# Why are there no triple affine Hecke algebras?

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.)

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This is probably unrelated, but I'm reminded of a quote (well, paraphrase) from Richard Borcherds in one of his Lie Theory classes. The context was the discussion of Kac–Moody groups. Recall that if the Cartan is positive definite ("of finite type") then the corresponding group is finite-dimensional and has a unique compact real form. If the Cartan has one-dimensional kernel ("of affine type"), then the corresponding group is a version of a loop group (up to some twisting) of a compact group. These groups are all very important. Richard ended his discussion by saying that one of the most ... –  Theo Johnson-Freyd May 14 '12 at 15:49
... important open problems in Lie Theory is to find any applications at all for the Kac–Moody groups beyond affine type. (Note in particular that they do not correspond to "double loop groups" or anything so simple.) –  Theo Johnson-Freyd May 14 '12 at 15:50
The Weyl group $W$ acts naturally on the root and coroot lattices, and my understanding is that the (double) affine Hecke algebra is a deformation of the group algebra of the semidirect product of $W$ with (the coroot lattice plus) the root lattice. It seems like a "triple affine Hecke algebra" would require the action of $W$ on a third lattice, but I don't know any natural candidate for such a lattice. –  Peter Samuelson May 14 '12 at 16:26
@Peter correct, after deformation we make q- difference operators from coroots which act on roots. In rational case we get differential operators. More generallly roots+coroots can be substituated by any symplectic vector space. And Weyl group by any finite group acting symplectically that what EtingofGinzburg did. –  Alexander Chervov May 14 '12 at 17:12
Looking 5min on paper it seems to me very interesting Kyoji Saito ideas deserving to be further explored however I did get their main idea. What they mean by triple group? –  Alexander Chervov May 14 '12 at 17:17