Daniel Funck: Geometry of Steinberg deformations

Number Theory Seminar: Daniel Funck (University of Tübingen)

Geometry of Steinberg deformations

Let p < \ell be primes, let F be a local field of residue characteristic p and G a reductive group over Z.  The moduli space of Langlands parameters Z^1(W_F,G), first constructed by Bellovin-Gee (characteristic zero), and more completely by Dat-Helm-Kurinczuk-Moss (over Z[1/p]), is a central object in the study of the Langlands programme. It has two main applications, the first is its (m-adic completions of) local rings are local deformation rings, which characterise liftings of mod \ell representations (valued in G) and are a key object in R=T theorems. The second is its role in the conjectural categorical Langlands programme.

It is a natural question (with certain automorphic applications) to study the local geometry of the irreducible components of this space, and their reductions modulo \ell. In this talk, I want to discuss certain results regarding the geometry of this space, I will present a characterisation of which irreducible components are smooth in the `considerate' case, and then turn our attention to a particular component of interest known as the Steinberg component, highlighting some results in the case G=GL_3 and some of the difficulties in studying the geometry of this space in higher generality.