Summary. The local Langlands programme predicts a correspondence between, on the one hand, arithmetic in the form of representations of the absolute Galois group of a p-adic field F and, on the other, the representation theory of reductive groups over F. An explicit understanding of the latter for the general linear groups has led to deep work on the functorial properties of this correspondence. In this project, we propose to investigate the smooth representation theory of symplectic, orthogonal and unitary groups over F (when p is not 2) and of the multiplicative group of central simple algebras over F. We will first explicitly construct all supercuspidal representations of these groups, which are the building blocks of the theory. Then we will compute certain Hecke algebras, whose module categories describe the category of smooth representations. An explicit description of these algebras should allow the reducibility of associated parabolically induced representations to be determined. This is also related to the poles and zeros of an L-function. By finding relationships between the Hecke algebras, we hope to obtain arithmetic information via these L-functions.

Publications

Final report