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?

Answer 2

This is a wonderful and highly suggestive question. In addition to the fascinating hints provided by Stephen's answer, there are also reasons from geometric representation theory (as well as from physics) to expect that a triple affine Hecke algebra theory might exist.

A very faint hint in representation theory can be seen for example from my paper Elliptic Springer Theory with David Nadler (I am sure there are better references for this faint hint but I'm sadly ignorant of the elliptic literature - just now learned of the beautiful paper of Ion-Sahi from reading this question!)

First recall that the Weyl group appears as endomorphisms of the Grothendieck-Springer sheaf, a remarkable perverse sheaf (or $D$-module) on the nilpotent cone of a Lie algebra. However, if you consider the ext-algebra of the Springer sheaf, you find instead the degenerate affine Hecke algebra - i.e., you "sprout" a polynomial ring (functions on a torus Lie algebra, or if you prefer, torus-equivariant cohomology of a point). Next we can consider the group version, the Hotta-Kashiwara sheaf, a $G$-equivariant sheaf on $G$ constructed from the group version of the Grothendieck-Springer resolution. It has an endomorphism algebra which is an affine Weyl group, i.e. we've added the group algebra of the lattice, but an Ext algebra which is a form of the degenerate DAHA (really we are missing all the interesting parameters here, one ought to consider a ``mixed" version to see something closer to the objects of interest).

Finally there's an elliptic version, living on the "elliptic group" -- the moduli of semistable $G$-bundles on an elliptic curve (recovering the previous two when the curve degenerates to a nodal, then cuspidal, Weierstrass cubic). We discuss an obvious elliptic version of the Springer sheaf there -- in fact a special case of the theory of geometric Eisenstein series (a much more thorough study is in Dragos Fratila's thesis). Its endomorphisms are identified with a double affine Weyl group (ie two copies of the same lattice, not duals)... but its Exts will again sprout another polynomial ring, and should be considered a very degenerate form of a "degenerate triple affine Hecke algebra". One really needs to make it much less degenerate to get something close to answering this question - i.e., add a "q" (mixed version) and a "t" (twisted $D$-module/quantum group parameter) at the least. But it's something natural and triple.

[I can't say anything reasonable about the physics in finite length, or maybe at all, but I'll allow myself just a vague hint: double affine Hecke algebras (at least in type A) arise (again in very degenerate form) in studying the Seiberg-Witten integrable systems (i.e. moduli space after compactification on a circle) of $N=4$ super Yang-Mills in four dimensions - which is itself a reduction on two circles of the mysterious (2,0) theory in 6 dimensions, aka "theory $\mathfrak X$".. this realization shows that there are not just two, but three, circles in the picture - switching around two gives an $SL_2Z$ symmetry, aka Langlands/S/electro-magnetic duality, but switching around all three ought to give some hint of $SL_3Z$ symmetry.]