Geometric Structure and the Local Langlands Conjecture

Let G be a reductive p-adic group which is  connected and split.
Examples are GL(n, F) ,  SL(n, F) ,  SO(n, F) , Sp(2n, F) ,
PGL(n, F) where n can be any positive integer and F can be any finite
extension of the field Q_p of p-adic numbers. The
smooth (or admissible) dual of G is the set of equivalence classes of
smooth irreducible representations of G. The
representations are on vector spaces over the complex numbers. Within
the smooth dual there are subsets known
as the Bernstein components, and the smooth dual is the disjoint union
of the Bernstein components. This talk will explain
a conjecture due to Aubert-Baum-Plymen-Solleveld (ABPS) which says that
each Bernstein component is a complex affine variety.
These affine varieties are explicitly identified as certain extended
quotients. The ABPS conjecture gives a method for proving the
local Langlands conjecture.  Via this method, local Langlands has been
proved for any Bernstein component in the principal series of G.