In the first part of the talk (above), we will discuss the computations of characters (automatically supercuspidal) for a p-adic quaternionic division algebra. We will assume a basic familiarity with the structure theory of p-adic fields and skew fields, and with elementary representation-theoretic methods such as compact induction. No other background will be assumed.
In the second part (below), we will discuss how the methods for p-adic division algebras can be generalised to handle supercuspidal characters of other reductive, p-adic groups. We will sketch the construction of J.-K. Yu; the existing character computations of Adler and Spice; and possible generalisations of the latter. In this part, we will assume a familiarity with more advanced topics in the structure theory of p-adic groups, such as Bruhat–Tits buildings and Moy–Prasad subgroups.