Joint IAS/Princeton University Number Theory Seminar

Let \(G\) be a connected split reductive \(p\)-adic group. Examples are \(\mathrm{GL}(n,F)\), \(\mathrm{SL}(n, F )\), \(\mathrm{SO}(n, F)\), \(\mathrm{Sp}(2n, F )\), \(\mathrm{PGL}(n, F )\) where \(n\) can be any positive integer and \(F\) can be any finite extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. The smooth 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. In 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 infinitesimal character of Bernstein and the L-packets which appear in the local Langlands conjecture are then described from this point of view. Granted a mild restriction on the residual characteristic of the field \(F\) over which \(G\) is defined, ABPS has been proved for any Bernstein component in the principal series of \(G\). A corollary is that the local Langlands conjecture is valid throughout the principal series of \(G\). The above is joint work with Anne-Marie Aubert, Roger Plymen, and Maarten Solleveld.