The Local Langlands Correspondence has motivated a great deal of Number Theory and Representation Theory over the last fifty years. It connects representations of (roughly) the absolute galois group of a p-adic field to the representations of a matrix group G over the field. In the original formulation (now proved for several families of matrix groups G, including general linear groups) it considered only complex representations but, more recently, representations with coefficients in other rings or fields have been considered also. The correspondence results in a partition of the irreducible representations of G into L-packets (which are singletons for general linear groups), each of which is conjecturally characterised by an equality of local factors (L-functions and -factors) – though a definition for such factors is only known for “generic” representations, of which there is (at least conjecturally) precisely one in each packet.
One of the difficulties in studying this is that the representations of G are (almost all) infinite-dimensional, so they are quite hard to get a handle on. One way for doing so, which has been very successful, is to restrict them to compact open subgroups K: we then look for representations of K whose presence in this restriction characterizes some property which representations of G might have; this property might be being “unramified”, ”tame”, “generic”,… Similarly, this restriction can be used to give very explicit descriptions of the representations of G, using arithmetic data. Such results also have interpretations via the Langlands correspondence, so consequences for the absolute galois group.
This PhD project will be in the area of the local Langlands correspondence, exploring some of the problems raised above, or related questions.