Moduli of Langlands parameters and LLIF – Robert Kurinczuk

The conjectural local Langlands correspondence connects representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters.  In recent joint work with Dat, Helm, and Moss, we have constructed moduli spaces of Langlands parameters over Z[1/p] and studied their geometry.  We expect this geometry is reflected in the representation theory of the p-adic group.  In particular, our main conjecture « local Langlands in families » describes the GIT quotient of the moduli space of Langlands parameters in terms of the centre of the category of representations of the p-adic group generalising a theorem of Helm-Moss for GL(n).  I will give an overview of this picture, finishing with current work in progress towards this conjecture.