Extended quotients in the principal series of reductive p-adic groups
  The geometric conjecture developed by the authors in [1,2,3,4] applies to the smooth dual Irr(G) of any reductive p-adic group G. It predicts a definite geometric structure - the structure of an extended quotient - for each component in the Bernstein decomposition of Irr(G).   In this article, we prove the geometric conjecture for the principal series in any split connected reductive p-adic group G. The proof proceeds via Springer parameters and Langlands parameters. As a consequence of this approach, we establish strong links with the local Langlands correspondence.   One important feature of our approach is the emphasis on two-sided cells in extended affine Weyl groups. 
Anne-Marie Aubert, Paul Baum and Roger Plymen
math.RT