This is something of a followup to the question "Kapranov's analogies", where a connection between Cherednik's double affine Hecke algebras (DAHA's) and Geometric Langlands program was mentioned.
I am interested to hear about known connections of the double affine Hecke algebras, or their various degenerations called rational and trigonometric degenerate DAHA's (or simply Cherednik algebras) to other areas of mathematics. Affine Hecke algebras answers are also interesting; I once asked a friend if he planned to attend a talk about DAHA, and was told "nah, that sounds like a little too much affine Hecke algebra for my tastes." As prompts for answers, I'll share my limited understanding in the hopes a kind reader will elaborate:
I understand that they were introduced to solve the so-called MacDonald Macdonald conjectures, though I don't know much about this and would be delighted to hear anyone's thoughts.
I understand that they can be defined in terms of certain "Dunkl-Opdam" differential operators, and can be thought of as the algebra of differential operators on a non-commutative resolution of $\mathfrak{h}/W$. I'd be interested to hear more about this.
I understand that they have an action by automorphisms of
$\widetilde{SL_2(Z)}$coming alternately from their connection to configurations of points on a torus, or from their description via differential operators. If anyone has another take on this action, that would be interesting to hear about.Apparently they also relate to the geometric Langlands program, which is what really prompts my post; can anyone elaborate on this?
