson-Freyd's comments to the question: the affine Hecke algebra has two realizations. First, the affine Hecke algebra is an affinization of a finite Hecke algebra; second, it is the Hecke algebra associated to the affine Weyl group (or, if you prefer, for a certain specialization it is the Hecke algebra corresponding to an Iwahori subgroup of a p-adic group). Starting with the second presentation, one affinizes again to obtain the DAHA. The point here is that

$$\{\text{affine Hecke algebras} \}=\{\text{Hecke algebras of affine groups} \}$$

and we know how to affinize the Hecke algebras on the RHS.

But so far the DAHA has no second realization as the Hecke algebra of something that can be affinized again. Perhaps recent work of Kazhdan and his collaborators could help here, but I have not read these papers carefully enough to know.

There is a second approach that is somewhat more geometrical. The rational Cherednik algebra is a deformation of the algebra $\mathbb{Q}[x_1,\dots,x_n,y_1,\dots,y_n] \rtimes S_n$, so one might look for nice deformations of the analogous objects in three sets of variables. Perhaps experts in Hochschild cohomology have done calculations suggesting where to look?