Abstract

The local Langlands conjectures predict a canonical bijection between certain (L-)packets of representations of a reductive group G over a p-adic field F, and the representations of the Weil group of F in the (Langlands) dual group of G; the case of GL_1 is given by Local Class Field Theory. The conjectures are now proven for general linear groups and, more recently, by relating it to this case, for symplectic and special orthogonal groups (Arthur). On the other hand, the representations of these p-adic groups G have been constructed in an explicit way so one can ask how the data involved in these constructions transfer under the Langlands correspondence. For general linear groups, this has been the subject of an ongoing programme of Bushnell and Henniart. I will talk about work in progress with Blondel and Henniart, where we have been considering what can then be said for symplectic groups.