Affine Hecke algebras are the key to many problems in representation theory. In this talk we will compare affine Hecke algebras H(q) and H(q') with the same underlying data, but with different parameters q and q'. This is an analogue of investigating one algebraic group over two different fields. We will show that, although these algebras are most probably not Morita equivalent, they are very close in several respects.

It is known from work of Lusztig and Baum--Nistor that H(q) and H(q') have the same periodic cyclic homology. We will prove that, with hindsight, this is in fact a consequence of an isomorphism between their respective Hochschild homology groups. We will explicitly determine these homology groups in terms of the irreducible representations of the algebras. To this end we will take a good look at the classical case q = 1, where everything can be described in simple geometric terms.