The affine Hecke algebra H' is a smash product of a finite Hecke algebra H with the group algebra of a certain lattice X, and it plays a key role in the representation theory of p-adic groups. In connection with the Macdonald conjectures, Cherednik introduced a double affine Hecke algebra H'', which is obtained by combining a finite Hecke algebra H with *two* lattices X,Y.

H' has an alternate description with generators corresponding to the nodes of an affine Coxeter graph. Each generator satisfies a quadratic polynomial relation, while pairs of generators satisfy braid relations that can be read off from the graph.

In recent joint work, Bogdan Ion and I have obtained a similar alternate description of H'' in terms of certain extended graphs, corresponding to extended affine Lie algebras (EALAs). This description reveals some hidden symmetries of H''. In particular, we obtain a natural morphism from a congruence subgroup of SL(2,Z) to the group of outer automorphisms of H''.