TITLE: Why are there no triple affine Hecke algebras?
QUESTION [17 upvotes]: 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.)

REPLY [6 votes]: 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.]