|Location:||MSRI: Simons Auditorium|
A basic problem when studying the category of l-adic (rational or integral) smooth representations of a p-adic group is to decompose it as a direct product of subcategories and show that each factor is equivalent to some previously known and supposedly simpler other abelian category.
In this talk we investigate what Langlands' functoriality principle can teach us on this problem, by making precise the following rough guess : the category of rational (resp integral) representations should have a decomposition indexed by L-parameters with source the inertia group (resp its pro-prime-to-l radical) and, under special circumstances, a morphism of L-group should "induce" an equivalence between categories associated to matching parameters.
Two techniques have already been succesfully used to produce equivalences of this kind : types theory (quite indirect and bound to rational coefficients at the moment) and non abelian Lubin-Tate theory (very specific). We will introduce a third technique, adapted to tame parameters, which roughly consists in gluing the cohomology of suitable Deligne-Lusztig varieties along a building. The upshot is that any tame integral block of a GLn is equivalent to the unipotent integral block of a suitable product of GLm's over extension fields.