I don't immediately recall the precise origin of Bernstein's presentation of the affine Iwahori-Hecke algebra, but two recent papers with references are
available on arXiv and would be worth looking at in any case: <a href="http://front.math.ucdavis.edu/0201.5176">haines-pettit</a> and <a href="http://front.math.ucdavis.edu/0610.5839">goertz</a>.